A052465 -id:A052465 - cp's OEIS frontend

A052462 a(n) is the minimal positive integral solution k to 24*k == 1 (mod 5^n).

4, 24, 99, 599, 2474, 14974, 61849, 374349, 1546224, 9358724, 38655599, 233968099, 966389974, 5849202474, 24159749349, 146230061849, 603993733724, 3655751546224, 15099843343099, 91393788655599, 377496083577474

Offset: 1

Eric W. Weisstein

Related to a Ramanujan congruence for the partition function P = A000041.

Extending work of Ramanujan, Watson (1938) proved that P(m) == 0 (mod 5^n) if 24*m == 1 (mod 5^n). In particular, P(a(n)) == 0 (mod 5^n). - Petros Hadjicostas, Jul 29 2020

			From _Petros Hadjicostas_, Jul 29 2020: (Start)
A000041(a(1)) = A000041(4) = 5 == 0 (mod 5).
A000041(a(2)) = A000041(24) = 1575 == 0 (mod 5^2).
A000041(a(3)) = A000041(99) = 169229875 == 0 (mod 5^3).
A000041(a(4)) = A000041(599) = 435350207840317348270000 == 0 (mod 5^4). (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Index entries for linear recurrences with constant coefficients, signature (1,25,-25).

Cf. A000041, A052463, A052465, A052466.

I:=[4, 24, 99]; [n le 3 select I[n] else Self(n-1)+25*Self(n-2)-25*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 01 2012

Table[PowerMod[24, -1, 5^a], {a, 21}]
CoefficientList[Series[(-25x^2+20x+4)/((1-x)(1-5x)(1+5x)),{x,0,30}],x] (* Vincenzo Librandi, Jul 01 2012 *)

a(n) = lift(Mod(24, 5^n)^-1) \\ David A. Corneth and Petros Hadjicostas, Jul 29 2020

G.f.: x*(-25*x^2 + 20*x + 4)/((1 - x)*(1 - 5*x)*(1 + 5*x)).
a(n) = (1 + (21 + 2*(-1)^n)*5^n)/24.  - Bruno Berselli, Apr 04 2011
a(n) = a(n-1) + 25*a(n-2) - 25*a(n-3). - Vincenzo Librandi, Jul 01 2012
A000041(a(n)) == 0 (mod 5^n). - Petros Hadjicostas, Jul 29 2020

Name edited by Petros Hadjicostas, Jul 29 2020

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

A052466 a(n) is the smallest positive solution k to 24*k == 1 (mod 13^n).

Original entry on oeis.org

6, 162, 1007, 27371, 170176, 4625692, 28759737, 781741941, 4860395546, 132114388022, 821406847267, 22327331575711, 138817757188116, 3773319036295152, 23460200964791597, 637690917133880681, 3964773963049779886, 107769764995625835082, 670046799755412800727

Offset: 1

Eric W. Weisstein

Related to a generalization of a Ramanujan congruence for the partition function P = A000041.
Atkin and O'Brien (1967) proved that for all integral n >= 1, there is an integral constant K(n) not divisible by 13 s.t. P(169*m - 7) == K(n)*P(m) (mod 13^n) for all integral m >= 1 that satisfy 24*m == 1 (mod 13^n). In particular, P(169*a(n) - 7) == K(n)*P(a(n)) (mod 13^n) for all n >= 1. Unfortunately, the calculation of the integral constants K(n) depends on several recursions found in the paper. (For each n, there are infinitely many such K(n)'s, but one may choose the smallest one that satisfies the above property.) See Theorem 2, p. 444, in their paper, even though their P is different that the P = A000041 here. - Petros Hadjicostas, Jul 29 2020
From Petros Hadjicostas, Aug 02 2020: (Start)
Assume n = 2*m, where m >= 1, and 24*k == 1 (mod 13^(2*m)), where k >= 1. Then there is an integer x = x(k) s.t. 24*k - 1 = 169^m*x. Then 1 = 24*k - 169^m*x == 0 - 1^m*x == -x (mod 24). With x = x(k) = 23, we find a(2*m), the smallest value of k >= 1 that satisfies 24*k == 1 (mod 13^(2*m)). Thus, a(2*m) = (1 + 23*13^(2*m))/24.
Assume now n = 2*m + 1, where m >= 0, and 24*k == 1 (mod 13^(2*m+1)), where k >= 1. Then there is an integer x = x(k) s.t. 24*k - 1 = 13*169^m*x. Then 1 = 24*k - 13*169^m*x == 0 - 13*1^m*x == -13*x (mod 24). With x = x(k) = 11, we find a(2*m+1), the smallest value of k >= 1 that satisfies 24*k == 1 (mod 13^(2*m)). Thus, a(2*m+1) = (1 + 11*13^(2*m+1))/24. (End)

			From _Petros Hadjicostas_, Jul 29 2020: (Start)
The only value of the constant K(n) that appears explicitly in Atkin and O'Brien (1967) is K(2) = 45 (see p. 453). We then have
P(169*a(2) - 7) - K(2)*P(a(2)) = P(169*162 - 7) - 45*P(162) = A000041(27371) - 45*A000041(162) = A000041(27371) - 5846125708665 == 0 (mod 13^2).
Thus, we must have A000041(27371) == 99 (mod 169). (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..900
A. O. L. Atkin and J. N. O'Brien, Some Properties of p(n) and c(n) Modulo Powers of 13, Trans. Amer. Math. Soc. 126 (1967), 442-459.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Index entries for linear recurrences with constant coefficients, signature (1,169,-169).

Cf. A000041, A052462, A052463, A052465.

I:=[6, 162, 1007]; [n le 3 select I[n] else Self(n-1)+169*Self(n-2)-169*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 01 2012

Table[PowerMod[24, -1, 13^d], {d, 20}]
CoefficientList[Series[(-169x^2+156x+6)/((1-x)(1-13x)(1+13x)),{x,0,40}],x] (* Vincenzo Librandi, Jul 01 2012 *)
LinearRecurrence[{1,169,-169},{6,162,1007},30] (* Harvey P. Dale, Mar 15 2015 *)

a(n) = lift(Mod(24, 13^n)^-1) \\ Petros Hadjicostas, Jul 29 2020

def a(n): return 24.inverse_mod(13^n)
print([a(n) for n in range(1, 20)]) # Peter Luschny, Jul 30 2020

G.f.: x*(-169*x^2 + 156*x + 6)/((1 - x)*(1 - 13*x)*(1 + 13*x)). - Vincenzo Librandi, Jul 01 2012
a(n) = a(n-1) + 169*a(n-2) - 169*a(n-3). - Vincenzo Librandi, Jul 01 2012
From Petros Hadjicostas, Aug 02 2020: (Start)
a(n) = (1 + 11*13^n)/24, if n is odd, and a(n) = (1 + 23*13^n)/24, if n is even.
a(n) - a(n-1) = 12*13^(n-1) for n even >= 2, and 5*13^(n-1) for n odd >= 3. (End)

Name edited by Petros Hadjicostas, Jul 29 2020

A340757 Counterexamples to a conjecture of Ramanujan about congruences related to the partition function.

Original entry on oeis.org

243, 586, 1272, 2301, 2644, 2987, 3673, 4702, 5045, 5388, 6074, 7103, 7446, 7789, 8475, 9504, 9847, 10190, 10876, 11905, 12248, 12591, 13277, 14306, 14649, 14992, 15678, 16707, 17050, 17393, 18079, 19108, 19451, 19794, 20480, 21509, 21852, 22195, 22881, 23910

Offset: 1

Washington Bomfim, Jan 19 2021

For b in 5,7,11, and all integers n,e >= 1, Ramanujan conjectured that if (24*n-1) is divisible by b^e, the partition function p(n) = A000041(n) is also divisible by b^e.
Chowla found the first counterexample a(1) = 243.  Watson showed the conjecture holds for b=5, and Atkin showed it holds for b=11.  Watson showed p(n) is divisible by 7^floor((d+2)/2) when 24n-1 is divisible by 7^d, so that exceptions here are restricted to 24n-1 == 0 (mod 7^3), which is n == 243 (mod 7^3).
See A340957 for the converse, those n == 243 (mod 7^3) where the conjecture does hold.

			243 is a term because for n = 243, the condition of Ramanujan (24*n - 1) divisible by b^e is true, and p(n) is not divisible by (b^e). [We have base b=7, and exponent e=3 in this case.] Since a(1) = A182719(91), 90 numbers satisfy the conjecture before the first counterexample a(1).

A. O. L. Atkin and P. Swinnerton-Dyer, Some Properties of Partitions,, Proceedings of the London Math. Soc., V. s3-4, Issue 1, pp. 84-106, (1954)
A. O. L. Atkin and S. M. Hussain, Some Properties of Partitions II, Trans. Amer. Math. Soc. 89, pp. 184-200 (1958).
Hansraj Gupta, Partitions - A Survey, Journal of Research of the National Bureau of Standards - B. Mathematical Sciences Vol. 74B, No. 1, January-March 1970. See section 6.1.
D. H. Lehmer, On a conjecture of Ramanujan, J. London Math. Soc. 11, 114-118 (1936).
D. H. Lehmer, On the Hardy-Ramanujan Series for the partition function,, J. London Math. Soc. 12, 171-176 (1937).
G. N. Watson, A New Proof of the Rogers-Ramanujan Identities, J. London Math. Soc., V. s1-4, 1, Pages 4-9. (1929).
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Index entries for related partition-counting sequences

Cf. A000041, A052462, A052465, A052466, A182719, A340957.

seq(x) = {my( n = -100, N=0); while(N < x, n += 343; if(valuation(numbpart(n),7) < valuation(24*n-1,7), print1(n", "); N++)) };
seq(100); \\ Gives the first 100 terms of the sequence.

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

A052462 a(n) is the minimal positive integral solution k to 24*k == 1 (mod 5^n).

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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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A052466 a(n) is the smallest positive solution k to 24*k == 1 (mod 13^n).

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A340757 Counterexamples to a conjecture of Ramanujan about congruences related to the partition function.

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