cp's OEIS Frontend

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

%I A369650 #40 Jun 15 2024 09:22:58
%S A369650 1,2,10,15,28,5005
%N A369650 Numbers k such that A003415(k) = A276085(k), where A003415 is the arithmetic derivative, and A276085 is the primorial base log-function.
%C A369650 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.
%C A369650 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.
%C A369650 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.
%C A369650 If it exists, a(7) > 2^19.
%H A369650 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%e A369650 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.
%o A369650 (PARI)
%o A369650 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A369650 A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1, primepi(f[k, 1]-1), prime(i))); };
%o A369650 isA369650(n) = (A003415(n) == A276085(n));
%Y A369650 Cf. A003415, A046337, A048103, A276085, A276086, A327859, A328110, A369002, A369668, A373144.
%Y A369650 Positions of 0's in A373146.
%Y A369650 Intersection of A373487 and A373490.
%Y A369650 Cf. also A351228.
%K A369650 nonn,hard,more
%O A369650 1,2
%A A369650 _Antti Karttunen_, Feb 05 2024