A268378 Numbers whose prime factorization includes at least one prime factor of form 4k+3 and any prime factor of the form 4k+1 has even multiplicity.

3, 6, 7, 9, 11, 12, 14, 18, 19, 21, 22, 23, 24, 27, 28, 31, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 66, 67, 69, 71, 72, 75, 76, 77, 79, 81, 83, 84, 86, 88, 92, 93, 94, 96, 98, 99, 103, 107, 108, 112, 114, 118, 121, 124, 126, 127, 129, 131, 132, 133, 134, 138, 139, 141, 142, 144, 147, 150

Offset

1,1

Comments

Closed under multiplication.

Links

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

Example

6 = 2*3 is included, as there is a prime factor of the form 4k+3 present.
75 = 3 * 5 * 5 is included, as there is a prime factor of the form 4k+3 present and the prime factor of the form 4k+1 (5) is present twice.

Mathematica

Select[Range@ 150, AnyTrue[#, Mod[First@ #, 4] == 3 &] && NoneTrue[#, And[Mod[First@ #, 4] == 1, OddQ@ Last@ #] &] &@ FactorInteger@ # &] (* Michael De Vlieger, Feb 04 2016, Version 10 *)

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(define A268378 (MATCHING-POS 1 1 (lambda (n) (and (even? (A267113 n)) (not (zero? (A065339 n)))))))
(PARI) isok(n) = {my(f = factor(n), nb3 = 0); for (i=1, #f~, if (((f[i, 1] % 4) == 1) && (f[i, 2] % 2), return (0)); if ((f[i, 1] % 4) == 3, nb3++); ); return (nb3); } \\ Michel Marcus, Feb 04 2016

Crossrefs

Cf. A065339, A267113.

Cf. A004431, A267099.

Subsequence of A268377.

Differs from A221264 for the first time at n=38, which here is a(38) = 75, a value missing from A221264.

Sequence in context: A186348 A071822 A268380 * A221264 A026415 A026406

Adjacent sequences: A268375 A268376 A268377 * A268379 A268380 A268381

Keyword

nonn

Author

Antti Karttunen, Feb 03 2016

Status

approved