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 1-8 of 8 results.

A366606 Number of divisors of 4^n+1.

Original entry on oeis.org

2, 2, 2, 4, 2, 6, 4, 8, 2, 16, 4, 8, 8, 16, 4, 48, 4, 16, 16, 16, 4, 64, 8, 32, 8, 64, 8, 64, 8, 8, 16, 32, 4, 64, 12, 96, 32, 32, 16, 768, 8, 32, 32, 32, 16, 1536, 4, 16, 8, 64, 64, 512, 4, 16, 64, 96, 32, 256, 8, 128, 64, 64, 16, 1024, 4, 768, 128, 64, 16
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(3)=4 because 4^3+1 has divisors {1, 5, 13, 65}.

Links

Crossrefs

Cf. A000005, A052539, A057940, A274903, A366602, A366605, A366607, A366608, A366609.
Cf. A046798, A366577, A366616, A366628, A366637, A366656, A366665, A344897, A366688, A366714.

Programs

  • Maple
    a:=n->numtheory[tau](4^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[0,4^Range[0,100]+1] (* Paolo Xausa, Oct 14 2023 *)
  • PARI
    a(n) = numdiv(4^n+1);
  • Python
    from sympy import divisor_count
def A366606(n): return divisor_count((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

Formula

a(n) = sigma0(4^n+1) = A000005(A052539(n)).
a(n) = A046798(2*n). - Max Alekseyev, Jan 08 2024

A366655 Number of distinct prime divisors of 8^n + 1.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 3, 3, 5, 4, 4, 3, 6, 5, 3, 5, 6, 4, 4, 5, 6, 4, 5, 6, 9, 6, 5, 4, 10, 4, 3, 7, 9, 10, 6, 6, 8, 5, 6, 6, 10, 5, 7, 9, 8, 6, 7, 6, 12, 9, 5, 5, 10, 10, 8, 6, 8, 7, 8, 3, 9, 10, 4, 10, 12, 7, 8, 6, 14, 7, 8, 5, 10, 10, 8, 11, 16, 5, 7, 10
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A062395, A001221, A057936, A274905, A366651, A366632, A366656, A366657, A366658, A366671.
Cf. A046799, A366580, A366605, A366615, A366627, A366636, A366664, A119704, A366686, A366712.

Programs

  • PARI
    for(n = 0, 100, print1(omega(8^n + 1), ", "))

Formula

a(n) = omega(8^n+1) = A001221(A062395(n)).
a(n) = A046799(3*n). - Max Alekseyev, Jan 09 2024

A366658 a(n) = phi(8^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 6, 48, 324, 3840, 19800, 186624, 1365336, 16515072, 84768120, 760320000, 5632621632, 64258375680, 366369658200, 3105655160832, 20140520400000, 280012271910912, 1495522910085120, 12824556668190720, 95907982079387520, 1080582572777472000, 5688765822212629632
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A062395, A000010, A057936, A274905, A366654, A366655, A366656, A366657, A366671.
Cf. A053285, A366579, A366608, A366618, A366630, A366639, A366667, A366669, A366690, A366716.

Programs

  • Mathematica
    EulerPhi[8^Range[0, 21] + 1] (* Paul F. Marrero Romero, Oct 17 2023 *)
  • PARI
    {a(n) = eulerphi(8^n+1)}
  • Python
    from sympy import totient
def A366658(n): return totient((1<<3*n)+1) # Chai Wah Wu, Oct 15 2023

Formula

a(n) = A000010(A062395(n)). - Paul F. Marrero Romero, Nov 06 2023
a(n) = A053285(3*n). - Max Alekseyev, Jan 09 2024

A366616 Number of divisors of 5^n+1.

Original entry on oeis.org

2, 4, 4, 12, 4, 8, 8, 16, 8, 32, 16, 32, 8, 16, 8, 96, 8, 16, 32, 32, 16, 576, 16, 16, 16, 32, 24, 320, 8, 16, 128, 32, 16, 384, 64, 128, 64, 32, 16, 192, 32, 64, 64, 64, 8, 512, 8, 32, 32, 128, 128, 768, 32, 32, 64, 128, 128, 384, 8, 64, 64, 64, 16, 24576, 16
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(3)=12 because 5^3+1 has divisors {1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126}.

Links

Crossrefs

Cf. A000005, A034474, A057939, A074478, A366612, A366607, A366615, A366617, A366618.
Cf. A046798, A366577, A366606, A366628, A366637, A366656, A366665, A344897, A366688, A366714.

Programs

  • Maple
    a:=n->numtheory[tau](5^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[0, 5^Range[0, 70] + 1] (* Paolo Xausa, Apr 20 2025 *)
  • PARI
    a(n) = numdiv(5^n+1);

Formula

a(n) = sigma0(5^n+1) = A000005(A034474(n)).

A366628 Number of divisors of 6^n+1.

Original entry on oeis.org

2, 2, 2, 4, 2, 8, 8, 12, 4, 8, 8, 4, 4, 16, 8, 32, 8, 8, 64, 8, 8, 48, 16, 8, 16, 16, 16, 32, 32, 16, 512, 4, 8, 64, 8, 1536, 32, 16, 8, 512, 32, 16, 128, 4, 8, 128, 32, 4, 128, 64, 64, 256, 16, 32, 1024, 192, 64, 128, 8, 4, 64, 8, 4, 768, 8, 256, 2048, 32, 32
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(3)=4 because 6^3+1 has divisors {1, 7, 31, 217}.

Links

Crossrefs

Cf. A062394, A000005, A057938, A366621, A366627, A366629, A366630, A366670.
Cf. A046798, A366577, A366606, A366616, A366637, A366656, A366665, A344897, A366688, A366714.

Programs

  • Maple
    a:=n->numtheory[tau](6^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[0, 6^Range[0, 70] + 1] (* Paolo Xausa, Apr 19 2025 *)
  • PARI
    a(n) = numdiv(6^n+1);

Formula

a(n) = sigma0(6^n+1) = A000005(A062394(n)).

A366637 Number of divisors of 7^n+1.

Original entry on oeis.org

2, 4, 6, 8, 4, 16, 24, 16, 8, 16, 32, 16, 32, 16, 12, 64, 8, 8, 48, 16, 16, 128, 48, 8, 16, 32, 24, 32, 64, 8, 512, 32, 16, 128, 48, 1024, 256, 16, 12, 256, 64, 64, 96, 512, 32, 2048, 96, 8, 64, 2048, 640, 128, 32, 64, 384, 3072, 256, 256, 96, 64, 512, 8, 48
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Examples

			a(4)=4 because 7^4+1 has divisors {1, 2, 1201, 2402}.

Links

Crossrefs

Cf. A034491, A000005, A057937, A227575, A366633, A366636, A366638, A366639.
Cf. A046798, A366577, A366606, A366616, A366628, A366656, A366665, A344897, A366688, A366714.

Programs

  • Maple
    a:=n->numtheory[tau](7^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[0, 7^Range[0, 62] + 1] (* Paul F. Marrero Romero, Oct 16 2023 *)
  • PARI
    a(n) = numdiv(7^n+1);

Formula

a(n) = sigma0(7^n+1) = A000005(A034491(n)).

A366657 Sum of the divisors of 8^n+1.

Original entry on oeis.org

3, 13, 84, 800, 4356, 51792, 351120, 3100240, 17041416, 211053040, 1494039792, 12611914848, 73234343952, 794382536272, 5936210280000, 60037292774400, 282937726148616, 3264911394064320, 24128875076496960, 208532141890460960, 1225825603154905104
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Examples

			a(4)=4356 because 8^4+1 has divisors {1, 17, 241, 4097}.

Links

Crossrefs

Cf. A062395, A000203, A057936, A274905, A366653, A366655, A366656, A366658.
Cf. A069061, A366578, A366607, A366617, A366629, A366638, A366666, A366668, A366689, A366715.

Programs

  • Maple
    a:=n->numtheory[sigma](8^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[1, 8^Range[0,20]+1] (* Paul F. Marrero Romero, Nov 19 2023 *)

Formula

a(n) = sigma(8^n+1) = A000203(A062395(n)).
a(n) = A069061(3*n). - Max Alekseyev, Jan 09 2024

A366665 Number of divisors of 9^n+1.

Original entry on oeis.org

2, 4, 4, 8, 8, 12, 8, 16, 4, 16, 8, 16, 64, 16, 16, 48, 4, 16, 16, 16, 32, 128, 32, 16, 16, 128, 16, 32, 64, 16, 128, 32, 4, 64, 32, 384, 256, 32, 64, 128, 32, 32, 1024, 128, 64, 384, 16, 16, 64, 512, 64, 256, 128, 64, 512, 192, 512, 512, 32, 8, 2048, 64, 16
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Examples

			a(2)=4 because 9^2+1 has divisors {1, 2, 41, 82}.

Links

Crossrefs

Cf. A062396, A000005, A057935, A366661, A366664, A366666, A366667.
Cf. A046798, A366577, A366606, A366616, A366628, A366637, A366656, A344897, A366688, A366714.

Programs

  • Maple
    a:=n->numtheory[tau](9^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[0, 9^Range[0,62] + 1] (* Paul F. Marrero Romero, Nov 13 2023 *)
  • PARI
    a(n) = numdiv(9^n+1);

Formula

a(n) = sigma0(9^n+1) = A000005(A062396(n)).
a(n) = A366577(2*n). - Max Alekseyev, Jan 08 2024
Showing 1-8 of 8 results.

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