A369650 - OEIS

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

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OFFSET

1,2

COMMENTS

Intersection with A048103 gives the fixed points (1, 2, 10, 15, 5005, ...) of A327859. Question: Does that set preclude numbers that are not squarefree (A371083)? Certainly it does not contain any multiples of 9. See also comments in A328110.

By considering the congruence A003415(k) == A276085(k) (mod 4), (see A392594), we deduce that any term of this sequence must either be (if it is odd) in A360110, or (if it is of the form 4m+2) in A017089 (which is the union of 2*A359161 and 2*A359151 = A369668), or if it is a multiple of 4, then in it must be one of the terms of 4*A373142. There are no terms of the form 8m+6 (A017137) in this sequence.

By considering known lower and upper bounds for arithmetic derivative, we know that this must be a subsequence of A392601.

By considering specific properties of A003415 and A276085, we conclude that if there are any terms of A371083 in this sequence, they must be included in A392592 (intersection of A391845 and A391866), and also in A392868, where the larger terms get noticeably more rare.

If it exists, a(7) > 5677350912 > 2^32. - Updated by Antti Karttunen, Jan 23 2026

If there is a term > 5005 which is also included in A048103, it is > 2^34 > A392868(25367) = 17178714237. - Antti Karttunen, Feb 04 2026

LINKS

Table of n, a(n) for n=1..6.

Index entries for sequences related to primorial base

EXAMPLE

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.

PROG

(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));

CROSSREFS

Cf. A003415, A046337, A048103, A276085, A276086, A327859, A328110, A359161, A369002, A369668, A373144.

Positions of 0's in A373146.

Intersection of A373487 and A373490.

Subsequence of the following sequences: A391864 (congruent mod 2310), A391865 (mod 5^5), A392593 (mod 3), A392594 (mod 4), A392601, A392867, A392869, A394895.

Intersection with A048103 is a subsequence of the following sequences: A370114, A370115, A392592, A392868 (which is the intersection of A391845 and A392867).

Cf. A371083, A392605.

Cf. also A351228 for a similar problem.

Sequence in context: A392601 A391864 A391865 * A392868 A181474 A392607

Adjacent sequences: A369647 A369648 A369649 * A369651 A369652 A369653

KEYWORD

nonn,hard,more

AUTHOR

Antti Karttunen, Feb 05 2024

EXTENSIONS

Comment-section rewritten by Antti Karttunen, Jan 22 2026

STATUS

approved