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.

Showing 1-4 of 4 results.

A091371 Smallest prime factor of n - number of prime factors of n with multiplicity.

Original entry on oeis.org

1, 1, 2, 0, 4, 0, 6, -1, 1, 0, 10, -1, 12, 0, 1, -2, 16, -1, 18, -1, 1, 0, 22, -2, 3, 0, 0, -1, 28, -1, 30, -3, 1, 0, 3, -2, 36, 0, 1, -2, 40, -1, 42, -1, 0, 0, 46, -3, 5, -1, 1, -1, 52, -2, 3, -2, 1, 0, 58, -2, 60, 0, 0, -4, 3, -1, 66, -1, 1, -1, 70, -3, 72, 0, 0, -1, 5, -1, 78, -3
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 04 2004

Keywords

Comments

a(n) = A020639(n) - A001222(n).
a(A091375(n)) < 0. a(A091376(n)) = 0. a(A091377(n)) > 0.

Links

Crossrefs

Cf. A020639 (lpf), A001222 (Omega).
Cf. A091372, A091373, A091374, A091375, A091376, A091377.

Programs

  • Maple
    with(numtheory); A091371:=n->`if`(n=1,1,min(op(factorset(n)))-bigomega(n)); seq(A091371(k), k=1..100); # Wesley Ivan Hurt, Oct 27 2013
  • Mathematica
    Array[FactorInteger[#][[1,1]]-PrimeOmega[#]&,80] (* Harvey P. Dale, May 25 2012 *)

Extensions

Definition clarified by Harvey P. Dale, May 25 2012

A091376 Numbers k with property that the number of prime factors of k (counted with repetition) equals the smallest prime factor of k.

Original entry on oeis.org

4, 6, 10, 14, 22, 26, 27, 34, 38, 45, 46, 58, 62, 63, 74, 75, 82, 86, 94, 99, 105, 106, 117, 118, 122, 134, 142, 146, 147, 153, 158, 165, 166, 171, 178, 194, 195, 202, 206, 207, 214, 218, 226, 231, 254, 255, 261, 262, 273, 274, 278, 279, 285, 298, 302, 314
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 04 2004

Keywords

Comments

A091371(a(n)) = 0: A001222(a(n))=A020639(a(n)).
Prime factors counted with multiplicity. - Harvey P. Dale, Nov 11 2012

Links

Crossrefs

Cf. A091373, A091375, A091377.
Cf. A002808.
Cf. A100484 (subsequence).

Programs

  • Haskell
    a091376 n = a091376_list !! (n-1)
a091376_list = [x | x <- a002808_list, a001222 x == a020639 x]
-- Reinhard Zumkeller, Nov 11 2012
  • Mathematica
    pfQ[n_]:=Module[{fi=Transpose[FactorInteger[n]]},fi[[1,1]] == Total[Last[fi]]]; Rest[Select[Range[400],pfQ]] (* Harvey P. Dale, Nov 11 2012 *)
Select[Range[400],PrimeOmega[#]==FactorInteger[#][[1,1]]&] (* Harvey P. Dale, Nov 26 2024 *)

Extensions

Definition edited by N. J. A. Sloane, Jan 21 2020

A091372 Number of numbers <= n having more prime factors than the value of their smallest prime factor.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 26, 26, 27
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 04 2004

Keywords

Comments

a(n) = #{m: A001222(m)>A020639(m), m<=n};
a(n) + A091373(n) + A091374(n) = n.

Links

Crossrefs

Cf. A091375, A091371.

Programs

  • Mathematica
    Accumulate@ Boole@ Map[Length@ Flatten[Table[#1, {#2}] & @@@ #] > #[[1, 1]] &@ FactorInteger@ # &, Range@ 80] (* Michael De Vlieger, Jul 06 2016 *)
  • PARI
    a(n)=sum(k=8,n, bigomega(k) > factor(k)[1,1]) \\ Charles R Greathouse IV, Jul 06 2016
  • PARI
    first(n)=my(v=vector(n),s); for(k=8,n, v[k] = s += bigomega(k) > factor(k)[1,1]); v \\ Charles R Greathouse IV, Jul 06 2016

Formula

For any k < 1, a(n) > kn for large enough k. For example, a(n) > n/2 for n > 26474. - Charles R Greathouse IV, Jul 06 2016

A091374 Number of numbers <= n having fewer prime factors than the value of their smallest prime factor.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 04 2004

Keywords

Comments

a(n) = #{m: A001222(m) < A020639(m), m<=n};
A091372(n) + A091373(n) + a(n) = n.

Crossrefs

Cf. A091377, A091371.

Programs

  • PARI
    isok(n) = {my(f=factor(n)); (#f~ == 0) || (bigomega(n) < f[1,1]);}
a(n) = sum(k=1, n, isok(k)); \\ Michel Marcus, Feb 05 2016
Showing 1-4 of 4 results.

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

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