A348283 Numbers that are multiples of their arithmetic derivative, A003415.

0, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Author

Wesley Ivan Hurt, Oct 09 2021

Keywords

• nonn

Comments

Here, m' denotes the arithmetic derivative of m (A003415).

Not the same as A211781 since this sequence does not contain 225, 252, etc.

All prime numbers p are in the sequence since p' = 1 | p.

Numbers k such that k' | k. - The original definition of the sequence.

Sequence consists of 0, primes, and the prime powers of the form p^p (A051674, that together with 0 give the only fixed points of A003415). This can be seen from theorems 4-6 given in the Ufnarovski & Åhlander paper. - Antti Karttunen, May 17 2025

Examples

			0 is in the sequence as A003415(0) = 0 and 0 is a multiple 0.
27 is in the sequence as A003415(27) = 27' = 27, and 27 is a multiple of 27.
127 (like any prime) is in the sequence since 127' = 1 | 127.
		

Links

• Antti Karttunen & R. J. Mathar, Table of n, a(n) for n = 1..20001
• Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

Crossrefs

Cf. A003415, A211781.

After the initial zero, gives the indices of 0's in A369049.

Disjoint union of {0}, A000040 and A051674.

Apart from term 2, a subsequence of A383300.

Programs

Maple

q:= n-> is(irem(n, n*add(i[2]/i[1], i=ifactors(n)[2]))=0):
select(q, [$2..300])[];  # Alois P. Heinz, Oct 11 2021

PARI

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = (0==k) || ((k>1) && !(k % ad(k))); \\ Michel Marcus, Oct 10 2021

Extensions

a(1) = 0 inserted because of a new, more inclusive definition. - Antti Karttunen, May 17 2025