+ i` Rt^RIHtHt^RIHt^RIHt^RIHt^RI H t ^RI H t ^RI Ht^RIHtR t!R R 4t]!4tR t]!R RRR7R4tRRltRRltRtRRltRtRRltR RltRtRtR#)!z" Generating and counting primes. )bisect bisect_leftcount)array)randint)sqrt)isprime) deprecated)as_intc0^RIHp\V!V44#)zWrapping ceiling in as_int will raise an error if there was a problem determining whether the expression was exactly an integer or not.)ceiling)#sympy.functions.elementary.integersr r )ar s& v/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/ntheory/generate.py_as_int_ceilingrs< '!* c~a]tRt^toRtRRltRtRRltRtRt Rt RR lt R t R t R tR tRtRtRtVtR#)SieveaA list of prime numbers, implemented as a dynamically growing sieve of Eratosthenes. When a lookup is requested involving an odd number that has not been sieved, the sieve is automatically extended up to that number. Implementation details limit the number of primes to ``2^32-1``. Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> 25 in sieve False >>> sieve._list array('L', [2, 3, 5, 7, 11, 13, 17, 19, 23]) c a^Sn\R.RO4Sn\R.R O4Sn\R.R O4SnV^8:d \ R4hVSn\;QJd@V3RlSPSPSP34F 'dK RM5 RM1!V3RlSPSPSP344'gQhR#) zInitial parameters for the Sieve class. Parameters ========== sieve_interval (int): Amount of memory to be used Raises ====== ValueError If ``sieve_interval`` is not positive. Liz+sieve_interval should be a positive integerc3T<"TFp\V4SP8HxK R#5iN)len_n).0rselfs& r !Sieve.__init__..Cs U.T3q6TWW$.Ts%(FTN) )r'r r )r&r'r)r&r))r_array_list_tlist_mlist ValueErrorsieve_intervalall)rr/sf&r__init__Sieve.__init__-sC!56 S"45 S"78 Q JK K,sUtzz4;; .TUsssUtzz4;; .TUUUUUrcRR\VP4VP^,VP^,VP^,VPR,VPR,R\VP4VP^,VP^,VP^,VPR,VPR,R\VP4VP^,VP^,VP^,VPR,VPR,3,#)zs<%s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i>primetotientmobiusr))rr+r,r-)rs&r__repr__Sieve.__repr__Es6c$**oA 1 tzz!}BB DKK(QQQR$++b/ s4;;'QQQR$++b/ :C C CrNc t\;QJdRWV34F 'dK RM RM!RWV344'dR;p;r#V'dVPRVPVnV'dVPRVPVnV'd!VPRVPVnR#R#)zQReset all caches (default). To reset one or more set the desired keyword to True.c3("TFqRJxK R#5ir)rrs& rrSieve._reset..Vs;":QDy":sFTN)r0r+rr,r-)rr4r5r6s&&&&r_reset Sieve._resetSs 3;56":;333;56":; ; ;'+ +E +G HTWW-DJ ++htww/DK ++htww/DK rc  j\V4pVPR,^,pW8dR#V^,pW18:d>V;P\RVPW#44, unY3^,r2KCV;P\RVPW!^,44, unR#)zGrow the sieve to cover all primes <= n. Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend(30) >>> sieve[10] == 29 True Nrr))intr+r* _primerange)rnnumnum2s&& rextend Sieve.extend_s Fjjnq  7 Avi JJ&d&6&6s&AB BJAg fS$"2"23A">?? rc # ."V^,'d V^,pW8d\VPW!, ^,4pR.V,pVP^\VP\ V^V,,^,44F7p\ V^,V,)^,V,W54FpRWF&K K9 \ V4F(wruV'gKV^V,,^,xK* V^V,, pKR#5i)aGenerate all prime numbers in the range (a, b). Parameters ========== a, b : positive integers assuming the following conditions * a is an even number * 2 < self._list[-1] < a < b < nextprime(self._list[-1])**2 Yields ====== p (int): prime numbers such that ``a < p < b`` Examples ======== >>> from sympy.ntheory.generate import Sieve >>> s = Sieve() >>> s._list[-1] 13 >>> list(s._primerange(18, 31)) [19, 23, 29] TFN)minr/r+rrrange enumerate)rrb block_sizeblockptidxs&&& rrBSieve._primerangexs4 q55 FAeT0015Q,?JFZ'EZZ&T!a*n:Lq:P5Q"RS!a%!)  1Q6 FA$EHGT$E*1a#g+/)+ Z As CD$1Dc \V4p\VP4V8d5VP\ VPR,R,44KNR#)axExtend to include the ith prime number. Parameters ========== i : integer Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend_to_no(9) >>> sieve._list array('L', [2, 3, 5, 7, 11, 13, 17, 19, 23]) Notes ===== The list is extended by 50% if it is too short, so it is likely that it will be longer than requested. g?Nr))r rr+rFrA)rrs&&r extend_to_noSieve.extend_to_nos@. 1I$**o! KKDJJrNS01 2"rc# &"Vf\V4p^pM \^\V44p\V4pW8dR#VPV4VP\ VPV4\ VPV4RjxL R#L5i)aGenerate all prime numbers in the range [2, a) or [a, b). Examples ======== >>> from sympy import sieve, prime All primes less than 19: >>> print([i for i in sieve.primerange(19)]) [2, 3, 5, 7, 11, 13, 17] All primes greater than or equal to 7 and less than 19: >>> print([i for i in sieve.primerange(7, 19)]) [7, 11, 13, 17] All primes through the 10th prime >>> list(sieve.primerange(prime(10) + 1)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] N)rmaxrFr+r)rrrLs&&&r primerangeSieve.primerangesz0 9"AAAq)*A"A 6  A::k$**a8)$**a8: : :sBBB Bc # "\^\V44p\V4p\VP4pW8dR#W#8:d*\ W4FpVPV,xK R#V;P\ R\ W244, un\ ^V4FpVPV,pWT^, 8XdhW4,^, V,V,p\ WbV4F:pVPV;;,VPV,V,,uu&K< WA8gKVxK \ W24FpVPV,pWT8XdK\ WBV4F:pVPV;;,VPV,V,,uu&K< WA8gKnVPV,xK R#5i)zGenerate all totient numbers for the range [a, b). Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.totientrange(7, 18)]) [6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16] Nr)rWrrr,rJr*)rrrLrCrti startindexjs&&& r totientrangeSieve.totientrangesM ?1% & A    6  V1[kk!n$! KK6#uQ{3 3K1a[[[^Q;"#%!)!1A!5J":!4 A$++a.A*==56H!1[[[^7"1^ A$++a.A*==,6++a.( !sD2G9A>> from sympy import sieve >>> print([i for i in sieve.mobiusrange(7, 18)]) [-1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1] Nr)rWrrr-rJr*)rrrLrCrmir\r]s&&& r mobiusrangeSieve.mobiusranges%& ?1% & A    6  V1[kk!n$! KK6#sAE{3 3K1a[[[^eaiA-1 za0AKKNb(N16H !1[[[^q1ua+AKKNb(N,6H !sDF A%F  F c 4\V4p\V4pV^8d\RV,4hWPR,8dVP V4\ VPV4pVPV^, ,V8XdW33#W3^,3#)a&Return the indices i, j of the primes that bound n. If n is prime then i == j. Although n can be an expression, if ceiling cannot convert it to an integer then an n error will be raised. Examples ======== >>> from sympy import sieve >>> sieve.search(25) (9, 10) >>> sieve.search(23) (9, 9) zn should be >= 2 but got: %sr))rr r.r+rFr)rrCtestrLs&& rsearch Sieve.search1s"q! 1I q5;a?@ @ zz"~  KKN 4::q ! ::a!e  $4K!e8Orc\V4pV^8gQhT^,^8XdT^8H#TPT4wr#Y#8H# \\3dR#i;i)r F)r r.AssertionErrorrf)rrCrrLs&& r __contains__Sieve.__contains__Ns` q A6M6 q5A:6M{{1~v N+  sAAAc#F"\^4F pW,xK R#5i)r'Nr)rrCs& r__iter__Sieve.__iter__YsqA'Ms!c \V\4'dVPVP4VPe VPM^pV^8d \ R4hVP V^, VP^, VP1,#V^8d \ R4h\V4pVPV4VP V^, ,#)zReturn the nth prime numberzSieve indices start at 1.) isinstanceslicerTstopstart IndexErrorr+stepr )rrCrss&& r __getitem__Sieve.__getitem__]s a     aff % ww2AGGEqy!!<==::eai 1669: :1u!!<==q A   a ::a!e$ $r)r+r-rr,r/)i@B)NNNr)__name__ __module__ __qualname____firstlineno____doc__r1r8r>rFrBrTrXr^rbrfrjrmrv__static_attributes____classdictcell__) __classdict__s@rrrsW$V0 C 0@2' R36":H#)J*X: %%rrc l\V4pV^8d \R4hV\\P48:d\V,#^RIHp^RIHpVR8d*\P^V,4\V,#^p\W!V4P4V!V!V44P4,,4pWE8d:WE,^, pV!V4P4V8dTpK4V^,pK?\V^, V\V^, 4, 4#)a  Return the nth prime number, where primes are indexed starting from 1: prime(1) = 2, prime(2) = 3, etc. Parameters ========== nth : int The position of the prime number to return (must be a positive integer). Returns ======= int The nth prime number. Examples ======== >>> from sympy import prime >>> prime(10) 29 >>> prime(1) 2 >>> prime(100000) 1299709 See Also ======== sympy.ntheory.primetest.isprime : Test if a number is prime. primerange : Generate all primes in a given range. primepi : Return the number of primes less than or equal to a given number. References ========== .. [1] https://en.wikipedia.org/wiki/Prime_number_theorem .. [2] https://en.wikipedia.org/wiki/Logarithmic_integral_function .. [3] https://en.wikipedia.org/wiki/Skewes%27_number z-nth must be a positive integer; prime(1) == 2loglii)r r.rsiever+&sympy.functions.elementary.exponentialr'sympy.functions.special.error_functionsrrFrAevalf nextprime_primepi)nthrCrrrrLmids& rr4r4ssT s A1uHII C Qx::4x QUQx A AQ#c!f+"3"3"55 67A %ul c7==?Q AaA QUAQ/ 00rzgThe `sympy.ntheory.generate.primepi` has been moved to `sympy.functions.combinatorial.numbers.primepi`.z1.13z%deprecated-ntheory-symbolic-functions)deprecated_since_versionactive_deprecations_targetc^RIHpV!V4#)aRepresents the prime counting function pi(n) = the number of prime numbers less than or equal to n. .. deprecated:: 1.13 The ``primepi`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.primepi` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. Algorithm Description: In sieve method, we remove all multiples of prime p except p itself. Let phi(i,j) be the number of integers 2 <= k <= i which remain after sieving from primes less than or equal to j. Clearly, pi(n) = phi(n, sqrt(n)) If j is not a prime, phi(i,j) = phi(i, j - 1) if j is a prime, We remove all numbers(except j) whose smallest prime factor is j. Let $x= j \times a$ be such a number, where $2 \le a \le i / j$ Now, after sieving from primes $\le j - 1$, a must remain (because x, and hence a has no prime factor $\le j - 1$) Clearly, there are phi(i / j, j - 1) such a which remain on sieving from primes $\le j - 1$ Now, if a is a prime less than equal to j - 1, $x= j \times a$ has smallest prime factor = a, and has already been removed(by sieving from a). So, we do not need to remove it again. (Note: there will be pi(j - 1) such x) Thus, number of x, that will be removed are: phi(i / j, j - 1) - phi(j - 1, j - 1) (Note that pi(j - 1) = phi(j - 1, j - 1)) $\Rightarrow$ phi(i,j) = phi(i, j - 1) - phi(i / j, j - 1) + phi(j - 1, j - 1) So,following recursion is used and implemented as dp: phi(a, b) = phi(a, b - 1), if b is not a prime phi(a, b) = phi(a, b-1)-phi(a / b, b-1) + phi(b-1, b-1), if b is prime Clearly a is always of the form floor(n / k), which can take at most $2\sqrt{n}$ values. Two arrays arr1,arr2 are maintained arr1[i] = phi(i, j), arr2[i] = phi(n // i, j) Finally the answer is arr2[1] Examples ======== >>> from sympy import primepi, prime, prevprime, isprime >>> primepi(25) 9 So there are 9 primes less than or equal to 25. Is 25 prime? >>> isprime(25) False It is not. So the first prime less than 25 must be the 9th prime: >>> prevprime(25) == prime(9) True See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime )primepi)%sympy.functions.combinatorial.numbersr)rC func_primepis& rrrspN ?rc0V^8dQhR\R\/#)r rCreturn)rA)formats"r __annotate__rs__s_s_rc LV^8d^#V\PR,8:d\PV4^,#\V4p\ V^,4Uu.uFq"^,^, NK pp^.\ ^V^,4Uu.uFq V,^,^, NK up,pR.V^,,p\ ^V^,^4EF/pWR,'dKW2^, ,p\ W!^,V4FpRWW&K \ ^\ WV,,V4^,^4FlpWW,'dKW',pW8:d$WG;;,WH,V, ,uu&KDWG;;,W0V,,V, ,uu&Kn \ V\ WV,^, 4R4F+pW7;;,W7V,,V, ,uu&K- EK2 V^,#uupiuupi)a1Represents the prime counting function pi(n) = the number of prime numbers less than or equal to n. Explanation =========== In sieve method, we remove all multiples of prime p except p itself. Let phi(i,j) be the number of integers 2 <= k <= i which remain after sieving from primes less than or equal to j. Clearly, pi(n) = phi(n, sqrt(n)) If j is not a prime, phi(i,j) = phi(i, j - 1) if j is a prime, We remove all numbers(except j) whose smallest prime factor is j. Let $x= j \times a$ be such a number, where $2 \le a \le i / j$ Now, after sieving from primes $\le j - 1$, a must remain (because x, and hence a has no prime factor $\le j - 1$) Clearly, there are phi(i / j, j - 1) such a which remain on sieving from primes $\le j - 1$ Now, if a is a prime less than equal to j - 1, $x= j \times a$ has smallest prime factor = a, and has already been removed(by sieving from a). So, we do not need to remove it again. (Note: there will be pi(j - 1) such x) Thus, number of x, that will be removed are: phi(i / j, j - 1) - phi(j - 1, j - 1) (Note that pi(j - 1) = phi(j - 1, j - 1)) $\Rightarrow$ phi(i,j) = phi(i, j - 1) - phi(i / j, j - 1) + phi(j - 1, j - 1) So,following recursion is used and implemented as dp: phi(a, b) = phi(a, b - 1), if b is not a prime phi(a, b) = phi(a, b-1)-phi(a / b, b-1) + phi(b-1, b-1), if b is prime Clearly a is always of the form floor(n / k), which can take at most $2\sqrt{n}$ values. Two arrays arr1,arr2 are maintained arr1[i] = phi(i, j), arr2[i] = phi(n // i, j) Finally the answer is arr2[1] Parameters ========== n : int FTr))rr+rfrrJrI) rClimrarr1arr2skiprOr]sts & rrrsx 1uEKKO||Aq!! q'C"'a. 1.QUqLL.D 1 35C!G+<=+>> from sympy import nextprime >>> [(i, nextprime(i)) for i in range(10, 15)] [(10, 11), (11, 13), (12, 13), (13, 17), (14, 17)] >>> nextprime(2, ith=2) # the 2nd prime after 2 5 See Also ======== prevprime : Return the largest prime smaller than n primerange : Generate all primes in a given range zith should be positiver7r))rAr r.rr+rfrr )rCithrl_nns&& rrrzsrP AAs AAv1221u  QEKKO||A 519s5;;' ';;quqy) ) KKO U[[! !! AqDB w Q 1:: FA Q 1 Q 1:: FA Q F 1:: FA Q 1:: FA Qrc 2\V4pV^8d \R4hV^8d^^^^^^^^^^/V,#V\PR,8:d?\P V4wrW8Xd\V^, ,#\V,#^V^,,pW, ^8:d'V^, p\ V4'dV#V^,pM V^,p\ V4'dV#V^,p\ V4'dV#V^,pK;)aReturn the largest prime smaller than n. Notes ===== Potential primes are located at 6*j +/- 1. This property is used during searching. >>> from sympy import prevprime >>> [(i, prevprime(i)) for i in range(10, 15)] [(10, 7), (11, 7), (12, 11), (13, 11), (14, 13)] See Also ======== nextprime : Return the ith prime greater than n primerange : Generates all primes in a given range zno preceding primesr))rr.rr+rfr )rCrurs& r prevprimers& A1u.//1u1aAq!Q1-a00EKKO||A 61: 8O AqDBv{ F 1::H Q F 1::H Q 1::H QrNc#"Vf^TrW8dR#\PR,pW8:d \PW4RjxL R#W8:d>\P\\PV4RRjxL V^,pMV^,'d V^,p\ W^,4pW8d \P W4RjxL TpW8:dR#\ V4pW8dVxKR#LLL,5i)aoGenerate a list of all prime numbers in the range [2, a), or [a, b). If the range exists in the default sieve, the values will be returned from there; otherwise values will be returned but will not modify the sieve. Examples ======== >>> from sympy import primerange, prime All primes less than 19: >>> list(primerange(19)) [2, 3, 5, 7, 11, 13, 17] All primes greater than or equal to 7 and less than 19: >>> list(primerange(7, 19)) [7, 11, 13, 17] All primes through the 10th prime >>> list(primerange(prime(10) + 1)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] The Sieve method, primerange, is generally faster but it will occupy more memory as the sieve stores values. The default instance of Sieve, named sieve, can be used: >>> from sympy import sieve >>> list(sieve.primerange(1, 30)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] Notes ===== Some famous conjectures about the occurrence of primes in a given range are [1]: - Twin primes: though often not, the following will give 2 primes an infinite number of times: primerange(6*n - 1, 6*n + 2) - Legendre's: the following always yields at least one prime primerange(n**2, (n+1)**2+1) - Bertrand's (proven): there is always a prime in the range primerange(n, 2*n) - Brocard's: there are at least four primes in the range primerange(prime(n)**2, prime(n+1)**2) The average gap between primes is log(n) [2]; the gap between primes can be arbitrarily large since sequences of composite numbers are arbitrarily large, e.g. the numbers in the sequence n! + 2, n! + 3 ... n! + n are all composite. See Also ======== prime : Return the nth prime nextprime : Return the ith prime greater than n prevprime : Return the largest prime smaller than n randprime : Returns a random prime in a given range primorial : Returns the product of primes based on condition Sieve.primerange : return range from already computed primes or extend the sieve to contain the requested range. References ========== .. [1] https://en.wikipedia.org/wiki/Prime_number .. [2] https://primes.utm.edu/notes/gaps.html Nr))rr+rXrrIrBr)rrLlargest_known_primetails&& rrXrXsV y!1v++b/##A)));;{5;;:;<<< ! # Q Q q* +Dx$$Q--- v aL 5G ) * = .s<ADD:DDD 9DD)DDDcW8dR#\\W34wr\V^, V4p\V4pW18d \ V4pW08d \ R4hV#)aReturn a random prime number in the range [a, b). Bertrand's postulate assures that randprime(a, 2*a) will always succeed for a > 1. Note that due to implementation difficulties, the prime numbers chosen are not uniformly random. For example, there are two primes in the range [112, 128), ``113`` and ``127``, but ``randprime(112, 128)`` returns ``127`` with a probability of 15/17. Examples ======== >>> from sympy import randprime, isprime >>> randprime(1, 30) #doctest: +SKIP 13 >>> isprime(randprime(1, 30)) True See Also ======== primerange : Generate all primes in a given range References ========== .. [1] https://en.wikipedia.org/wiki/Bertrand's_postulate Nz&no primes exist in the specified range)maprArrrr.)rrLrCrOs&& r randprimeresZ@ v sQF DAAqA! Av aLuABB HrcV'd \V4pM \V4pV^8d \R4h^pV'd/\^V^,4FpV\ V4,pK V#\ ^V^,4F pW#,pK V#)a Returns the product of the first n primes (default) or the primes less than or equal to n (when ``nth=False``). Examples ======== >>> from sympy.ntheory.generate import primorial, primerange >>> from sympy import factorint, Mul, primefactors, sqrt >>> primorial(4) # the first 4 primes are 2, 3, 5, 7 210 >>> primorial(4, nth=False) # primes <= 4 are 2 and 3 6 >>> primorial(1) 2 >>> primorial(1, nth=False) 1 >>> primorial(sqrt(101), nth=False) 210 One can argue that the primes are infinite since if you take a set of primes and multiply them together (e.g. the primorial) and then add or subtract 1, the result cannot be divided by any of the original factors, hence either 1 or more new primes must divide this product of primes. In this case, the number itself is a new prime: >>> factorint(primorial(4) + 1) {211: 1} In this case two new primes are the factors: >>> factorint(primorial(4) - 1) {11: 1, 19: 1} Here, some primes smaller and larger than the primes multiplied together are obtained: >>> p = list(primerange(10, 20)) >>> sorted(set(primefactors(Mul(*p) + 1)).difference(set(p))) [2, 5, 31, 149] See Also ======== primerange : Generate all primes in a given range zprimorial argument must be >= 1)r rAr.rJr4rX)rCrrOrs&& r primorialrsd 1I F1u:;; A q!a%A qMA! HAq1u%A FA& Hrc#"\T;'g^4p^;rEY!V4rv^pV'dVxWg8wdJV'dW8d<V^, pWE8XdTpV^,p^pV'dVxV!V4pV^, pKOV'dW8XdV'dR#VR3xR#V'gH^p T;rg\V4F pV!V4pK Wg8wdV!V4pV!V4pV ^, p K!WY3xR#R#5i)aFor a given iterated sequence, return a generator that gives the length of the iterated cycle (lambda) and the length of terms before the cycle begins (mu); if ``values`` is True then the terms of the sequence will be returned instead. The sequence is started with value ``x0``. Note: more than the first lambda + mu terms may be returned and this is the cost of cycle detection with Brent's method; there are, however, generally less terms calculated than would have been calculated if the proper ending point were determined, e.g. by using Floyd's method. >>> from sympy.ntheory.generate import cycle_length This will yield successive values of i <-- func(i): >>> def gen(func, i): ... while 1: ... yield i ... i = func(i) ... A function is defined: >>> func = lambda i: (i**2 + 1) % 51 and given a seed of 4 and the mu and lambda terms calculated: >>> next(cycle_length(func, 4)) (6, 3) We can see what is meant by looking at the output: >>> iter = cycle_length(func, 4, values=True) >>> list(iter) [4, 17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14] There are 6 repeating values after the first 3. If a sequence is suspected of being longer than you might wish, ``nmax`` can be used to exit early (and mu will be returned as None): >>> next(cycle_length(func, 4, nmax = 4)) (4, None) >>> list(cycle_length(func, 4, nmax = 4, values=True)) [4, 17, 35, 2] Code modified from: https://en.wikipedia.org/wiki/Cycle_detection. N)rArJ) fx0nmaxvaluespowerlamtortoiseharermus &&&& r cycle_lengthrsf tyyq>DOE2d A   DAH Q <H QJEC Jw q   *    sAT7D{HT7D !GBg  s#TpKB\ V4'd V^,pV#^RIHp^RIH p^p\W!V4V!V!V44,,4pW48d9W4,^, pWW!V4, ^, V8dTpK3V^,pK>V\ V4, ^, pW8d&\ V4'g V^,pV^,pK+\ V4'd V^,pV#)aReturn the nth composite number, with the composite numbers indexed as composite(1) = 4, composite(2) = 6, etc.... Examples ======== >>> from sympy import composite >>> composite(36) 52 >>> composite(1) 4 >>> composite(17737) 20000 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range primepi : Return the number of primes less than or equal to n prime : Return the nth prime compositepi : Return the number of positive composite numbers less than or equal to n z1nth must be a positive integer; composite(1) == 4rr) r( r)) r r.rr+rr rrrrrA) rrC composite_arrrrLrrr n_compositess & r compositer+sO0 s A1uLMM8MBwU## ekk"oq Oa a%i5Q,CXc]"Q&* 1:: FA:: A As1vCF # $%A %ul C=1 q AaAx{?Q&L  qzz A L Qqzz Q Hrc\\V4pV^8d^#V\V4, ^, #)aReturn the number of positive composite numbers less than or equal to n. The first positive composite is 4, i.e. compositepi(4) = 1. Examples ======== >>> from sympy import compositepi >>> compositepi(25) 15 >>> compositepi(1000) 831 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime primepi : Return the number of primes less than or equal to n composite : Return the nth composite number )rAr)rCs&r compositepirls*, AA1u x{?Q r)r'r)T)NF)r|rr itertoolsrrr*sympy.core.randomrsympy.external.gmpyr primetestr sympy.utilities.decoratorr sympy.utilities.miscr rrrr4rrrrrXrrrrrr<rrrs '"%$0'T%T%n F1R  kBDU DUp_DPf,^fR) X? DUp> Br