A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

0, 1, 6, 2315, 510510

Offset: 1

Author

Antti Karttunen, Feb 07 2024

Keywords

• nonn
• hard
• more

Comments

For the general dynamics of this phenomenon, see the scatter plots of A351231 and A351233.

Question: Are the terms by necessity all squarefree?

As a subsequence this sequence includes all primorials with indices k such that A024451(k) is a multiple of A000040(1+k). See A369972 and A369973.

872415232 < a(6) <= 13082761331670030 [= A369973(4)].

Examples

			2315 is included as A003415(2315) = 5+463 = 468 = 2^2 * 3^2 * 13 (note that 2315 is a semiprime = 5*463, thus its arithmetic derivative is the sum of its two prime factors), and because that 468 is a multiple of A276086(2315) = 234 = 2 * 3^2 * 13 [the exponents of primes are here read from the primorial base expansion of 2315, A049345(2315) = 100021].
510510 is included because A003415(510510) = 19*37693, which is a multiple of A276086(510510) = 19.
		

Links

• Index entries for sequences related to primorial base

Crossrefs

Cf. A000040, A003415, A024451, A276086, A369972, A369973 (subsequence).

Positions of 1's in A351231, positions of 0's in A351233 and in A369971.

After the two initial terms, a subsequence of A351228.

Cf. also A358221.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA369970(n) = !(A003415(n)%A276086(n));