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A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

%I A369970 #24 Feb 07 2024 20:38:06
%S A369970 0,1,6,2315,510510
%N A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.
%C A369970 For the general dynamics of this phenomenon, see the scatter plots of A351231 and A351233.
%C A369970 Question: Are the terms by necessity all squarefree?
%C A369970 As a subsequence this sequence includes all primorials with indices k such that A024451(k) is a multiple of A000040(1+k). See A369972 and A369973.
%C A369970 872415232 < a(6) <= 13082761331670030 [= A369973(4)].
%H A369970 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%e A369970 2315 is included as A003415(2315) = 5+463 = 468 = 2^2 * 3^2 * 13 (note that 2315 is a semiprime = 5*463, thus its arithmetic derivative is the sum of its two prime factors), and because that 468 is a multiple of A276086(2315) = 234 = 2 * 3^2 * 13 [the exponents of primes are here read from the primorial base expansion of 2315, A049345(2315) = 100021].
%e A369970 510510 is included because A003415(510510) = 19*37693, which is a multiple of A276086(510510) = 19.
%o A369970 (PARI)
%o A369970 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A369970 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A369970 isA369970(n) = !(A003415(n)%A276086(n));
%Y A369970 Cf. A000040, A003415, A024451, A276086, A369972, A369973 (subsequence).
%Y A369970 Positions of 1's in A351231, positions of 0's in A351233 and in A369971.
%Y A369970 After the two initial terms, a subsequence of A351228.
%Y A369970 Cf. also A358221.
%K A369970 nonn,hard,more
%O A369970 1,3
%A A369970 _Antti Karttunen_, Feb 07 2024