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%I #27 Nov 17 2019 16:01:22
%S 3,7,9,33,37,38,211,213,218,241,242,246,247,249,2313,2317,2319,2341,
%T 2342,2346,2521,2523,2526,2529,2550,2553,2559,30031,30038,30039,30061,
%U 30062,30063,30066,30069,30242,30243,30249,30270,30278,30279,32341,32342,32347,32370,32373,32377,32379,32551,32553,510513,510518,510519
%N Numbers n such that the arithmetic derivative of A276086(n) is prime.
%C Numbers n for which A327860(n) = A003415(A276086(n)) is a prime.
%C Numbers n such that A276086(n) is in A157037.
%C Terms come in distinct "batches", where in each batch they are "slightly more" than the nearest primorial (A002110) below. This is explained by the fact that for A276086(n) to be a squarefree (which is the necessary condition for A157037), n's primorial base expansion (A049345) must not contain digits larger than 1. Thus this is a subsequence of A276156.
%C Numbers n such that A327860(A276086(n)) = A003415(A276087(n)) is a prime [A276087(n) is in A157037] are much rarer: 2, 4, 30, 212, 421, 30045, 510511, 512820, 9729723, ...
%C For all terms k in this sequence, A327969(k) <= 4, and particularly A327969(k) = 2 when k is a prime. Otherwise, when k is not a prime, but A003415(k) is, A327969(k) = 3, while for other cases (when k is neither prime nor in A157037), we have A327969(k) = 4.
%H Antti Karttunen, <a href="/A328233/b328233.txt">Table of n, a(n) for n = 1..269</a>
%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%o (PARI)
%o A327860(n) = { my(m=1, i=0, s=0, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), my(e=((n%nextpr)/pr)); m *= (prime(i)^e); s += (e / prime(i)); n-=(n%nextpr)); pr=nextpr); (s*m); };
%o isA328233(n) = isprime(A327860(n));
%Y Cf. A002110, A003415, A051674, A157037, A276086, A327860, A327969, A327978, A328232, A328240.
%Y Subsequence of A276156, of A328116, and of A328242.
%K nonn
%O 1,1
%A _Antti Karttunen_, Oct 09 2019
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