cp's OEIS Frontend

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

%I A258401 #66 Aug 11 2018 22:01:51
%S A258401 4030,5830,45356,91388,243892,254012,338572,343876,388076,1713592,
%T A258401 4199030,8812312,9928792,11339816,11547352,15126992,17999992,29465852,
%U A258401 29581424,38546576,74899952,85389368,89283592,95327216,120888092,141659096,146764264,162079768,173482552
%N A258401 Primitive weird numbers (A002975) of the form 2^k*p*q*x with k >= 0 and odd p, q, x >= 3.
%C A258401 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.
%C A258401 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.
%C A258401 More rare are the primitive weird numbers in which there is an odd prime squared factor, for example:
%C A258401 a(40) = A002975(156) = 1550860550 = 2 * 5^2 * 29 * 37 * 137 * 211,
%C A258401 a(45) = A002975(179) = 2319548096 = 2^6 * 137^2 * 1931,
%C A258401 a(117) = A002975(483) = 66072609790 = 2 * 5 * 11 * 127^2 * 167 * 223,
%C A258401 a(123) = A002975(508) = 114141404156 = 2^2 * 13^2 * 19 * 383 * 23203,
%C A258401 a(143) = A002975(725) = 232374697216 = 2^8 * 797^2 * 1429.
%C A258401 These PWN with an odd square factor are now listed as A273815. - _M. F. Hasler_, Jul 10 2016
%H A258401 Robert G. Wilson v, <a href="/A258401/b258401.txt">Table of n, a(n) for n = 1..186</a> (52 terms were from M. F. Hasler)
%H A258401 Douglas E. Iannucci, <a href="http://arxiv.org/abs/1504.02761"> On primitive weird numbers of the form 2^k*p*q</a>, arXiv:1504.02761 [math.NT], 2015.
%t A258401 (* copy the terms from A002975, assign them equal to 'pwn' and then *) fQ[n_] :=
%t A258401 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 *)
%o A258401 (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
%Y A258401 Cf. A002975, A258401, A258374, A258375, A258882, A258883, A258884, A258885.
%K A258401 nonn
%O A258401 1,1
%A A258401 _Robert G. Wilson v_, May 28 2015
%E A258401 Edited and definition corrected by _M. F. Hasler_, Jul 10 2016