A258401 Primitive weird numbers (A002975) of the form 2^k*p*q*x with k >= 0 and odd p, q, x >= 3.

4030, 5830, 45356, 91388, 243892, 254012, 338572, 343876, 388076, 1713592, 4199030, 8812312, 9928792, 11339816, 11547352, 15126992, 17999992, 29465852, 29581424, 38546576, 74899952, 85389368, 89283592, 95327216, 120888092, 141659096, 146764264, 162079768, 173482552

Offset: 1

Author

Robert G. Wilson v, May 28 2015

Keywords

• nonn

Comments

The complement of A258882 in A002975, i.e., primitive weird numbers not of the form 2^k*p*q with primes p, q. Equivalently, subsequence of A002975 for numbers with at least 3 odd prime factors, counting multiplicity. (No weird number is of the form 2^k*p^m.) Note that, e.g., a(40) = 2^6 * 137^2 * 1931 and a(143) = 2^8 * 797^2 * 1429 have only 3 distinct prime factors.

Primitive weird numbers of the excluded set (of the form 2^k*p*q, cf. A258882) are well studied and comparably easier to produce, see the Douglas E. Iannucci link; therefore this sequence is noteworthy and harder to produce.

More rare are the primitive weird numbers in which there is an odd prime squared factor, for example:

a(40) = A002975(156) = 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211,

a(45) = A002975(179) = 2319548096 = 2^6 * 137^2 * 1931,

a(117) = A002975(483) = 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223,

a(123) = A002975(508) = 114141404156 = 2^2 * 13^2 * 19 * 383 * 23203,

a(143) = A002975(725) = 232374697216 = 2^8 * 797^2 * 1429.

These PWN with an odd square factor are now listed as A273815. - M. F. Hasler, Jul 10 2016

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..186 (52 terms were from M. F. Hasler)
• Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.

Crossrefs

Cf. A002975, A258401, A258374, A258375, A258882, A258883, A258884, A258885.

Programs

Mathematica

(* copy the terms from A002975, assign them equal to 'pwn' and then *) fQ[n_] :=
Block[{m = n}, While[ Mod[m, 2] == 0, m /= 2]; Total[Last@# & /@ FactorInteger@ m] > 2]; Select[pwn, fQ] (* Robert G. Wilson v, May 28 2015 and modified Mar 30 2017 *)

PARI

select(t->factor(t)[,2][^1]<>[1,1]~, A002975) \\ Assuming that A002975 is defined as set or vector. - M. F. Hasler, Jul 11 2016

Extensions

Edited and definition corrected by M. F. Hasler, Jul 10 2016