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

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, 71, 73, 74, 79, 83, 86, 89, 97, 101, 103, 107, 109, 113, 122, 127, 131, 134, 137, 139, 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

Offset: 1

Author

Antti Karttunen, Feb 01 2025

Keywords

• nonn

Comments

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.

The least terms k for which A001222(k) = 0, 1, 2, ..., are given in A380475.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to primorial base.

Crossrefs

Cf. A048103, A276086, A359550, A380459, A380467 (characteristic function), A380475.

Setwise difference A005117 \ A380470.

Subsequences: A380474, A380478 (nonprime terms).

Programs

PARI

is_A380468 = A380467;