A331625 Numbers k such that both k and k+1 are exceptional (A072066).

1215, 98415, 273375, 413343, 846368, 1987983, 2302911, 6082047, 6200144, 8089712, 9034496, 9861183, 11868848, 13010463, 13325391, 13955247, 16159743, 16592768, 17537552, 18482336, 20686832, 20883663, 21198591, 22143375, 22891328, 23206256, 24347871, 25607583

Offset: 1

Author

Jianing Song, Jan 22 2020

Keywords

• nonn

Comments

Conjecture: for every p > 0, there exist infinitely many k such that k, k+1, ..., k+p-1 are all exceptional numbers. In specific, there exist infinitely many k such that both k and k+1 are exceptional.

Examples

			1215 = 3^5 * 5, 1216 = 2^6 * 19;
thus A037019(1215) = 2^4 * 3^2 * 5^2 * 7^2 * 11^2 * 13^2 = 3607203600, A037016(1216) = 2^18 * 3 * 5 * 7 * 11 * 13 * 17 = 66913566720;
but the smallest number with 1215 divisors is 3073593600 = 2^8 * 3^4 * 5^2 * 7^2 * 11^2, the smallest number with 1216 divisors is 35424829440 = 2^18 * 3^3 * 5 * 7 * 11 * 13;
so both 1215 and 1216 are exceptional, so 1215 is a term.
		

Crossrefs

Cf. A005179, A037019, A072066.

Programs

PARI

isA331625(n) = A037019(n+1) > A005179(n+1) && A037019(n) > A005179(n) \\ See A037019 and A005179 for their programs. Warning: this is extremely inefficient

Extensions

More terms from Jinyuan Wang, Jan 21 2025