A348284 Numbers k such that k | k" where k" is the 2nd arithmetic derivative of k.

1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 27, 29, 31, 37, 41, 43, 47, 48, 53, 54, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 108, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 162, 163, 167, 168, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset

1,2

Comments

As 1" = 0 and p" = 0 when p is prime, 1 and every prime are terms, hence A008578 is a subsequence. - Bernard Schott, Oct 12 2021

Links

Table of n, a(n) for n=1..64.

Example

8 is in the sequence since 8" = 16 and 8 | 16.

Maple

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
q:= n-> is(irem(d(d(n)), n)=0):
select(q, [$1..250])[];  # Alois P. Heinz, Oct 15 2021

Prog

(PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = !(ad(ad(k)) % k); \\ Michel Marcus, Oct 10 2021

Crossrefs

Cf. A003415 (1st derivative), A068346 (2nd derivative).

Cf. A008578, A072873.

Sequence in context: A307824 A081730 A162408 * A162721 A378441 A176176

Adjacent sequences: A348281 A348282 A348283 * A348285 A348286 A348287

Keyword

nonn

Author

Wesley Ivan Hurt, Oct 09 2021

Status

approved