A369650 Numbers k such that A003415(k) = A276085(k), where A003415 is the arithmetic derivative, and A276085 is the primorial base log-function.

1, 2, 10, 15, 28, 5005

Offset: 1

Author

Antti Karttunen, Feb 05 2024

Keywords

• nonn
• hard
• more

Comments

Intersection with A048103 gives the fixed points (1, 2, 10, 15, 5005, ...) of A327859. Question: Does that set preclude nonsquarefree numbers? Certainly it does not contain any multiples of 9. See also comments in A328110.

If k == 2 (mod 4), then both A003415(k) and A276085(k) are odd, and the latter is of the form 4m+1 (if k has an odd number of prime factors), or of the form 4m+3 (if k has an even number of prime factors). Therefore, for k of the form 4m+2 to be included in this sequence, a necessary condition is that it must be either in the intersection of A026424 and A358772 (like, for example, 2 is) or in A369668 (the intersection of A028260 and A358774), like for example, 10 is.

If k is odd, then A276085(k) is even, and for A003415(k) to be even with k odd, then k has to be in A046337 (odd numbers with an even number of prime factors, counted with multiplicity). But A276085(A046337(n)) == 0 (mod 4) for all n, so also A003415(k) has to be a multiple of 4, so k has to be in A360110 (itself a subsequence of A369002), like for example k=15 and k=5005 are.

If it exists, a(7) > 2^19.

Examples

			As 5005 = 5*7*11*13, A003415(5*7*11*13) = (5*7*11) + (5*7*13) + (5*11*13) + (7*11*13) = 2556 = 2^2 * 3^2 * 71 = A276085(5005) = A002110(2) + A002110(3) + A002110(4) + A002110(5) [as 5, 7, 11 and 13 are prime(3) .. prime(6)], therefore 5005 is included in this sequence.
		

Links

• Index entries for sequences related to primorial base

Crossrefs

Cf. A003415, A046337, A048103, A276085, A276086, A327859, A328110, A369002, A369668, A373144.

Positions of 0's in A373146.

Intersection of A373487 and A373490.

Cf. also A351228.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1, primepi(f[k, 1]-1), prime(i))); };
isA369650(n) = (A003415(n) == A276085(n));