A273815 Primitive weird numbers (pwn) with nonsquarefree odd part.

1550860550, 2319548096, 66072609790, 114141404156, 232374697216

Offset: 1

Author

M. F. Hasler, Jul 08 2016

Keywords

• nonn
• hard
• more

Comments

Equivalently, primitive weird numbers (A002975) with at least one odd prime factor with multiplicity > 1. A subsequence of A258401.

Although some of them have only few prime divisors, these primitive weird numbers are not in the sequences A258882, A258883, A258884 defined to list "pwn of the form 2^k p*...*r with primes p<... squarefree). Sequence A258885 (pwn with 6 prime divisors) does not have this restriction. - M. F. Hasler, Jul 26 2016

a(6) <= 2^10*2081^2*129083 = 572417848896512, which is also in the sequence. - M. F. Hasler, Feb 15 2018

Examples

			a(1) = 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211 = A258885(1): the smallest pwn with 6 (distinct) prime divisors.
a(2) = 2319548096 = 2^6 * 137^2 * 1931 = A258401(45), but not in A258882 nor A258883, cf. comment.
a(3) = 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223 = A258885(3).
a(4) = 114141404156 = 2^2 * 13^2 * 19 * 383 * 23203 = A258401(123), but not in A258884, cf. comment.
a(5) = 232374697216 = 2^8 * 797^2 * 1429 = A258401(143), but not in A258882 nor A258883, cf. comment.
		

Links

• Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton, Primitive abundant and weird numbers with many prime factors, Journal of Number Theory vol. 201 (2019) pp. 436-459. DOI: 10.1016/j.jnt.2019.02.027. (Preprint: arXiv:1802.07178.)

Crossrefs

Cf. A002975, A258401, A258882, A258883, A258884, A258885.

Programs

PARI

select(t->vecmax(factor(t)[,2][^1])>1, A002975) \\ Assuming that A002975 is defined as vector holding enough terms of that sequence