A327770 -id:A327770 - 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

A052463 a(n) is the smallest nonnegative solution k to 24k == 1 (mod 7^(2n-2)).

Original entry on oeis.org

0, 47, 2301, 112747, 5524601, 270705447, 13264566901, 649963778147, 31848225129201, 1560563031330847, 76467588535211501, 3746911838225363547, 183598680073042813801, 8996335323579097876247

Offset: 1

Eric W. Weisstein

Related to a Ramanujan congruence for the partition function P = A000041.
In other words, a(n) = k such that 24*k (mod 7^(2*n-2) ) == 1. - N. J. A. Sloane, Oct 08 2019
If b(n) = a(n) + 7^(2*n-2)*r, where r is a nonnegative integer, then there is an integer s >= 0 such that 24*b(n) = 24*a(n) + 24*7^(2*n-2)*r = 7^(2*n-2)*s + 1 + 24*7^(2*n-2)*r = 7^(2*n-2)*(24*r+s) + 1 == 1 (mod 7^(2*n-2)). Thus, we insist that a(n) is the smallest k >= 0 such that 24*k == 1 (mod 7^(2*n-2)). - Petros Hadjicostas, Oct 09 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..600
G. K. Patil, Ramanujan's Life And His Contributions In The Field Of Mathematics, International Journal of Scientific Research and Engineering Studies (IJSRES), 1(6) (2014), ISSN: 2349-8862.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Index entries for linear recurrences with constant coefficients, signature (50,-49).

Cf. A000041, A052462, A052465, A052466, A327770.

I:=[0, 47]; [n le 2 select I[n] else 49*Self(n-1)-2: n in [1..20]]; // Vincenzo Librandi, Jul 01 2012

Table[PowerMod[24, -1, 7^(2b-2)], {b, 20}]
CoefficientList[Series[(-49x^2+47x)/((1-x)(1-49x)),{x,0,30}],x] (* Vincenzo Librandi, Jul 01 2012 *)
LinearRecurrence[{50,-49},{0,47,2301},20] (* Harvey P. Dale, Aug 23 2021 *)

G.f.: x^2*(-49*x + 47)/((1 - x)*(1 - 49*x)).
a(n) = 49*a(n-1) - 2. - Vincenzo Librandi, Jul 01 2012
a(n) = 23*49^n/1176 + 1/24, n > 1. - R. J. Mathar, Oct 09 2019

Name edited by Petros Hadjicostas, Oct 09 2019

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)

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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A052463 a(n) is the smallest nonnegative solution k to 24*k == 1 (mod 7^(2*n-2)).

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

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Formula

A052463 a(n) is the smallest nonnegative solution k to 24k == 1 (mod 7^(2n-2)).