A336548 Numbers k such that at least one pair sigma(p_i^e_i), sigma(p_j^e_j) [with i != j] share a prime factor, when k = p_1^e_1 * ... * p_h^e_h, where each p_i^e_i is the maximal power of prime p_i dividing k.

10, 15, 21, 22, 30, 33, 34, 35, 39, 40, 42, 46, 51, 52, 55, 57, 58, 60, 65, 66, 69, 70, 77, 78, 82, 84, 85, 87, 88, 90, 91, 93, 94, 95, 98, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 123, 129, 130, 132, 133, 135, 136, 138, 140, 141, 142, 143, 145, 152, 154, 155, 156, 159, 160, 161, 164, 165, 166, 168, 170

Offset: 1

Author

Antti Karttunen, Jul 25 2020

Keywords

• nonn

Comments

Numbers k for which A353802(k) = Product_{p^e||k} A051027(p^e) > A051027(k), i.e. numbers at which points A051027 is not multiplicative. The notation p^e||k means that p^e divides k, but p^(1+e) does not.

If x is present, then also multiples y*x are present for all y for which gcd(x,y) = 1.

Also numbers at which points A062401 and A353750 are not multiplicative. - Antti Karttunen, May 09 2022

Examples

			10 = 2*5 is present as sigma(2) = 3 and sigma(5) = 6, and 3 and 6 share a prime factor (gcd(3,6) = 3). Also we see that sigma(sigma(2))*sigma(sigma(5)) = 4*12 = 48 > sigma(sigma(10)) = 39.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..25000

Crossrefs

Cf. A051027, A062401, A336546, A336547 (complement), A336549, A353750, A353802.

Cf. A336357, A336558, A336560, A336561, A353807 (subsequences).

Positions of nonzero terms in A336562, in A353753 and in A353803.

Positions of terms larger than 1 in A353755, in A353784 and in A353806.

Subsequence of A024619.

Programs

PARI

isA336548(n) = !A336546(n);

Formula

{k | A336562(k) > 0}. - Antti Karttunen, May 09 2022

Extensions

The old definition moved to comments and replaced with a more generic, but equivalent definition by Antti Karttunen, May 09 2022