A071734 -id:A071734 - cp's OEIS frontend

A160521 Coefficients in the expansion of C^7/B^8, in Watson's notation of page 106.

1, 8, 44, 192, 726, 2457, 7648, 22220, 60993, 159478, 399906, 966600, 2261630, 5139897, 11378988, 24598683, 52033372, 107890610, 219630050, 439535138, 865784403, 1680352500, 3216454360, 6077280123, 11343018559, 20928404349, 38194869384, 68989715838

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			x^27+8*x^51+44*x^75+192*x^99+726*x^123+2457*x^147+7648*x^171+...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Watson, G. N., Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), this sequence (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^7/(1 - x^k)^8, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

See Maple code in A160458 for formula.

a(n) ~ sqrt(11) * exp(Pi*sqrt(22*n/5)) / (2500*n). - Vaclav Kotesovec, Nov 28 2016

A213260 p(5n+4) where p(k) = number of partitions of k = A000041(k).

Original entry on oeis.org

5, 30, 135, 490, 1575, 4565, 12310, 31185, 75175, 173525, 386155, 831820, 1741630, 3554345, 7089500, 13848650, 26543660, 49995925, 92669720, 169229875, 304801365, 541946240, 952050665, 1653668665, 2841940500, 4835271870, 8149040695, 13610949895, 22540654445, 37027355200, 60356673280, 97662728555, 156919475295

Offset: 0

N. J. A. Sloane, Jun 07 2012

nonn

It is known that a(n) is divisible by 5 (see A071734).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
James Grime and Brady Haran, Partitions, Numberphile video (2016).
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6 1969 56--59. MR0236136 (38 #4434). - From _N. J. A. Sloane_, Jun 07 2012

Cf. A000041, A071734, A213256, A076394.

Table[PartitionsP[5n+4],{n,0,40}] (* Harvey P. Dale, May 30 2013 *)

a(n) = numbpart(5*n+4); \\ Michel Marcus, Jan 07 2015

from sympy.functions import partition
def a(n): return partition(5*n+4)
print([a(n) for n in range(33)]) # Michael S. Branicky, May 30 2021

a(n) = A000041(A016897(n)). - Omar E. Pol, Jan 18 2013

A220505 a(n) = spt(5n+4)/5 where spt(n) = A092269(n).

Original entry on oeis.org

2, 16, 88, 364, 1309, 4126, 11992, 32368, 82590, 200487, 467152, 1049224, 2283364, 4829302, 9959035, 20069790, 39612612, 76703340, 145945332, 273224940, 503888206, 916373028, 1644925432, 2916814954, 5113148026, 8866911378, 15220453704

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

That spt(5n+4) == 0 (mod 5) is one of the congruences stated by George E. Andrews. See theorem 2 in the Andrews' paper. See also A220507 and A220513.

G. E. Andrews, The number of smallest parts in the partitions of n
G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function
K. C. Garrett, C. McEachern, T. Frederick, O. Hall-Holt, Fast computation of Andrews' smallest part statistic and conjectured congruences, Discrete Applied Mathematics, 159 (2011), 1377-1380.
F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2010.
K. Ono, Congruences for the Andrews spt-function

Cf. A016897, A071734, A092269, A220485, A220507, A220513.

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i j, i - 1], {j, 0, n/i}]];
spt[n_] := b[n, n];
a[n_] := spt[5n+4]/5;
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jan 30 2019, after Alois P. Heinz in A092269 *)

a(n) = A092269(A016897(n))/5 = A220485(n)/5.

A327713 Exceptional class of numbers k such that p(25*k + 24) == 0 (mod 125), where p() = A000041().

Original entry on oeis.org

6, 26, 60, 65, 70, 81, 96, 126, 135, 141, 175, 176, 196, 205, 206, 226, 305, 310, 330, 340, 346, 371, 380, 435, 436, 440, 460, 480, 481, 516, 595, 611, 646, 650, 665, 666, 685, 696, 700, 710, 716, 725, 730, 736, 745, 751, 760, 765, 775, 780, 811, 826, 841, 860, 871

Offset: 1

Petros Hadjicostas, Sep 23 2019

nonn

The unexceptional class consists of the numbers k == (2, 3, or 4) (mod 5). Watson (1938, p. 111) proved that such numbers k satisfy p(25*k + 24) == 0 (mod 125).

(p(25*a(m) + 24)/125: m >= 1) = (3177000598, 140513239982045202108972, 23104937422373952975695974907848646058, ...).

			p(25*6 + 24) = p(174) = 397125074750 = 3177000598 * 125 (the only example in Watson (1938)).

David A. Corneth, Table of n, a(n) for n = 1..10000
Watson, G. N., Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle) 179 (1938), 97-128; see pp. 111-113.

Cf. A000041, A071734, A110375, A160524, A327714.

is(n) = n % 5 < 2 && numbpart(25*n+24)%125==0 \\ David A. Corneth, Sep 23 2019

More terms from David A. Corneth, Sep 23 2019

A160524 Exceptional class of numbers k such that p(5k+4) == 0 (mod 25), where p() = A000041().

Original entry on oeis.org

8, 15, 17, 37, 41, 46, 51, 53, 55, 65, 75, 77, 102, 106, 110, 116, 130, 131, 138, 140, 147, 157, 158, 165, 166, 167, 178, 180, 183, 192, 197, 217, 222, 225, 233, 235, 251, 258, 285, 287, 302, 310, 315, 321, 325, 328, 333, 336, 340, 355, 368, 371, 377, 380, 393, 416, 418, 420, 430, 432, 441, 447

Offset: 1

N. J. A. Sloane, Nov 13 2009

nonn

The unexceptional class consists of the numbers k == 4 (mod 5).

(p(5*a(m) + 4)/25: m >= 1) = (3007, 553946, 1999837, 61090943985, 341143252095, 2634063438811, 18381830017947, 38993374797785, 81633034103003, ...) - Petros Hadjicostas, Sep 23 2019

Watson, G. N., Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle) 179 (1938), 97-128; see p. 113.

Cf. A000041, A071734, A327713, A327714.

isA160524 := n -> 0 = modp(combinat:-numbpart(5*n + 4), 25) and 4 <> modp(n, 5):
select(isA160524, [$1..200]); # Petros Hadjicostas, Sep 23 2019

More terms from Petros Hadjicostas, Sep 23 2019

A333435 Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of y_n.

Original entry on oeis.org

4, 5, 6, 237, 2623, 815655

Offset: 3

Frank Ellermann, Mar 21 2020

Grime notes that Ramanujan's pattern for a(3), a(4), a(5) or prime(3), prime(4), prime(5) cannot be directly extended to prime(6) = 13, and shows solutions for 13, 17, 19.

			All {partition( 5k+4)} are divisible by prime(3) = 5, so a(3) = 4.
All {partition( 7k+5)} are divisible by prime(4) = 7, so a(4) = 5.
All {partition(11k+6)} are divisible by prime(5) = 11, so a(5) = 6.

James Grime and Brady Haran, Partitions, Numberphile video (2016).
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6 1969 56--59. MR0236136 (38 #4434).

Cf. A333436 (y_n), A000040 (primes), A000041 (partitions).
Cf. A071734 (p(5k+4)/5), A071746 (p(7k+5)/7), A076394 (p(11k+6)/11).
Cf. A213260 (p(5k+4)).

A333436 Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of x_n.

Original entry on oeis.org

5, 7, 11, 17303, 206839, 1977147619

Offset: 3

Frank Ellermann, Mar 21 2020

Grime notes that Ramanujan's pattern for a(3), a(4), a(5) or prime(3), prime(4), prime(5) cannot be directly extended to prime(6) = 13, and shows solutions for 13, 17, 19.

			All {partition( 5k+4)} are divisible by prime(3) = 5, so a(3) = 5.
All {partition( 7k+5)} are divisible by prime(4) = 7, so a(4) = 7.
All {partition(11k+6)} are divisible by prime(5) = 11, so a(5) = 11.

James Grime and Brady Haran, Partitions, Numberphile video (2016).
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6 1969 56--59. MR0236136 (38 #4434).

Cf. A333435 (x_n), A000040 (primes), A000041 (partitions).
Cf. A071734 (p(5k+4)/5), A071746 (p(7k+5)/7), A076394 (p(11k+6)/11).
Cf. A213260 (p(5k+4)).

cp's OEIS Frontend

A160521 Coefficients in the expansion of C^7/B^8, in Watson's notation of page 106.

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A213260 p(5n+4) where p(k) = number of partitions of k = A000041(k).

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A220505 a(n) = spt(5n+4)/5 where spt(n) = A092269(n).

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A327713 Exceptional class of numbers k such that p(25*k + 24) == 0 (mod 125), where p() = A000041().

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A160524 Exceptional class of numbers k such that p(5k+4) == 0 (mod 25), where p() = A000041().

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A333435 Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of y_n.

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A333436 Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of x_n.

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