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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.

%I A328110 #43 Mar 24 2025 09:26:03
%S A328110 0,1,7,8,2556
%N 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.
%C A328110 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.
%C A328110 No more terms below <= 2550136832.
%C A328110 From _Antti Karttunen_, Feb 09 2024: (Start)
%C A328110 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.
%C A328110 Even terms here must be multiples of 4, see comment in A327860.
%C A328110 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.
%C A328110 (End)
%e A328110 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
%o A328110 (PARI)
%o A328110 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); };
%o A328110 isA328110(n) = (A327860(n) == n);
%Y A328110 Cf. A002110, A002620, A003415, A047257, A048103, A100716, A276085, A276086, A327859, A327860, A369650.
%Y A328110 After 0, the intersection of A351087 and A380527, thus like the latter, also this is conjectured to be a subsequence of A276156.
%Y A328110 After two initial terms (0 & 1), a subsequence of A328118. Subsequence of A351088.
%K A328110 nonn,hard,more
%O A328110 1,3
%A A328110 _Antti Karttunen_, Oct 08 2019