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%I #38 Aug 31 2022 09:09:30
%S 6,10,14,15,21,22,26,33,34,35,38,39,46,51,55,57,58,62,65,69,74,77,82,
%T 85,86,87,91,93,94,95,106,111,115,118,119,122,123,129,133,134,141,142,
%U 143,145,146,155,158,159,161,166,177,178,180,183,185,187,192,194
%N Numbers that have an equal number of even- and odd-length unordered factorizations and also an equal number of even- and odd-length unordered factorizations into distinct factors.
%C First differs from A006881 at a(53) = 180.
%C By length, we mean the number of factors in a particular factorization.
%C Intersection of A319240 (factors are not necessarily distinct) and A319238 (factors are distinct).
%C Numbers k such that A316441(k) = A114592(k) = 0.
%C There are infinitely many terms in this sequence since all squarefree semiprimes (listed in A006881) are always such numbers.
%C There are no terms of the form p^k with p prime (listed in A000961).
%C Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881) and p*q^6 (A189987) are terms.
%C Similar methods can be applied to all prime signatures.
%H L. Euler, <a href="https://arxiv.org/abs/math/0505373">On the remarkable properties of the pentagonal numbers</a>, arXiv:math/0505373 [math.HO], 2005.
%H L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 2
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>
%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Pentagonal number theorem">Pentagonal number theorem</a>
%e 6=2*3 (unrestricted) has an equal number (1) of even-length factorizations and odd-length factorizations, and 6=2*3 (distinct) has an equal number (1) of even-length factorizations and odd-length factorizations.
%t facs[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[facs[n/d], Min @@ # >= d &]], {d, Rest[Divisors[n]]}]]; Intersection @@ First@Flatten[Position[#, 0] & /@ Transpose@Table[Sum[(-1)^Length[f], {f, #}] & /@ {facs[n], Select[facs[n], UnsameQ @@ # &]}, {n, #1, #2}], {3}]&[1,194] (* _Robert P. P. McKone_, Jan 05 2022, from _Gus Wiseman_ in A319238 and A319240 *)
%Y Cf. A006881, A114592, A189987, A316441, A319240, A319238.
%K nonn,easy
%O 1,1
%A _Tian Vlasic_, Jan 01 2022
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