A182719 -id:A182719 - cp's OEIS frontend

A131210 Numbers k such that 24k - 1 is prime.

1, 2, 3, 7, 8, 10, 11, 13, 15, 16, 18, 20, 21, 25, 27, 30, 31, 35, 36, 37, 38, 41, 43, 46, 48, 51, 55, 57, 60, 62, 63, 65, 66, 67, 76, 77, 78, 85, 86, 87, 88, 92, 98, 100, 101, 102, 106, 108, 111, 112, 113, 120, 121, 122, 125, 126, 130, 132, 133, 140, 142, 147, 151, 153

Offset: 1

Zak Seidov, Oct 29 2007

The complement is 4, 5, 6, 9, 12, 14, 17, 19, 22, 23, 24, ... and contains numbers of the form == 4 (mod 5), == 5 (mod 7), == 6 (mod 11), == 6 (mod 13), == 5 (mod 17) etc., so the complement is a superset of A182719. - R. J. Mathar, Jun 10 2020

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A182719.

Filtered([1..160],n->IsPrime(24*n-1)); # Muniru A Asiru, Feb 19 2018

[n: n in [0..500]|IsPrime(24*n-1)] // Vincenzo Librandi, Nov 23 2010

select(n->isprime(24*n-1),[$1..160]); # Muniru A Asiru, Feb 19 2018

Select[Range[150], PrimeQ[24*# - 1] &] (* Amiram Eldar, Mar 28 2024 *)

is(n)=isprime(24*n-1) \\ Charles R Greathouse IV, Jun 06 2017

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.

A194699 a(n) = floor((p - 1)/12) - floor((p^2 - 1)/(24*p)), where p = prime(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16

Offset: 1

Omar E. Pol, Jan 18 2012

nonn

Sequence related to Ramanujan's famous partition congruences modulo powers of 5, 7 and 11. Ramanujan wrote: "It appears there are no equally simple properties for any moduli involving primes other than these three". On the other hand the Folsom-Kent-Ono theorem said: for a prime L >= 5, the partition numbers are L-adically fractal. Moreover, the Hausdorff dimension is <= floor((L - 1)/12) - floor((L^2 - 1)/(24*L)). Also, the Folsom-Kent-Ono corollary said: the dim is 0 only for L = 5, 7, 11 and so we have: 1) Ramanujan's congruences powers of 5, 7 and 11. 2) There are no simple properties for any other primes.

			For primes 5, 7, 11 the Hausdorff dimension = 0, so a(3)..a(5) = 0.
For primes 13, 17, 19, 23, 29, 31 the Hausdorff dimension = 1, so a(6)..a(11) = 1.

S. Ahlgren and K. Ono, Addition and counting: the arithmetic of partitions
A. Folsom, Z. A. Kent and K. Ono, l-adic properties of the partition function, preprint.
A. Folsom, Z. A. Kent and K. Ono, l-adic properties of the partition function, Advances in Mathematics, 229 (2012), pages 1586-1609.
Ken Ono (with Jan Bruinier, Amanda Folsom and Zach Kent), Emory University, Adding and counting
Wikipedia, Ramanujan's congruences

Cf. A000040, A000041, A182719, A194698.

a(n) = A194698(A000040(n)).

a(n) ~ 0.125 n log n. [Charles R Greathouse IV, Jan 25 2012]

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A131210 Numbers k such that 24k - 1 is prime.

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

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A194699 a(n) = floor((p - 1)/12) - floor((p^2 - 1)/(24*p)), where p = prime(n).

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