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

1,3

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)].

Links

Table of n, a(n) for n=1..5.

Index entries for sequences related to primorial base

Example

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.

Prog

(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));

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.

Sequence in context: A254005 A279654 A198403 * A279533 A069643 A264801

Adjacent sequences: A369967 A369968 A369969 * A369971 A369972 A369973

Keyword

nonn,hard,more

Author

Antti Karttunen, Feb 07 2024

Status

approved