cp's OEIS Frontend

A380468 Numbers k such that A380459(k) has no divisors of the form p^p, for any prime p.

%I A380468 #25 Feb 03 2025 17:02:21
%S A380468 1,2,3,5,6,7,11,13,14,17,19,23,26,29,31,37,38,41,43,47,53,59,61,62,67,
%T A380468 71,73,74,79,83,86,89,97,101,103,107,109,113,122,127,131,134,137,139,
%U A380468 146,149,151,157,158,163,167,173,179,181,186,191,193,194,197,199,206,211,218,223,227,229,233,239,241,251,254
%N A380468 Numbers k such that A380459(k) has no divisors of the form p^p, for any prime p.
%C A380468 Proof that this is a subsequence of squarefree numbers (A005117): Let's write A380459(n) = Product_{d|n} A276086(n/d)^A349394(d). Then suppose that we have a prime p such that p^e || n, with e > 1 (the maximal exponent e for which p^e divides n). We set d = p^e, and for that d, factor A276086(n/(p^e))^A349394(p^e) = A276086(n/(p^e))^(p^(e-1)) is contributed to the product A380459(n). But n/(p^e) is not divisible by p, so A020639(A276086(n/(p^e))) <= p as either p or some lesser prime q < p is the first prime missing from the factorization of n/(p^e) [see comments in A276086], so in the former case there will be a factor p^(p^(e-1)) [thus also p^p], and in the latter case a factor q^(p^(e-1)) [thus also q^q as p^(e-1) > q] present in the product, so in any such case the product will not be in A048103, and therefore each term must be squarefree.
%C A380468 The least terms k for which A001222(k) = 0, 1, 2, ..., are given in A380475.
%H A380468 Antti Karttunen, <a href="/A380468/b380468.txt">Table of n, a(n) for n = 1..20000</a>
%H A380468 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.
%o A380468 (PARI) is_A380468 = A380467;
%Y A380468 Cf. A048103, A276086, A359550, A380459, A380467 (characteristic function), A380475.
%Y A380468 Setwise difference A005117 \ A380470.
%Y A380468 Subsequences: A380474, A380478 (nonprime terms).
%K A380468 nonn
%O A380468 1,2
%A A380468 _Antti Karttunen_, Feb 01 2025