A377869 Numbers k such that A276085(k) has no divisors of the form p^p, where A276085 is fully additive with a(p) = p#/p.

2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 18, 19, 22, 23, 24, 26, 27, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 42, 43, 45, 46, 47, 48, 50, 53, 54, 56, 58, 59, 60, 61, 62, 63, 64, 66, 67, 70, 71, 72, 73, 74, 75, 78, 79, 80, 82, 83, 84, 86, 88, 89, 90, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 109

Offset: 1

Author

Antti Karttunen, Nov 10 2024

Keywords

• nonn

Comments

Numbers k for which A276085(k) is in A048103, i.e., in the range of A276086.

Numbers k such that A276086(A276085(A276085(k))) is equal to A276085(k).

This is a subsequence of A369003 (numbers k for which A276085(k) is not a multiple of 4), from which it differs for the first time at n=122, where a(122) = 175, as A369003(122) = 174 is not included in this sequence.

From Antti Karttunen, Nov 17 2024: (Start)

More generally, this is equal to setwise difference A000027 \ (A369002 U A377872 U A377878 U ...).

Even semiprimes (A100484) is a subsequence, but the odd semiprimes (A046315) are all in the complement (A377873), because they are included in A369002.

For k=1..6, there are 8, 70, 656, 6531, 64773, 645301 terms <= 10^k. Question: What is the asymptotic density of this sequence, if it has one?

(End)

Examples

			A276085(11) = 210 = 2*3*5*7, which has no divisor of the form p^p, therefore 11 is included in this sequence.
A276085(15) = 8 = 2^2 * 2, which has a divisor of the form p^p, therefore 15 is NOT included in this sequence.
A276085(25) = 12 = 2^2 * 3, which has a divisor of the form p^p, therefore 25 is NOT included.
A276085(34) = 30031 = A002110(1-1)+A002110(7-1) (as 34 = 2*17 = prime(1)*prime(7)), and because 30031 = 59*509 (an odd semiprime), 34 is included.
A276085(60) = 10 = 2*5, which has no divisors of the form p^p, therefore 60 is included.
A276085(102) = 30033 = 3^2 * 47 * 71, which has no p^p divisors, therefore 102 is included.
A276085(174) = 223092873 = 3^3 * 3 * 1063 * 2591, which thus has a divisor of the form p^p, and therefore 174 is NOT included in this sequence.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to primorial base

Crossrefs

Cf. A046315, A048103, A276085, A276086, A377868 (characteristic function), A377873 (complement).

Setwise difference A369003 \ A377875.

Cf. A000040, A100484, A377871, A377989 (subsequences).

Cf. A369002, A377872, A377878.

Programs

PARI

\\ See A377868.