

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



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
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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 IntSeqlibrary)
(define A268378 (MATCHINGPOS 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



