A344120 For n >= 0, let N = 243 + n*343, let v(x) be the maximum power of 7 dividing x, and let p(N) be the partition function A000041(N). If v(p(N)) >= v(24*N-1) then a(n)=1, otherwise a(n)=0.

0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1

Offset

0

Comments

For n == 2,4, or 5 mod 7, a(n) = 1 [K. G. Ramanathan, p. 149, Corollary 1].

For n == 0,1,3, or 6 mod 7, it appears that a(n) = 0 in about 80% of the cases.

Links

Table of n, a(n) for n=0..81.

Washington Bomfim, For n mod 7 = 2,4,5, a(n) = 1. WHY?

K. G. Ramanathan, Ramanujan and the congruence properties of partitions, Prec. Indian Acad. Sci. (Math. Sci.), Vol. 89, Number 3, (1980), 133-157.

Example

a(0) = 0 because N = 243, p(243) = 133978259344888 = 2^3 * 7^2 * 97 * 5783 * 609289, so v(p(N)) = 2. Also 24*243 - 1 = 7^3 * 17, and v(24*N-1) = 3.

Prog

(PARI) a(n) = my(N = 243 + n*343); (n%7==2)||(n%7==4)||(n%7==5) || valuation(numbpart(N), 7) >= valuation(24*N-1, 7);

Crossrefs

Cf. A000041, A214411, A340757, A340957, A344121.

Sequence in context: A056051 A143541 A359583 * A090173 A214090 A072784

Adjacent sequences: A344117 A344118 A344119 * A344121 A344122 A344123

Keyword

nonn

Author

Washington Bomfim, May 09 2021

Status

approved