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 1-10 of 13 results. Next

A366709 Number of divisors of 12^n-1.

Original entry on oeis.org

2, 4, 4, 16, 4, 32, 8, 64, 16, 16, 12, 256, 8, 64, 64, 512, 8, 512, 4, 192, 32, 48, 16, 4096, 16, 192, 64, 1024, 32, 8192, 32, 2048, 192, 64, 512, 16384, 8, 64, 128, 12288, 16, 12288, 32, 3072, 4096, 256, 8, 262144, 32, 1024, 64, 6144, 128, 65536, 192, 8192
Offset: 1

Views

Author

Sean A. Irvine, Oct 17 2023

Keywords

Examples

			a(3)=4 because 12^3-1 has divisors {1, 11, 157, 1727}.

Links

Crossrefs

Cf. A000005, A024140, A046801, A070528, A366707, A366708, A366710, A366683.

Programs

  • Maple
    a:=n->numtheory[tau](12^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[0, 12^Range[100]-1]
  • PARI
    a(n) = numdiv(12^n-1);

Formula

a(n) = sigma0(12^n-1) = A000005(A024140(n)).

A366712 Number of distinct prime divisors of 12^n + 1.

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 3, 3, 3, 5, 3, 2, 4, 4, 4, 7, 2, 3, 5, 4, 6, 8, 6, 4, 6, 6, 5, 10, 3, 3, 5, 3, 2, 9, 4, 5, 7, 4, 5, 10, 4, 5, 6, 5, 4, 12, 3, 5, 5, 5, 8, 9, 5, 5, 10, 7, 9, 10, 7, 6, 10, 5, 6, 14, 5, 8, 11, 4, 5, 12, 9, 3, 10, 5, 8, 17, 5, 6, 9, 4, 6, 15, 3
Offset: 0

Views

Author

Sean A. Irvine, Oct 17 2023

Keywords

Links

Crossrefs

Cf. A178248, A001221, A046799, A119704, A366713, A366714, A366707, A366686.

Programs

  • PARI
    for(n = 0, 100, print1(omega(12^n + 1), ", "))

Formula

a(n) = omega(12^n+1) = A001221(A178248(n)).

A366681 Number of distinct prime divisors of 11^n - 1.

Original entry on oeis.org

2, 3, 4, 4, 3, 6, 4, 5, 5, 5, 4, 9, 4, 6, 6, 7, 3, 8, 3, 7, 9, 9, 5, 12, 6, 8, 6, 10, 4, 11, 5, 9, 9, 7, 7, 12, 6, 8, 12, 10, 9, 13, 4, 12, 8, 10, 5, 18, 7, 10, 9, 10, 6, 11, 9, 15, 7, 8, 5, 16, 5, 10, 15, 12, 7, 19, 6, 12, 10, 15, 7, 18, 3, 9, 13, 11, 8, 20
Offset: 1

Views

Author

Sean A. Irvine, Oct 16 2023

Keywords

Links

Crossrefs

Cf. A024127, A001221, A274910, A366682, A366683, A366684, A366685.
Cf. A046800, A133801, A102347, A366707.

Programs

  • PARI
    for(n = 1, 100, print1(omega(11^n - 1), ", "))

Formula

a(n) = omega(11^n-1) = A001221(A024127(n)).

A366710 Sum of the divisors of 12^n-1.

Original entry on oeis.org

12, 168, 1896, 30240, 271464, 4247040, 39156480, 636854400, 5817876000, 72749094432, 852203639280, 15743437516800, 116720110574544, 1518251476008960, 17220536137159296, 292933954031846400, 2420303924088730368, 38936041113123840000, 348523635677043192936
Offset: 1

Views

Author

Sean A. Irvine, Oct 17 2023

Keywords

Examples

			a(3)=1896 because 12^3-1 has divisors {1, 11, 157, 1727}.

Links

Crossrefs

Cf. A024140, A000203, A366707, A366708, A366709, A366711, A366684.

Programs

  • Maple
    a:=n->numtheory[sigma](12^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[1, 12^Range[30]-1]

Formula

a(n) = sigma(12^n-1) = A000203(A024140(n)).

A366711 a(n) = phi(12^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

10, 120, 1560, 13440, 226200, 2021760, 32518360, 274391040, 4534807680, 51953616000, 646094232960, 4662793175040, 97266341877120, 1070382142166400, 13666309113600000, 109897747141754880, 2016918439151095000, 17518491733377024000, 290436363064202660760
Offset: 1

Views

Author

Sean A. Irvine, Oct 17 2023

Keywords

Links

Crossrefs

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), this sequence (k=12).
Cf. A000010, A024140, A366707, A366708, A366709, A366710, A366717, A366718.

Programs

  • Mathematica
    EulerPhi[12^Range[30] - 1]
  • PARI
    {a(n) = eulerphi(12^n-1)}

A366620 Number of distinct prime divisors of 6^n - 1.

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A024062, A001221, A057955, A059888, A274907, A366621, A366622, A366623.
Cf. A046800, A133801, A366604, A366611, A366632, A366651, A366660, A102347, A366681, A366707.

Programs

  • PARI
    for(n = 1, 100, print1(omega(6^n - 1), ", "))

Formula

a(n) = omega(6^n-1) = A001221(A024062(n)).

A366632 Number of distinct prime divisors of 7^n - 1.

Original entry on oeis.org

2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 4, 7, 3, 6, 6, 6, 4, 7, 4, 8, 6, 6, 5, 11, 5, 5, 9, 8, 5, 10, 5, 8, 8, 5, 7, 11, 5, 6, 7, 11, 5, 11, 4, 10, 10, 6, 4, 14, 8, 8, 9, 8, 5, 12, 6, 13, 8, 6, 6, 17, 6, 8, 9, 11, 9, 13, 6, 9, 9, 15, 4, 18, 7, 7, 10, 8, 9, 13, 4, 16, 13
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A024075, A001221, A057954, A059889, A074249, A218358, A366620, A366633, A366634, A366635.
Cf. A046800, A133801, A366604, A366611, A366620, A366651, A366660, A102347, A366681, A366707.

Programs

  • PARI
    for(n = 1, 100, print1(omega(7^n - 1), ", "))

Formula

a(n) = omega(7^n-1) = A001221(A024075(n)).

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

A366604 Number of distinct prime divisors of 4^n - 1.

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A024036, A001221, A057957, A133801, A366602, A366603.
Cf. A046800, A133801, A366611, A366620, A366632, A366651, A366660, A102347, A366681, A366707.

Programs

  • Mathematica
    PrimeNu[4^Range[100]-1] (* Paolo Xausa, Oct 14 2023 *)
  • PARI
    for(n = 1, 100, print1(omega(4^n - 1), ", "))
  • Python
    from sympy import primenu
def A366604(n): return primenu((1<<(n<<1))-1) # Chai Wah Wu, Oct 15 2023

Formula

a(n) = omega(4^n-1) = A001221(A024036(n)).
a(n) = A046800(2*n) = A046799(n) + A046800(n). - Max Alekseyev, Jan 07 2024

A366651 Number of distinct prime divisors of 8^n - 1.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 6, 3, 6, 4, 8, 4, 6, 6, 9, 5, 6, 4, 11, 6, 8, 4, 12, 7, 7, 6, 12, 6, 11, 3, 12, 8, 10, 10, 12, 6, 8, 9, 15, 5, 11, 5, 14, 10, 8, 6, 17, 5, 13, 8, 16, 8, 12, 10, 17, 7, 10, 6, 21, 5, 7, 9, 15, 8, 15, 6, 19, 9, 20, 7, 18, 7, 12, 14, 16, 9
Offset: 1

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A024088, A001221, A057953, A366652, A366653, A366654.
Cf. A046800, A133801, A366604, A366611, A366620, A366632, A366660, A102347, A366681, A366707.

Programs

  • PARI
    for(n = 1, 100, print1(omega(8^n - 1), ", "))

Formula

a(n) = omega(8^n-1) = A001221(A024088(n)).
a(n) = A046800(3*n). - Max Alekseyev, Jan 09 2024
Showing 1-10 of 13 results. Next

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