A340957 Numbers k such that 24*k-1 has at least three factors 7 and the partition function evaluated at k has at least the same number of factors 7 as 24*k-1.

929, 1615, 1958, 3330, 4016, 4359, 5731, 6417, 6760, 8132, 8818, 9161, 10533, 11219, 11562, 12934, 13620, 13963, 15335, 16021, 16364, 17736, 18422, 18765, 20137, 20823, 21166, 22538, 23224, 23567, 24939, 25282, 25625, 25968, 27340, 28026, 28369, 29741, 30427

Offset: 1

Author

Washington Bomfim, Feb 01 2021

Keywords

• nonn

Comments

Note that [ 24*k-1 == 0 (mod 7^3) ] iff [ k == 243 (mod 7^3) ].

Related to a general congruence of Ramanujan about the partition function. See A340757.

Examples

			929 is a term because A000041(929) = 1378663636778122744608506419570 = 2 * 5 * 7^3 * 17 * 19 * 139 * 8952552536856941584067 and 24 * 929 - 1 = 5 * 7^3 * 13.
		

Links

• Index entries for related partition-counting sequences

Crossrefs

Cf. A000041, A214411, A340757.

Programs

Mathematica

Select[243 + 343*Range[100], IntegerExponent[PartitionsP[#], 7] >= IntegerExponent[24*# - 1, 7] &] (* Amiram Eldar, Feb 05 2021 *)

PARI

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

Formula

{ k : A214411(A000041(k)) >= A214411(24*k-1) >= 3 }. - Alois P. Heinz, Feb 01 2021