A002557 Odd squarefree numbers with an even number of prime factors that have no prime factors greater than 31. (Formerly M4959 N2126)

1, 15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 115, 119, 133, 143, 145, 155, 161, 187, 203, 209, 217, 221, 247, 253, 299, 319, 323, 341, 377, 391, 403, 437, 493, 527, 551, 589, 667, 713, 899, 1155, 1365, 1785, 1995, 2145, 2415, 2805, 3003

Offset

1,2

Comments

Original name: A subset of A056913, definition unclear.

The definition is given on page 70 of Gupta (1943), but is hard to understand.

A variant of A056913, which has terms that also have prime factors > 31. - Arkadiusz Wesolowski, Jan 21 2016

The b-file contains the full sequence. - Robert Israel, Jan 21 2016

The sequence is closed under the commutative binary operation A059897(.,.). As integers are self-inverse under A059897, it forms a subgroup of the positive integers considered as a group under A059897. A subgroup of A056913. - Peter Munn, Jan 16 2020

References

H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Robert Israel, Table of n, a(n) for n = 1..512

H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71. [Annotated scanned copy]

Maple

S:= select(t -> (nops(t)::even), combinat:-powerset(select(isprime, [seq(i, i=3..31, 2)]))):
sort(map(convert, S, `*`)); # Robert Israel, Jan 21 2016

Prog

(Magma) a:= func< n | Factorization(n)>; [1] cat [n: n in [3..3003 by 2] | IsSquarefree(n) and (-1)^&+[p[2]: p in a(n)] eq 1 and f[#f][1] le 31 where f is a(n)]; // Arkadiusz Wesolowski, Jan 21 2016
(Python) powerset = lambda lst: reduce(lambda result, x: result + [subset + [x] for subset in result], lst, [[]])
product = lambda lst: reduce(lambda x, y: x*y, lst, 1)
primes = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31]
sequence = sorted(product(s) for s in powerset(primes) if len(s) % 2 == 0) # David Radcliffe, Jan 21 2016

Crossrefs

Cf. A002556, A046337, A059897. Subset of A056913.

Sequence in context: A024556 A046388 A056913 * A128907 A321644 A225709

Adjacent sequences: A002554 A002555 A002556 * A002558 A002559 A002560

Keyword

nonn,full,fini

Author

N. J. A. Sloane

Extensions

Name changed and sequence extended by Arkadiusz Wesolowski, Jan 21 2016

Status

approved