a kh##@sddlmZmZmZmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZddlmZmZddlmZddlmZddlZddlmZdd Zd d Zd d ZeddgeedddZeedddZ edddddddZ!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_memoNcountcCsd}dg|adg|at|dd}td|D]:}t|dkr2t||||D]}t|dkrR|t|<qRq2td|D]h}t|dkr|t|<|dt|<qxt|}||}||dkrt||t|<qxt||dt|<qxdS)N順rr?)_factor_totientintrange)ZmaxnZlimijxyr%G/var/www/auris/lib/python3.9/site-packages/sympy/ntheory/partitions_.py_pre s&        r'cCsd|dkr tS|}d}t|}||dkr:||}|d7}q||}dd|}t|}|dkr|dkrd|} | || }|td|d| | }t|d|dd} tttd| ||t| |} tttd |d| dtt|||t | ||S|dkrd|} | || }|dkrF|td |dd| | }t|d|dd} tttd| ||t| |} tttdd |ddd| dtt|d||t | ||S|||}|d d vrd nd} ||dkr|dkrtt| tt|||St St ||s*t S||d|d} |td | d|}t|||d} tttd| ||t||} tttd| tt|||t | ||S|dks|dkrt |dt |d}}d||}||||dd|t||||t|d||}||||dd|t||||t|d||}tt|||t||||S|dkrd|dtdt|d||}d|d|ddd|ddd}tttd t||||t|||Sd|dtdt|d||}d||ddddd}tt|||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 rr rrrrmathgcdr_a)nkprecZk1epZk2vpimodmargZjacobi3Z_phid1Zd2Zn1Zn2r%r%r&r6"s           "        $,  $ r6c Cst|}t|}t|||}tt|tdd||}t||}tt|||\} } tt||tt||||} tt|| t| |||} t| | S)z Compute the sinh term in the outer sum of the HRR formula. The constants sqrt(2/3*pi) and sqrt(8) must be precomputed. rr()rrr r rr rr ) r7r"r9sq23pisqrt8r=abcchshDEr%r%r&_dys  rK)r7returncCszd}d}tD]f}|d|d7}||}|dkr6qv||}||d8}d|kr^|||7}||drn| n|7}q|S)zt Calculate the partition function P(n) Parameters ========== n : int nonnegative integer rr+rrr)r7prevr<Zpentar!npsr%r%r&_partition_recs    rPcsB|dkr dS|dkr$|tdks8tdkr@|dkr@t|SdtvrPtttjd|ddtd |td d td }t|d d }}dtjddt dtjt ddtjt dfdd}t dt |d}|||dksJ|dkrB|||dkrB|d}q|}|d}||d kr||d}|||}dkr|}n|dkrN|}qN|} | dkrt dt } tttdd||t||} ttd|} td | D]H} t|| |}t|| || | }t| t|||} ttt|d}qttt| t|S)zX Calculate the partition function P(n) Parameters ========== n : int ri@ Fri@8rg@rr, rg?d,r+KgUUUUUU?cs<tj}|||||dt|||S)Nr)r4sqrtsinh)r7NrWZc1c2c3r%r&_Msz_partition.._Mr*(rz Input too bigr)2)rPZ cache_lengthglobalsr'rr4r=loglog2rWmaxceil ValueErrorrr r rrrr r6rKr rabsrr)r7Zpbitsr9r;r]bigsmallrYZerMrOrBrCqrDdr%rZr& _partitionsb         rlzpThe `sympy.ntheory.partitions_.npartitions` has been moved to `sympy.functions.combinatorial.numbers.partition`.z1.13z%deprecated-ntheory-symbolic-functions)Zdeprecated_since_versionZactive_deprecations_targetFcCsddlm}||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)Z%sympy.functions.combinatorial.numbersrm)r7verboseZfunc_partitionr%r%r& npartitionss! ro)F)"Z mpmath.libmprrrrrrrr r r r r rrrrZresidue_ntheoryrrZsympy.utilities.decoratorrZsympy.utilities.memoizationrr4 itertoolsrr'r6rKrrPrlror%r%r%r&s H   W M