A327714 -id:A327714 - cp's OEIS frontend

A071746 a(n) = p(7n+5)/7 where p(k) denotes the k-th partition number.

1, 11, 70, 348, 1449, 5334, 17822, 55165, 160215, 441105, 1159752, 2929465, 7142275, 16873472, 38749850, 86737678, 189672868, 405991500, 852077072, 1756048833, 3558408287, 7098041203, 13951818365, 27047831797, 51760979985

Offset: 0

Benoit Cloitre, Jun 24 2002

One of the congruences related to the partition numbers stated by Ramanujan.

Berndt and Rankin, "Ramanujan: letters and commentaries", AMS-LMS, History of mathematics, vol. 9, pp. 192-193.
G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940. - From N. J. A. Sloane, Jun 07 2012

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. L. Drost, A Shorter Proof of the Ramanujan Congruence Modulo 5, Amer. Math. Monthly 104(10) (1997), 963-964.
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6(1) (1969), 56-59. MR0236136 (38 #4434). - From _N. J. A. Sloane_, Jun 07 2012

Cf. A000041, A017041, A071734, A076394, A213261, A327582, A327714, A327770.

a:= func< n | NumberOfPartitions((7*n+5)) div 7 >; [ a(n) : n in [0..30]]; // Vincenzo Librandi, Nov 30 2015

Table[PartitionsP[7n+5]/7, {n, 0, 24}] (* Jean-François Alcover, Nov 30 2015 *)

a(n)=if(n<0, 0, n=7*n+5; polcoeff(1/eta(x+x*O(x^n)),n)/7)

{a(n)=local(A,B); if(n<0, 0, A=x*O(x^n); B=eta(x^7+A); A=eta(x+A); polcoeff( B^3/A^4 +x*7*B^7/A^8, n))} /* Michael Somos, Jan 01 2006 */

a(n) = numbpart(7*n+5)/7; \\ Michel Marcus, Nov 30 2015

a(n) = (1/7)*A000041(7n+5).

a(n) = A000041(A017041(n))/7 = A213261(n)/7. - Omar E. Pol, Jan 18 2013

A213261 a(n) = p(7*n + 5), where p(k) = number of partitions of k = A000041(k).

Original entry on oeis.org

7, 77, 490, 2436, 10143, 37338, 124754, 386155, 1121505, 3087735, 8118264, 20506255, 49995925, 118114304, 271248950, 607163746, 1327710076, 2841940500, 5964539504, 12292341831, 24908858009, 49686288421, 97662728555, 189334822579, 362326859895, 684957390936, 1280011042268, 2366022741845, 4328363658647, 7840656226137

Offset: 0

N. J. A. Sloane, Jun 07 2012

nonn

It is known that a(n) is divisible by 7 (see A071746).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Ho-Hon Leung, Another Identity for Complete Bell Polynomials based on Ramanujan's Congruences, arXiv:1802.08443 [math.CO], 2018.
Ho-Hon Leung, Another Identity for Complete Bell Polynomials based on Ramanujan's Congruences, J. Integer Seq. 21 (2018), Article 18.6.4.
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6(1) (1969), 56-59. MR0236136 (38 #4434). - From _N. J. A. Sloane_, Jun 07 2012

Cf. A000041, A017041, A071734, A071746, A076394, A213256, A213260, A213261, A327582, A327714, A327770.

Table[PartitionsP[7 n + 5], {n, 0, 29}] (* Jean-François Alcover, Nov 12 2018 *)

a(n) = numbpart(7*n+5); \\ Michel Marcus, Jan 07 2015

a(n) = A000041(A017041(n)). - Omar E. Pol, Jan 18 2013

a(n) = 7 * A071746(n). - Joerg Arndt, Nov 06 2016

A327770 a(n) = (23 * 7^(2*n) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

Original entry on oeis.org

1, 47, 2301, 112747, 5524601, 270705447, 13264566901, 649963778147, 31848225129201, 1560563031330847, 76467588535211501, 3746911838225363547, 183598680073042813801, 8996335323579097876247, 440820430855375795936101, 21600201111913414000868947

Offset: 0

Petros Hadjicostas, Sep 24 2019

If p(n) = A000041(n) is the partition function, Watson (1938) proved that p(7^(2*m)*n + a(m)) == 0 mod 7^(m+1) for n >= 0 and m >= 1. (Obviously, this is not always true for m = 0).
For m=1 and n=0, p(7^(2*1)*0 + a(1)) = p(47) = 7^(1+1) * 2546.
For m=1 and n=1, p(7^(2*1)*1 + a(1)) = p(96) = 7^(1+1) * 2410496.
For m=1 and n=2, p(7^(2*1)*2 + a(1)) = p(145) = 7^(1+1) * 508344041.
For m=2 and n=0, p(7^(2*2)*0 + a(2)) = p(2301) = 7^(2+1) * 49629361905981812695622866669844910256876089360.
Essentially the same as A052463. - R. J. Mathar, Oct 08 2019

Colin Barker, Table of n, a(n) for n = 0..500
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see pp. 118 and 124.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Wikipedia, G. N. Watson.
Index entries for linear recurrences with constant coefficients, signature (50,-49).

Cf. A052463, A071746, A213261, A327714, A327582.

CoefficientList[Series[(1 - 3 x)/((1 - x) (1 - 49 x)), {x, 0, 15}], x] (* Michael De Vlieger, Sep 27 2019 *)
LinearRecurrence[{50,-49},{1,47},20] (* Harvey P. Dale, Mar 09 2023 *)

a(n) = (23 * 7^(2*n) + 1)/24; \\ Michel Marcus, Sep 25 2019

Vec((1 - 3*x) / ((1 - x)*(1 - 49*x)) + O(x^20)) \\ Colin Barker, Sep 25 2019

From Colin Barker, Sep 25 2019: (Start)
G.f.: (1 - 3*x) / ((1 - x)*(1 - 49*x)).
a(n) = 50*a(n-1) - 49*a(n-2) for n>1.
(End)

A327582 a(n) = (17 * 7^(2*n+1) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

Original entry on oeis.org

5, 243, 11905, 583343, 28583805, 1400606443, 68629715705, 3362856069543, 164779947407605, 8074217422972643, 395636653725659505, 19386196032557315743, 949923605595308471405, 46546256674170115098843, 2280766577034335639843305, 111757562274682446352321943

Offset: 0

Petros Hadjicostas, Sep 23 2019

If p(n) = A000041(n) is the partition function, Watson (1938) proved that p(7^(2*m+1)*n + a(m)) == 0 mod 7^(m+1) for n >= 0 and m >= 1.

It is well-known that this result is true even for m = 0 (cf. A071746 and the references there).

			For m=1 and n=0, p(7^(2*1+1)*0 + a(1)) = p(243) = 133978259344888 = 7^2 * 2734250190712.
For m=1 and n=1, p(7^(2*1+1)*1 + a(1)) = p(586) = 224282898599046831034631 = 7^2 * 4577202012225445531319.

Colin Barker, Table of n, a(n) for n = 0..500
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see pp. 118 and 124.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Wikipedia, G. N. Watson.
Index entries for linear recurrences with constant coefficients, signature (50,-49).

Cf. A000041, A071746, A110375, A213261, A327714, A327770.

a(n) = (17 * 7^(2*n+1) + 1)/24; \\ Michel Marcus, Sep 25 2019

Vec((5 - 7*x) / ((1 - x)*(1 - 49*x)) + O(x^15)) \\ Colin Barker, Sep 27 2019

From Colin Barker, Sep 27 2019: (Start)
G.f.: (5 - 7*x) / ((1 - x)*(1 - 49*x)).
a(n) = 50*a(n-1) - 49*a(n-2) for n>1.
(End)

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

A327771 a(n) = p(49*n + 47)/49, where p(k) denotes the k-th partition number (i.e., A000041).

Original entry on oeis.org

2546, 2410496, 508344041, 48286178405, 2734250190712, 106823899382728, 3143746885297470, 73830872731991927, 1440681502991063990, 24058683492974200054, 351628923073820626951, 4577202012225445531319, 53811955397591074514675, 577896157936323089053580

Offset: 0

Petros Hadjicostas, Sep 24 2019

nonn

Watson (1938), p. 120, proved that p(7*n + 5) == 0 (mod 7) and p(49*n + 47) == 0 (mod 49) for n >= 0, where p() = A000041(). For more general congruence results modulo a power of 7 by George Neville Watson regarding the partition function, see A327582 and A327770.

G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see p. 120.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Wikipedia, G. N. Watson.

Cf. A000041, A052462, A052463, A052465, A052466, A071746, A213261, A327714, A327582, A327770.

Table[PartitionsP[49n+47]/49,{n, 0, 13}] (* Metin Sariyar, Sep 25 2019 *)

a(n) = numbpart(49*n + 47)/49; \\ Michel Marcus, Sep 25 2019

a(n) = A000041(49*n + 47)/49.

cp's OEIS Frontend

A071746 a(n) = p(7n+5)/7 where p(k) denotes the k-th partition number.

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A213261 a(n) = p(7*n + 5), where p(k) = number of partitions of k = A000041(k).

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A327770 a(n) = (23 * 7^(2*n) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

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A327582 a(n) = (17 * 7^(2*n+1) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

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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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A327771 a(n) = p(49*n + 47)/49, where p(k) denotes the k-th partition number (i.e., A000041).

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