A331625 - OEIS

%I #14 Jan 21 2025 09:01:57

%S 1215,98415,273375,413343,846368,1987983,2302911,6082047,6200144,

%T 8089712,9034496,9861183,11868848,13010463,13325391,13955247,16159743,

%U 16592768,17537552,18482336,20686832,20883663,21198591,22143375,22891328,23206256,24347871,25607583

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

%C 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.

%e 1215 = 3^5 * 5, 1216 = 2^6 * 19;

%e 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;

%e 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;

%e so both 1215 and 1216 are exceptional, so 1215 is a term.

%o (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

%Y Cf. A005179, A037019, A072066.

%K nonn

%O 1,1

%A _Jianing Song_, Jan 22 2020

%E More terms from _Jinyuan Wang_, Jan 21 2025