A369970 - OEIS

%I #24 Feb 07 2024 20:38:06

%S 0,1,6,2315,510510

%N 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 For the general dynamics of this phenomenon, see the scatter plots of A351231 and A351233.

%C Question: Are the terms by necessity all squarefree?

%C 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 872415232 < a(6) <= 13082761331670030 [= A369973(4)].

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%e 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 510510 is included because A003415(510510) = 19*37693, which is a multiple of A276086(510510) = 19.

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o isA369970(n) = !(A003415(n)%A276086(n));

%Y Cf. A000040, A003415, A024451, A276086, A369972, A369973 (subsequence).

%Y Positions of 1's in A351231, positions of 0's in A351233 and in A369971.

%Y After the two initial terms, a subsequence of A351228.

%Y Cf. also A358221.

%K nonn,hard,more

%O 1,3

%A _Antti Karttunen_, Feb 07 2024