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.

Previous Showing 11-16 of 16 results.

A366661 Number of divisors of 9^n-1.

Original entry on oeis.org

4, 10, 16, 24, 24, 80, 16, 112, 128, 180, 64, 384, 16, 160, 768, 256, 128, 1280, 64, 864, 768, 640, 32, 14336, 384, 160, 4096, 1536, 256, 23040, 128, 576, 2048, 1280, 768, 12288, 128, 640, 12288, 16128, 128, 61440, 32, 12288, 196608, 320, 512, 131072, 2048
Offset: 1

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Examples

			a(2)=10 because 9^2-1 has divisors {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}.

Links

Crossrefs

Cf. A024101, A000005, A057952, A366660, A366662, A366663.
Cf. A046801, A366575, A366602, A366612, A366621, A366633, A366652, A070528, A366683, A366709.

Programs

  • Maple
    a:=n->numtheory[tau](9^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[0, 9^Range[100]-1]
  • PARI
    a(n) = numdiv(9^n-1);

Formula

a(n) = sigma0(9^n-1) = A000005(A024101(n)).
a(n) = A366575(2*n) = A366575(n) * A366577(n) * (4 + A007814(n)) / (2 * (3 + A007814(n))). - Max Alekseyev, Jan 07 2024

A059891 a(n) = |{m : multiplicative order of 9 mod m = n}|.

Original entry on oeis.org

4, 6, 12, 14, 20, 58, 12, 88, 112, 150, 60, 290, 12, 138, 732, 144, 124, 1088, 60, 670, 740, 570, 28, 13864, 360, 138, 3968, 1362, 252, 22058, 124, 320, 1972, 1146, 732, 10704, 124, 570, 12260, 15176, 124, 60470, 28, 11634, 195728, 282, 508, 116592, 2032
Offset: 1

Views

Author

Vladeta Jovovic, Feb 06 2001

Keywords

Comments

The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m).
a(n) = number of orders of degree-n monic irreducible polynomials over GF(9).
Also, number of primitive factors of 9^n - 1. - Max Alekseyev, May 03 2022

Links

Crossrefs

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), this sequence (b=9), A059892 (b=10).
Cf. A000005, A008683, A027381, A053452, A057952, A058946, A274909.
Column k=9 of A212957.

Programs

  • Maple
    with(numtheory):
a:= n-> add(mobius(n/d)*tau(9^d-1), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 12 2012
  • Mathematica
    a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 9^d-1], {d, Divisors[n]}];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 13 2025, after Alois P. Heinz *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d) * numdiv(9^d-1)); \\ Amiram Eldar, Jan 25 2025

Formula

a(n) = Sum_{d|n} mu(n/d)*tau(9^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A366660 Number of distinct prime divisors of 9^n - 1.

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 3, 5, 6, 5, 5, 7, 3, 6, 8, 6, 6, 9, 5, 7, 8, 8, 4, 12, 7, 6, 11, 9, 7, 12, 6, 7, 10, 9, 8, 12, 6, 8, 12, 11, 6, 14, 4, 12, 16, 7, 8, 15, 10, 12, 13, 9, 6, 15, 11, 14, 13, 10, 5, 18, 5, 10, 16, 8, 9, 15, 6, 13, 13, 15, 7, 19, 7, 10, 19, 13, 11
Offset: 1

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A024101, A001221, A057952, A366661, A366662, A366663.
Cf. A046800, A133801, A366604, A366611, A366620, A366632, A366651, A102347, A366681, A366707.

Programs

  • PARI
    for(n = 1, 100, print1(omega(9^n - 1), ", "))

Formula

a(n) = omega(9^n-1) = A001221(A024101(n)).
a(n) = A133801(2*n) = A133801(n) + A366580(n) - 1. - Max Alekseyev, Jan 07 2024

A085034 Number of prime factors of cyclotomic(n,9), which is A019327(n), the value of the n-th cyclotomic polynomial evaluated at x=9.

Original entry on oeis.org

3, 2, 2, 2, 3, 1, 2, 3, 3, 2, 4, 1, 2, 2, 3, 2, 5, 1, 4, 1, 4, 2, 3, 3, 4, 2, 5, 2, 6, 1, 5, 2, 3, 2, 3, 1, 5, 2, 8, 2, 5, 2, 3, 3, 5, 2, 7, 2, 7, 4, 5, 2, 5, 1, 5, 3, 6, 2, 4, 3, 4, 3, 5, 2, 4, 1, 5, 3, 7, 3, 6, 2, 6, 3, 7, 4, 4, 2
Offset: 1

Views

Author

T. D. Noe, Jun 19 2003

Keywords

Comments

The Mobius transform of this sequence yields A057952, number of prime factors of 9^n-1.

References

Links

Crossrefs

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), this sequence (x=9), A085035 (x=10).
Cf. A019327, A057952, A059891, A274909.

Programs

  • Mathematica
    Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 9]]][[2]], {n, 1, 100}]

A366708 Number of prime factors of 12^n - 1 (counted with multiplicity).

Original entry on oeis.org

1, 2, 2, 4, 2, 5, 3, 6, 4, 4, 4, 8, 3, 6, 6, 9, 3, 9, 2, 8, 5, 6, 4, 12, 4, 8, 6, 10, 5, 13, 5, 11, 8, 6, 9, 14, 3, 6, 7, 14, 4, 14, 5, 12, 12, 8, 3, 18, 5, 10, 6, 13, 7, 16, 8, 13, 7, 8, 4, 19, 4, 8, 8, 13, 8, 17, 5, 10, 7, 14, 4, 21, 3, 7, 11, 11, 11, 18, 4
Offset: 1

Views

Author

Sean A. Irvine, Oct 17 2023

Keywords

Links

Crossrefs

Cf. A024140, A001222, A046051, A057951, A057952, A057953, A057954, A057955, A057956, A057957, A057958, A250288, A252170, A366707, A366709, A366682.

Programs

  • Mathematica
    PrimeOmega[12^Range[70]-1]
  • PARI
    a(n)=bigomega(12^n-1)

Formula

a(n) = bigomega(12^n-1) = A001222(A024140(n)).

A366682 Number of prime factors of 11^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 5, 4, 7, 4, 9, 4, 9, 5, 8, 4, 13, 4, 8, 7, 12, 3, 12, 3, 11, 10, 11, 5, 17, 8, 10, 6, 13, 4, 15, 5, 15, 9, 9, 8, 17, 6, 10, 12, 15, 9, 17, 4, 15, 9, 12, 5, 24, 7, 14, 9, 13, 6, 16, 10, 19, 8, 10, 5, 21, 5, 12, 16, 19, 8, 22, 6, 15, 10, 19, 7, 24, 3, 11, 15
Offset: 1

Views

Author

Sean A. Irvine, Oct 16 2023

Keywords

Links

Crossrefs

Cf. A024127, A001222, A366681, A366683, A366684, A366685, A366687.
Cf. A046051, A057958, A057957, A057956, A057955, A057954, A057953, A057952, A057951, A366708.

Programs

  • Mathematica
    PrimeOmega[11^Range[70]-1]
  • PARI
    a(n)=bigomega(11^n-1)

Formula

a(n) = bigomega(11^n-1) = A001222(A024127(n)).
Previous Showing 11-16 of 16 results.

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