OFFSET
1,1
COMMENTS
Numbers of the form 4^i * 9^j * (6k+5), i, j, k >= 0.
1/5 of each multiple of 5 in A352272.
The product of any two terms is in A352272.
The product of a term of this sequence and a term of A352272 is a term of this sequence.
The positive integers are usefully partitioned as {A352272, 2*A352272, 3*A352272, 6*A352272, {a(n)}, 2*{a(n)}, 3*{a(n)}, 6*{a(n)}}. There is a table in the example section giving sequences formed from unions of the parts.
The parts correspond to the cosets of A352272 considered as a subgroup of the positive integers under the operation A059897(.,.). Viewed another way, the parts correspond to the intersection of the integers with the cosets of the multiplicative subgroup of the positive rationals generated by the terms of A352272.
The asymptotic density of this sequence is 1/4. - Amiram Eldar, Apr 03 2022
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree Part.
FORMULA
EXAMPLE
The squarefree part of 11 is 11, which is congruent to 5 (mod 6), so 11 is in the sequence.
The squarefree part of 15 is 15, which is congruent to 3 (mod 6), so 15 is not in the sequence.
The squarefree part of 20 = 2^2 * 5 is 5, which is congruent to 5 (mod 6), so 20 is in the sequence.
The table below lists OEIS sequences that are unions of the cosets described in the initial comments, and indicates the cosets included in each sequence. A352272 (as a subgroup) is denoted H, and this sequence (as a coset) is denoted H/5, in view of its terms being one fifth of the multiples of 5 in A352272.
H 2H 3H 6H H/5 2H/5 3H/5 6H/5
A003159 X X X X
A036554 X X X X
.
A007417 X X X X
A145204\{0} X X X X
.
A026225 X X X X
A026179\{1} X X X X
.
A036668 X X X X
A325424 X X X X
.
A055047 X X
A055048 X X
A055041 X X
A055040 X X
.
A189715 X X X X
A189716 X X X X
.
A225837 X X X X
A225838 X X X X
.
A339690 X X
A329575 X X
.
A352274 X X
(The sequence groupings in the table start with the subgroup of the quotient group of H, followed by its cosets.)
MATHEMATICA
q[n_] := Module[{e2, e3}, {e2, e3} = IntegerExponent[n, {2, 3}]; EvenQ[e2] && EvenQ[e3] && Mod[n/2^e2/3^e3, 6] == 5]; Select[Range[250], q] (* Amiram Eldar, Apr 03 2022 *)
PROG
(PARI) isok(m) = core(m) % 6 == 5;
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Munn, Mar 10 2022
STATUS
approved