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

A091375 Numbers k with property that the number of prime factors of k (counted with repetition) exceeds the smallest prime factor of k.

Original entry on oeis.org

8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 100, 102, 104, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 144, 148, 150, 152, 154, 156, 160, 162
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 04 2004

Keywords

Comments

A091371(a(n)) < 0: A001222(a(n))>A020639(a(n)).
Numbers of the form m*i + n*j = k*(i+j), where i and j are > 1. - Giovanni Teofilatto, Aug 29 2007

Links

Crossrefs

Cf. A091372, A091376, A091377.

Programs

  • Mathematica
    Select[Range@ 162, Length@ Flatten[Table[#1, {#2}] & @@@ #] > #[[1, 1]] &@ FactorInteger@ # &] (* Michael De Vlieger, Jul 06 2016 *)
  • PARI
    is(n)=n>7 && bigomega(n) > factor(n)[1,1] \\ Charles R Greathouse IV, Jul 06 2016
  • PARI
    bigomegaAtLeast(n,k,startAt=2)=if(k<3, return(if(k==2,n>1&&!isprime(n),k==0||n>1))); forprime(p=startAt,logint(n,k), if(n%p==0, k-=valuation(n,p);n/=p^valuation(n,p); return(bigomegaAtLeast(n,k)))); 0
is(n)=if(n%2==0,return(bigomegaAtLeast(n/2,2))); if(n%3==0,return(bigomegaAtLeast(n/3,3,3))); if(n<9,return(0)); forprime(p=5,log(n)/lambertw(log(n)), if(n%p==0, return(bigomegaAtLeast(n/p,p,p)))); 0 \\ Charles R Greathouse IV, Jul 06 2016

Extensions

Missing a(10001)-a(10424) inserted into b-file by Andrew Howroyd, Feb 25 2018
Definition clarified by N. J. A. Sloane, Jan 20 2020

A091373 Number of numbers <= n having exactly as many prime factors as the value of their smallest prime factor.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 04 2004

Keywords

Comments

a(n) = #{m: A001222(m)=A020639(m), m<=n};
A091372(n) + a(n) + A091374(n) = n.

Crossrefs

Cf. A091376, A091371.

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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