A331625 - OEIS

%I #10 Nov 14 2022 23:05:14

%S 1215,98415,273375,413343,846368

%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. A072066, A005179, A037019.

%K nonn,hard,more

%O 1,1

%A _Jianing Song_, Jan 22 2020