cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A046337 Odd numbers with an even number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, 9, 15, 21, 25, 33, 35, 39, 49, 51, 55, 57, 65, 69, 77, 81, 85, 87, 91, 93, 95, 111, 115, 119, 121, 123, 129, 133, 135, 141, 143, 145, 155, 159, 161, 169, 177, 183, 185, 187, 189, 201, 203, 205, 209, 213, 215, 217, 219, 221, 225, 235, 237, 247, 249, 253, 259
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Links

Crossrefs

Intersection of A005408 and A028260.
Setwise difference A005408 \ A067019.
Setwise difference A028260 \ A063745.
Union of A359161 and A359163.
Union of A327862 and A360110.
Subsequence of A345452, of A356312 and of A359371.
Positions of positive terms in A166698, positions of even terms in A327858 and A356299.
Subsequences: A002557, A046315 (odd semiprimes), A056913, A359596, A359607, A359608 (without its term 2).
Cf. A000035, A008836, A046338, A046470, A353557 (characteristic function), A358777.
Cf. also A036349, A297845.

Programs

  • Mathematica
    Select[Range[1,301,2],EvenQ[PrimeOmega[#]]&] (* Harvey P. Dale, Jul 25 2011 *)
  • PARI
    lista(nn) = {forstep(n=1, nn, 2, if (bigomega(n) % 2 == 0, print1(n, ", ")));} \\ Michel Marcus, Jul 04 2015

Formula

{k | A000035(k) > 0 and A008836(k) > 0}. - Antti Karttunen, Jan 13 2023

A002556 Odd squarefree numbers with an odd number of prime factors that have no prime factors greater than 31.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 105, 165, 195, 231, 255, 273, 285, 345, 357, 385, 399, 429, 435, 455, 465, 483, 561, 595, 609, 627, 651, 663, 665, 715, 741, 759, 805, 897, 935, 957, 969, 1001, 1015, 1023, 1045, 1085, 1105, 1131, 1173, 1209, 1235, 1265
Offset: 1

Views

Author

N. J. A. Sloane, Oct 07 2015

Keywords

Comments

Original name: A subset of A056912, definition unclear.
The definition is given on page 70 of Gupta (1943), but is hard to understand.
A variant of A056912, 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

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

Crossrefs

Cf. A002557, A067019. Subset of A056912.

Programs

  • Magma
    a:= func< n | Factorization(n)>; [n: n in [3..1265 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)::odd), combinat:-powerset(select(isprime, [seq(i,i=3..31,2)]))):
sort(map(convert,S,`*`)); # Robert Israel, Jan 21 2016
  • Mathematica
    osfnQ[n_]:=SquareFreeQ[n]&&OddQ[PrimeOmega[n]]&&Max[FactorInteger[n][[All, 1]]]<32; Select[Range[1,1301,2],osfnQ] (* Harvey P. Dale, Jul 19 2019 *)
  • PARI
    isok(n) = (n % 2) && issquarefree(n) && (omega(n) % 2) && (vecmax(factor(n)[,1]) <= 31); \\ Michel Marcus, Jan 21 2016

Extensions

Name changed and sequence extended by Arkadiusz Wesolowski, Jan 21 2016
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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