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

A091377 Numbers having fewer prime factors than the value of their smallest prime factor.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 143
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 04 2004

Keywords

Comments

A091371(a(n)) > 0: A001222(a(n)) < A020639(a(n)).

Links

Crossrefs

Cf. A091374, A091376, A091375.

Programs

  • Mathematica
    Select[Range[143],PrimeOmega[#]James C. McMahon, Dec 28 2024 *)
  • PARI
    is(n)=if(n%2==0, return(n==2)); if(n<27, return(1)); forprime(p=2,bigomega(n), if(n%p==0, return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

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

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.
Showing 1-4 of 4 results.

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