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 1-7 of 7 results.

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

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

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.

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

A377723 Numbers whose number of prime factors (counted with repetition) is greater than or equal to its smallest prime factor.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116
Offset: 1

Views

Author

James C. McMahon, Dec 28 2024

Keywords

Comments

Numbers k such that A001222(k) >= A020639(k).
Complement of A091377.
A091371(a(n)) < 1: A001222(a(n)) => A020639(a(n)).

Examples

			4 is a term because bigomega(4) = spf(4) = 2.
12 is a term because bigomega(12) = 3 > spf(12) = 2.
3 is not a term because bigomega(3) = 1 < spf(3) = 3.

Links

Crossrefs

Cf. A001222, A020639, A091371, A091375, A091376, A091377.

Programs

  • Mathematica
    Select[Range[116], PrimeOmega[#]>=FactorInteger[#][[1, 1]]&]
Showing 1-7 of 7 results.

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