Otj##ddlmZmZmZmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZddlmZmZddlmZddlmZddlZddlmZdZd Zd Zeddgd ed efd Zd ed efdZ edddddZ!dS))fzerofrom_int from_rationalfonefhalfbitcountto_intmpf_mulmpf_divmpf_submpf_addmpf_sqrtmpf_pi mpf_cosh_sinhmpf_cosmpf_sin)_sqrt_mod_prime_poweris_quad_residue) deprecated)recurrence_memoNcountcd}dg|zadg|zat|dzdz}td|D]E}t|dkr2t||z||D]}t|dkr |t|<Ftd|D]z}t|dkr|t|<|dz t|<+t|}||z}||zdkrt||zt|<_t||dz zt|<{dS)N順rr?)_factor_totientintrange)maxnlimijxys W/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/ntheory/partitions_.py_prer* s Dc$hGs4xH dCi..1 C 1c]]## 1:??1Q3a(( # #1:??!"GAJ 1d^^ . . 1:??GAJA#HQK  AJ qD q5A::"1+a-HQKK"1+q1u-HQKK . .c  |dkrtS|}d}t|}||zdkr||z}|dz }||zdk||z}dd|zz }t|}|dkr,|dkrd|z} | || zz}|td|dz | z| z}t |d|dzd} t t td| z||t| |} t t td |zd| dzz ztt|||t| ||S|dkrd|z} | || zz}|dkr|td |dzdz | z| z}t |d|dzd} t t td| z||t| |} t t tdd |dzzzdd| dzzz ztt|dz||t| ||S|||zz}|d zd vrd nd} ||zdkrF|dkr9t t| tt|||StSt||stS||dz z|dz z} |td | dz |z}t |||d} t t td| z||t||} t t td| ztt|||t| ||S|dks|dkrtj|dtj|d}}d||zz}||z|z|dzdz |zzt||z|z|zt|dz |z|z}||z|z|dzdz |zzt||z|z|zt|dz |z|z}t t!|||t!||||S|dkrd|zdztdt|dz |z|z}d|dz |dzdz dzz |dzzdzzdz}t t td t!||||t!|||Sd|zdztdt|dz |z|z}d||dzdz dzz dzzdz}t t!|||t!||||S)z Compute the inner sum in HRR formula [1]_ References ========== .. [1] https://msp.org/pjm/1956/6-1/pjm-v6-n1-p18-p.pdf rrr @ )ri@r )rrrpowrr r rrrrrrmathgcdr _a)nkpreck1epk2vpimodmargjacobi3_phid1d2n1n2s r)r;r;"s Avv B A A q&A++ q Q q&A++ BB BqDA B Qww 66A#Ca#g A3q!a%%%%,A%aAE2215A'1 r4))*23--??C7"q!q1u+.//!d++T33T""D** * 66A#Ca#g A1uus2q!tax---4%aAE2215A'(1Q3--T:: t%%C7B!a%=!aQi-899!Q$..66T""D** * AIB&((""a q5A::AvvW%%Xa[[$//777Lq!$$ L1q5z1q5! s3q!$$ $ !!Q * *1 - HQqSMM2t , , QKKw QwY   Xa[[$ ' '// C  && &  Avva"b!!48B#3#3B BK!tAvQR' "R HRL1,b 1 12578!tAvQR' "R HRL1,b 1 12578r"b$''BD)9)94@@@AvvsQwC"!12666" <AERUQYN*RU3q88A =w RLL r2t  d$$ r2t    Q37CHRL1,b11 1R 7B qBEAI>!Q& &! +B 2b"d##RB%5%5t < <sq23pisqrt8rDabcchshDEs r)_drXys  A B4  A ]1b$77>>ADA 71a==$ / /FBDq!!2&& . .A 2At 4 4d;;A 1a==r+r<returncd}d}tD]H}|d|zdzz }||z }|dkrn/||}||dzz}d|kr |||z }||dzr| n|z }I|S)zt Calculate the partition function P(n) Parameters ========== n : int nonnegative integer rr0rrr)r<prevrCpentar%npss r)_partition_recr_s A E WW    1q Y 66 E H a!e  77 bMA 1q5 aRRa Hr+c |dkrdS|dkr |tz dks#tdkr|dkrt|Sdtvrtt t jd|zdz dzzt jd |zz t jd z d zt jd z}t |d zd zx}}dt jdzzdt j dzz t jt j dzdz t jt j dzfd}tdt j |dz}|||dksJ|dkr+|||dkr|dz}|dkr|||dk|}|dz}||z d kr.||zdz}|||x}dkr|}n|dkr|}||z d k.|} | dkrtdt} ttt!dd||t#||} tt%d|} t'd | D]q} t)|| |}t+|| || | }t-| t|||} t/t1t3|dz}rt t3t-| t4|S)zX Calculate the partition function P(n) Parameters ========== n : int ri@ Fri@8rg@rr1 rg?d,r0KgUUUUUU?ctj}||z |||dz z ztj||z|z zzS)Nr)r9sqrtsinh)r<Nrhc1c2c3s r)_Mz_partition.._MsUy$$q''zBttAq1uI.tyDDGGA/F/FFFFr+r/(rz Input too bigr.2)r_ cache_lengthglobalsr*r!r9rDloglog2rhmaxceil ValueErrorrr rrrrr"r;rXr rabsr r)r<pbitsr>rArnbigsmallrjerMr^rOrPqrQdrkrlrms @@@r) _partitionrs_ 1uuq W ^88:::R??  ' ' ) )Q . .1v::a    !!   1R#  1  x||$&'( ) ) "  E59s?###D1 DGQJDIaLL( )B 1 b B 3 BGGGGGGG a1c6"" # #C 2a::     ((rr!SzzC''   ((rr!SzzC'' E 'C +// 5[1 "Q((NBc ! !CC 3YYE +// A 5yy))) A XmAq!44a88&))Q G GF Xa[[! $ $E 1a[[** q!QKK q!Q & & Awq!}}d + + S^^ $ $r ) vga--.. / //r+zpThe `sympy.ntheory.partitions_.npartitions` has been moved to `sympy.functions.combinatorial.numbers.partition`.z1.13z%deprecated-ntheory-symbolic-functions)deprecated_since_versionactive_deprecations_targetFc$ddlm}||S)a  Calculate the partition function P(n), i.e. the number of ways that n can be written as a sum of positive integers. .. deprecated:: 1.13 The ``npartitions`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.partition` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. P(n) is computed using the Hardy-Ramanujan-Rademacher formula [1]_. The correctness of this implementation has been tested through $10^{10}$. Examples ======== >>> from sympy.functions.combinatorial.numbers import partition >>> partition(25) 1958 References ========== .. [1] https://mathworld.wolfram.com/PartitionFunctionP.html r) partition)%sympy.functions.combinatorial.numbersr)r<verbosefunc_partitions r) npartitionsrs(BRQQQQQ >!  r+)F)" mpmath.libmprrrrrrr r r r r rrrrrresidue_ntheoryrrsympy.utilities.decoratorrsympy.utilities.memoizationrr9 itertoolsrr*r;rXr!r_rrr+r)rs@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@DCCCCCCC000000777777 ....U=U=U=n$!Q c C    4J0#J0#J0J0J0J0Z  tBDDD DDr+