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 1-6 of 6 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

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

A139270 Twice nonprime numbers.

Original entry on oeis.org

2, 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, 84, 88, 90, 92, 96, 98, 100, 102, 104, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150
Offset: 1

Views

Author

Omar E. Pol, May 16 2008

Keywords

Comments

Besides a(1), is this sequence a subset of A091375? - Bill McEachen, Dec 18 2016

Crossrefs

Cf. A018252.

Programs

  • Mathematica
    2*Select[Range[75],!PrimeQ[#]&] (* Stefano Spezia, Nov 18 2023 *)
  • PARI
    is(n)=n%2==0 && !isprime(n/2) && n \\ Charles R Greathouse IV, Feb 21 2017
  • Python
    from sympy import composite
def A139270(n): return composite(n-1)<<1 if n>1 else 2 # Chai Wah Wu, Oct 15 2024

Formula

a(n) = A018252(n)*2.

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 1-6 of 6 results.

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