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%I #28 Nov 28 2023 10:58:25
%S 1,6,10,14,15,21,22,26,33,34,35,36,38,39,46,51,55,57,58,62,65,69,74,
%T 77,82,85,86,87,91,93,94,95,100,106,111,115,118,119,122,123,129,133,
%U 134,141,142,143,145,146,155,158,159,161,166,177,178,183,185,187,194
%N Numbers k such that each integer that appears in the prime signature of k appears an even number of times.
%C Values of n for which all numbers in row A238747(n) are even. Also, numbers n such that A000005(n^m) is a perfect square for all nonnegative integers m; numbers n such that A181819(n) is a perfect square; numbers n such that A182860(n) is odd.
%C The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 3, 33, 314, 3119, 31436, 315888, 3162042, 31626518, 316284320, 3162915907, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3162... . - _Amiram Eldar_, Nov 28 2023
%H Antti Karttunen, <a href="/A238748/b238748.txt">Table of n, a(n) for n = 1..10000</a>
%e The prime signature of 36 = 2^2 * 3^2 is {2,2}. One distinct integer (namely, 2) appears in the prime signature, and it appears an even number of times (2 times). Hence, 36 appears in the sequence.
%e The prime factorization of 1260 = 2^2 * 3^2 * 5^1 * 7^1. Exponent 2 occurs twice (an even number of times), as well as exponent 1, thus 1260 is included. It is also the first term k > 1 in this sequence for which A182850(k) = 4, not 3. - _Antti Karttunen_, Feb 06 2016
%t q[n_] := n == 1 || AllTrue[Tally[FactorInteger[n][[;; , 2]]][[;; , 2]], EvenQ]; Select[Range[200], q] (* _Amiram Eldar_, Nov 28 2023 *)
%o (Scheme)
%o (define A238748 (MATCHING-POS 1 1 (lambda (n) (square? (A181819 n)))))
%o (define (square? n) (not (zero? (A010052 n))))
%o ;; Requires also MATCHING-POS macro from my IntSeq-library - _Antti Karttunen_, Feb 06 2016
%o (PARI) is(n) = {my(e = factor(n)[, 2], m = #e); if(m%2, return(0)); e = vecsort(e); forstep(i = 1, m, 2, if(e[i] != e[i+1], return(0))); 1;} \\ _Amiram Eldar_, Nov 28 2023
%Y Subsequences include A006881, A030229, A046386, A077448, A162142, A179690, A190108, A190465.
%Y Subsequence of A000379, A007422, A063774 and A268390.
%Y Cf. A000005, A181819, A182850, A182860, A238747.
%K nonn
%O 1,2
%A _Matthew Vandermast_, May 08 2014
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