A002557 Odd squarefree numbers with an even number of prime factors that have no prime factors greater than 31.

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

Author

N. J. A. Sloane

Keywords

• nonn
• full
• fini

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]

Crossrefs

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

Programs

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

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

Mathematica

npfQ[n_]:=With[{pf=FactorInteger[n][[;;,1]]},SquareFreeQ[n]&&EvenQ[PrimeOmega[n]]&&Max[pf]<32]; Select[Range[1,3003,2],npfQ] (* Harvey P. Dale, May 03 2025 *)

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

Extensions

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