A238748 Numbers k such that each integer that appears in the prime signature of k appears an even number of times.

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194

Offset

1,2

Comments

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.

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Example

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.
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

Mathematica

q[n_] := n == 1 || AllTrue[Tally[FactorInteger[n][[;; , 2]]][[;; , 2]], EvenQ]; Select[Range[200], q] (* Amiram Eldar, Nov 28 2023 *)

Prog

(Scheme)
(define A238748 (MATCHING-POS 1 1 (lambda (n) (square? (A181819 n)))))
(define (square? n) (not (zero? (A010052 n))))
;; Requires also MATCHING-POS macro from my IntSeq-library - Antti Karttunen, Feb 06 2016
(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

Crossrefs

Subsequences include A006881, A030229, A046386, A077448, A162142, A179690, A190108, A190465, A367590.

Subsequence of A000379, A028260, A063774 and A268390.

Cf. A000005, A181819, A182850, A182860, A238747.

Sequence in context: A371781 A362617 A374472 * A367590 A268390 A265693

Adjacent sequences: A238745 A238746 A238747 * A238749 A238750 A238751

Keyword

nonn

Author

Matthew Vandermast, May 08 2014

Status

approved