A328110 Fixed points of A327860: numbers k for which A003415(A276086(k)) = k, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

0, 1, 7, 8, 2556

Offset: 1

Author

Antti Karttunen, Oct 08 2019

Keywords

• nonn
• hard
• more

Comments

Applying A276086 to these terms gives the fixed points of A327859: 1, 2, 10, 15, 5005, ..., i.e., A369650 without any of the terms of A100716.

No more terms below <= 2550136832.

From Antti Karttunen, Feb 09 2024: (Start)

The known five terms are all members of A276156, which is equal to the claim that the intersection of A048103 and A369650 is squarefree. See the example, and also comments in A351088 and in A380527.

Even terms here must be multiples of 4, see comment in A327860.

No terms of A047257 may occur in this sequence, which is equal to the claim that A276086(a(n)) is never a multiple of 9. See comment in A327859.

(End)

Examples

			Computing A327860(2556) is easy, because it is a member of A276156, as 2556 = 6 + 30 + 210 + 2310. Therefore A327860(2556) = A003415(5*7*11*13) = (5*7*11) + (5*7*13) + (5*11*13) + (7*11*13) = 2556, and 2556 is included in this sequence. - _Antti Karttunen_, Feb 04 2024
		

Crossrefs

Cf. A002110, A002620, A003415, A047257, A048103, A100716, A276085, A276086, A327859, A327860, A369650.

After 0, the intersection of A351087 and A380527, thus like the latter, also this is conjectured to be a subsequence of A276156.

After two initial terms (0 & 1), a subsequence of A328118. Subsequence of A351088.

Programs

PARI

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
isA328110(n) = (A327860(n) == n);