A225857 Numbers of the form 2^i*3^j*(12k+1) or 2^i*3^j*(12k+5), i, j, k >= 0.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17, 18, 20, 24, 25, 26, 27, 29, 30, 32, 34, 36, 37, 39, 40, 41, 45, 48, 49, 50, 51, 52, 53, 54, 58, 60, 61, 64, 65, 68, 72, 73, 74, 75, 77, 78, 80, 81, 82, 85, 87, 89, 90, 96, 97, 98, 100, 101, 102, 104, 106, 108, 109, 111

Offset

1,2

Comments

From Peter Munn, Nov 11 2023: (Start)

Numbers k whose 5-rough part, A065330(k), is congruent to 1 modulo 4.

Contains all nonzero squares.

Positive integers in the multiplicative subgroup of rationals generated by 2, 3, 5 and integers congruent to 1 modulo 12. Thus, the sequence is closed under multiplication and, provided the result is an integer, under division.

This subgroup has index 2 and does not include -1, so is the complement of its negation. In respect of the sequence, the index 2 property implies we can take any absent positive integer m, and divide by m all terms that are multiples of m to get the complementary sequence, A225858.

Likewise, the sequence forms a subgroup of index 2 of the positive integers under the operation A059897(.,.).

(End)

The asymptotic density of this sequence is 1/2. - Amiram Eldar, Nov 14 2023

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Mathematica

Select[Range[120], Mod[#/Times @@ ({2, 3}^IntegerExponent[#, {2, 3}]), 4] == 1 &] (* Amiram Eldar, Nov 14 2023 *)

Prog

(PARI) for(n=1, 200, t=n/(2^valuation(n, 2)*3^valuation(n, 3)); if((t%4==1), print1(n, ", ")))
(Magma) [n: n in [1..200] | d mod 4 eq 1 where d is n div (2^Valuation(n, 2)*3^Valuation(n, 3))]; // Bruno Berselli, May 16 2013

Crossrefs

Complement of A225858.

Cf. A059897, A065330, A225837.

Sequence in context: A176845 A196127 A194467 * A168134 A245030 A245027

Adjacent sequences: A225854 A225855 A225856 * A225858 A225859 A225860

Keyword

nonn,easy

Author

Ralf Stephan, May 18 2013

Extensions

Name clarified by Peter Munn, Nov 10 2023

Status

approved