A063774 -id:A063774 - cp's OEIS frontend

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

Matthew Vandermast, May 08 2014

nonn

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

			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

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

Subsequences include A006881, A030229, A046386, A077448, A162142, A179690, A190108, A190465, A367590.
Subsequence of A000379, A028260, A063774 and A268390.
Cf. A000005, A181819, A182850, A182860, A238747.

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

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

(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

A175391 Perfect squares having a square number of divisors.

Original entry on oeis.org

1, 36, 100, 196, 225, 256, 441, 484, 676, 1089, 1156, 1225, 1296, 1444, 1521, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929, 6561, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 10000, 11236, 12321, 13225, 13924, 14161, 14884, 15129

Offset: 1

Leroy Quet, Apr 27 2010

nonn

From Robert Israel, Mar 20 2018: (Start)
If m and n are coprime members of the sequence, then m*n is in the sequence.
Includes all numbers of the forms p^(4*i*(i+1)) and p^(2*i)*q^(2*i) where p, q are distinct primes and i is a positive integer. (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A063774, A175050. - Leroy Quet, May 16 2010

with(numtheory): a := proc (n) if type(sqrt(tau(n^2)), integer) = true then n^2 else end if end proc: seq(a(n), n = 1 .. 130); # Emeric Deutsch, May 11 2010

Select[Range[150], IntegerQ[Sqrt[DivisorSigma[0, #^2]]]&]^2 (* Vincenzo Librandi, Mar 21 2018 *)
Select[Range[150]^2,IntegerQ[Sqrt[DivisorSigma[0,#]]]&] (* Harvey P. Dale, Aug 16 2025 *)

isok(n) = issquare(n) && issquare(numdiv(n)); \\ Michel Marcus, Mar 21 2018

a(n) = A063774(n)^2. - Leroy Quet, May 16 2010

Extended by Emeric Deutsch and Jon E. Schoenfield, May 11 2010

cp's OEIS Frontend

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

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