cp's OEIS Frontend

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

%I A002557 M4959 N2126 #43 May 03 2025 15:28:28
%S A002557 1,15,21,33,35,39,51,55,57,65,69,77,85,87,91,93,95,115,119,133,143,
%T A002557 145,155,161,187,203,209,217,221,247,253,299,319,323,341,377,391,403,
%U A002557 437,493,527,551,589,667,713,899,1155,1365,1785,1995,2145,2415,2805,3003
%N A002557 Odd squarefree numbers with an even number of prime factors that have no prime factors greater than 31.
%C A002557 Original name: A subset of A056913, definition unclear.
%C A002557 The definition is given on page 70 of Gupta (1943), but is hard to understand.
%C A002557 A variant of A056913, which has terms that also have prime factors > 31. - _Arkadiusz Wesolowski_, Jan 21 2016
%C A002557 The b-file contains the full sequence. - _Robert Israel_, Jan 21 2016
%C A002557 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
%D A002557 H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71.
%D A002557 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002557 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002557 Robert Israel, <a href="/A002557/b002557.txt">Table of n, a(n) for n = 1..512</a>
%H A002557 H. Gupta, <a href="/A002556/a002556.pdf"> A formula for L(n)</a>, J. Indian Math. Soc., 7 (1943), 68-71. [Annotated scanned copy]
%p A002557 S:= select(t -> (nops(t)::even), combinat:-powerset(select(isprime, [seq(i,i=3..31,2)]))):
%p A002557 sort(map(convert,S,`*`)); # _Robert Israel_, Jan 21 2016
%t A002557 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 *)
%o A002557 (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
%o A002557 (Python) powerset = lambda lst: reduce(lambda result, x: result + [subset + [x] for subset in result], lst, [[]])
%o A002557 product = lambda lst: reduce(lambda x, y: x*y, lst, 1)
%o A002557 primes = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31]
%o A002557 sequence = sorted(product(s) for s in powerset(primes) if len(s) % 2 == 0) # _David Radcliffe_, Jan 21 2016
%Y A002557 Cf. A002556, A046337, A059897. Subset of A056913.
%K A002557 nonn,full,fini
%O A002557 1,2
%A A002557 _N. J. A. Sloane_
%E A002557 Name changed and sequence extended by _Arkadiusz Wesolowski_, Jan 21 2016