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 1-5 of 5 results.

A087436 Number of odd prime factors of n, counted with repetitions.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 0, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 3, 0, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 1, 2
Offset: 1

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Author

Reinhard Zumkeller, Sep 03 2003

Keywords

Comments

Number of parts larger than 1 in the partition with Heinz number n. The Heinz number of an integer partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: a(9) = 2 because the partition with Heinz number 9 (=3*3) is [2,2]. - Emeric Deutsch, Oct 02 2015
Totally additive because both A001222 and A007814 are. a(2) = 0, and a(p) = 1 for odd primes p, a(m*n) = a(m)+a(n) for m, n > 1. - Antti Karttunen, Jul 10 2020

Examples

			a(9) = 2 because 9 = 3*3 has 2 odd prime factors. - _Emeric Deutsch_, Oct 02 2015

Links

Crossrefs

Cf. A000244 (the first occurrence of each n, and also the positions of records).
Cf. A000265, A001222, A005087, A007814, A083342, A215366, A336118, A336161.

Programs

  • Maple
    seq(bigomega(n) - padic[ordp](n, 2), n=1..102); # Peter Luschny, Dec 06 2017
  • Mathematica
    Join[{0},Table[Length[Select[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[ n]],OddQ]],{n,2,110}]] (* Harvey P. Dale, Feb 01 2013 *)
  • PARI
    a(n) = bigomega(n) - valuation(n, 2); \\ Michel Marcus, Sep 10 2019
  • PARI
    A087436(n) = (bigomega(n>>valuation(n,2))); \\ Antti Karttunen, Jul 10 2020

Formula

a(n) = A001222(n) - A007814(n).
a(n) = A001222(A000265(n)). - Antti Karttunen, Jul 10 2020
Sum_{k=1..n} a(k) = n * (log(log(n)) + B_2 - 1) + O(n/log(n)), where B_2 = A083342. - Amiram Eldar, May 16 2025

A335915 Fully multiplicative with a(2) = 1, and a(p) = A000265(p-1)*A000265(p+1) = A000265(p^2 - 1), for odd primes p.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 3, 1, 1, 3, 15, 1, 21, 3, 3, 1, 9, 1, 45, 3, 3, 15, 33, 1, 9, 21, 1, 3, 105, 3, 15, 1, 15, 9, 9, 1, 171, 45, 21, 3, 105, 3, 231, 15, 3, 33, 69, 1, 9, 9, 9, 21, 351, 1, 45, 3, 45, 105, 435, 3, 465, 15, 3, 1, 63, 15, 561, 9, 33, 9, 315, 1, 333, 171, 9, 45, 45, 21, 195, 3, 1, 105, 861, 3, 27, 231, 105, 15, 495, 3, 63, 33, 15, 69
Offset: 1

Views

Author

Antti Karttunen, Jul 09 2020

Keywords

Comments

For all i, j: A324400(i) = A324400(j) => a(i) = a(j) => A336118(i) = A336118(j).

Links

Crossrefs

Cf. A000265.
Cf. also A167344, A324400, A335904, A336118, A336455, A336456, A336466, A336467.

Programs

  • PARI
    A000265(n) = (n>>valuation(n,2));
A335915(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1)*A000265(f[k,1]+1))^f[k,2])); };

Formula

Completely multiplicative with a(2) = 1, and for odd primes p, a(p) = A000265(p-1)*A000265(p+1).
For all n >= 1, A335904(a(n)) = A336118(n).
For all n >= 0, a(2^n) = a(3^n) = 1, a(5^n) = a(7^n) = 3^n.
a(n) = A336466(n) * A336467(n). - Antti Karttunen, Jan 31 2021

A335904 Fully additive with a(2) = 0, and a(p) = 1+a(p-1)+a(p+1), for odd primes p.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 2, 2, 4, 1, 4, 2, 3, 0, 3, 2, 5, 2, 3, 4, 6, 1, 4, 4, 3, 2, 6, 3, 4, 0, 5, 3, 4, 2, 8, 5, 5, 2, 6, 3, 8, 4, 4, 6, 8, 1, 4, 4, 4, 4, 8, 3, 6, 2, 6, 6, 10, 3, 8, 4, 4, 0, 6, 5, 9, 3, 7, 4, 7, 2, 11, 8, 5, 5, 6, 5, 8, 2, 4, 6, 10, 3, 5, 8, 7, 4, 9, 4, 6, 6, 5, 8, 7, 1, 6, 4, 6, 4, 9, 4, 9, 4, 5
Offset: 1

Views

Author

Antti Karttunen, Jun 29 2020

Keywords

Links

Crossrefs

Cf. A000244, A052126, A087436, A171462, A335875, A335876, A335881, A335884, A335885, A335905, A335906, A335915, A336118.

Programs

  • PARI
    A335904(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A335904(f[k,1]-1)+A335904(f[k,1]+1)))); };

Formula

Totally additive with a(2) = 0, and for odd primes p, a(p) = 1 + a(p-1) + a(p+1).
a(n) = A336118(n) + A087436(n).
For all n >= 1, a(A335915(n)) = A336118(n).
For all n >= 1, a(n) >= A335884(n) >= A335881(n) >= A335875(n) >= A335885(n).
For all n >= 0, a(3^n) = n.

A336396 a(n) = A329697(n) - A087436(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 0, 1, 1, 2, 0, 0, 1, 0, 1, 2, 0, 2, 0, 1, 0, 1, 0, 2, 2, 1, 0, 1, 1, 3, 1, 0, 2, 3, 0, 2, 0, 0, 1, 2, 0, 1, 1, 2, 2, 3, 0, 2, 2, 1, 0, 1, 1, 3, 0, 2, 1, 3, 0, 2, 2, 0, 2, 2, 1, 3, 0, 0, 1, 2, 1, 0, 3, 2, 1, 2, 0, 2, 2, 2, 3, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Jul 24 2020

Keywords

Comments

Totally additive because both A087436 and A329697 are.

Links

Crossrefs

Cf. A000265, A046660, A087436, A329697, A336118, A336466, A336469.

Programs

Formula

a(n) = A329697(n) - A087436(n).
a(n) = A329697(A336466(n)).
a(n) = A336469(n) - A046660(A000265(n)).
For all n >= 1, a(n) <= A336118(n).

A336921 a(n) = A331410(n) - A087436(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 3, 1, 0, 0, 1, 2, 1, 0, 3, 2, 1, 1, 2, 0, 2, 1, 1, 1, 1, 0, 0, 2, 2, 1, 3, 0, 2, 0, 2, 3, 3, 1, 1, 0, 0, 0, 2, 1, 3, 2, 1, 1, 2, 0, 4, 3, 2, 2, 1, 1, 2, 1, 0, 2, 2, 0, 3, 2, 3, 1, 4, 1, 1, 1, 0, 1, 3, 0, 2, 0, 1, 2, 4, 2, 2, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 09 2020

Keywords

Comments

Totally additive because both A087436 and A331410 are.

Links

Crossrefs

Cf. A000265, A046660, A087436, A331410, A336467, A336922.
Cf. also A336118, A336396.

Programs

Formula

a(n) = A331410(A336467(n)).
a(n) = A331410(n) - A087436(n).
a(n) = A336922(n) - A046660(A000265(n)).
For all n >= 1, a(n) <= A336118(n).
Showing 1-5 of 5 results.

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