cp's OEIS Frontend

A328233 Numbers n such that the arithmetic derivative of A276086(n) is prime.

%I A328233 #27 Nov 17 2019 16:01:22
%S A328233 3,7,9,33,37,38,211,213,218,241,242,246,247,249,2313,2317,2319,2341,
%T A328233 2342,2346,2521,2523,2526,2529,2550,2553,2559,30031,30038,30039,30061,
%U A328233 30062,30063,30066,30069,30242,30243,30249,30270,30278,30279,32341,32342,32347,32370,32373,32377,32379,32551,32553,510513,510518,510519
%N A328233 Numbers n such that the arithmetic derivative of A276086(n) is prime.
%C A328233 Numbers n for which A327860(n) = A003415(A276086(n)) is a prime.
%C A328233 Numbers n such that A276086(n) is in A157037.
%C A328233 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 A328233 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 A328233 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 A328233 Antti Karttunen, <a href="/A328233/b328233.txt">Table of n, a(n) for n = 1..269</a>
%H A328233 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%o A328233 (PARI)
%o A328233 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 A328233 isA328233(n) = isprime(A327860(n));
%Y A328233 Cf. A002110, A003415, A051674, A157037, A276086, A327860, A327969, A327978, A328232, A328240.
%Y A328233 Subsequence of A276156, of A328116, and of A328242.
%K A328233 nonn
%O A328233 1,1
%A A328233 _Antti Karttunen_, Oct 09 2019