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

Author

Wesley Ivan Hurt, Oct 09 2021

Keywords

• nonn

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

Examples

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

Crossrefs

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

Cf. A008578, A072873.

Programs

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

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