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 1-9 of 9 results.

A366608 a(n) = phi(4^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 4, 16, 48, 256, 800, 3840, 12544, 65536, 186624, 986880, 3345408, 16515072, 52306176, 252645120, 760320000, 4288266240, 13628740608, 64258375680, 218462552064, 1095233372160, 3105655160832, 16510446886912, 56000724240384, 280012271910912, 869940000000000
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A000010, A052539, A057940, A274903, A295501, A366605, A366606, A366607, A366609.
Cf. A053285, A366579, A366618, A366630, A366639, A366658, A366667, A366669, A366690, A366716.

Programs

  • Mathematica
    EulerPhi[4^Range[0,30]+1] (* Paolo Xausa, Oct 14 2023 *)
  • PARI
    {a(n) = eulerphi(4^n+1)}
  • Python
    from sympy import totient
def A366608(n): return totient((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

Formula

a(n) = A053285(2*n). - Max Alekseyev, Jan 08 2024

A366607 Sum of the divisors of 4^n+1.

Original entry on oeis.org

3, 6, 18, 84, 258, 1302, 4356, 20520, 65538, 351120, 1110276, 5048232, 17041416, 82623888, 284225796, 1494039792, 4301668356, 20788904016, 73234343952, 332019460560, 1103789883396, 5936210280000, 18679788287496, 84884999116320, 282937726148616
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(3)=84 because 4^3+1 has divisors {1, 5, 13, 65}.

Links

Crossrefs

Cf. A000203, A052539, A057940, A274903, A366603, A366605, A366606, A366608, A366609.
Cf. A069061, A366578, A366617, A366629, A366638, A366657, A366666, A366668, A366689, A366715.

Programs

  • Maple
    a:=n->numtheory[sigma](4^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[1,4^Range[0,30]+1] (* Paolo Xausa, Oct 14 2023 *)
  • Python
    from sympy import divisor_sigma
def A366607(n): return divisor_sigma((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

Formula

a(n) = sigma(4^n+1) = A000203(A052539(n)).
a(n) = A069061(2*n). - Max Alekseyev, Jan 08 2024

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

A366615 Number of distinct prime divisors of 5^n + 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 3, 6, 3, 4, 5, 5, 4, 8, 4, 4, 4, 5, 4, 7, 3, 4, 7, 5, 4, 8, 6, 7, 6, 5, 4, 7, 5, 6, 6, 6, 3, 8, 3, 5, 5, 7, 7, 9, 5, 5, 6, 7, 7, 8, 3, 6, 6, 6, 4, 13, 4, 8, 7, 3, 7, 8, 7, 5, 6, 5, 5, 12, 5, 9, 9, 6, 6, 10, 6, 5, 7, 9
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A001221, A034474, A057939, A074478, A366611, A366616, A366617, A366618.
Cf. A046799, A366580, A366605, A366627, A366636, A366655, A366664, A119704, A366686, A366712.

Programs

  • Mathematica
    Table[PrimeNu[5^n+1],{n,0,90}] (* Harvey P. Dale, Apr 06 2025 *)
  • PARI
    for(n = 0, 100, print1(omega(5^n + 1), ", "))

Formula

a(n) = omega(5^n+1) = A001221(A034474(n)).

A366627 Number of distinct prime divisors of 6^n + 1.

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A062394, A001221, A057938, A366620, A366628, A366629, A366630.
Cf. A046799, A366580, A366605, A366615, A366636, A366655, A366664, A119704, A366686, A366712.

Programs

  • Mathematica
    PrimeNu[6^Range[0,84] + 1] (* Paul F. Marrero Romero, Nov 11 2023 *)
  • PARI
    for(n = 0, 100, print1(omega(6^n + 1), ", "))

Formula

a(n) = omega(6^n+1) = A001221(A062394(n)).

A366636 Number of distinct prime divisors of 7^n + 1.

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A034491, A001221, A057937, A227575, A366632, A366637, A366638, A366639.
Cf. A046799, A366580, A366605, A366615, A366627, A366655, A366664, A119704, A366686, A366712.

Programs

  • Mathematica
    PrimeNu[7^Range[0,84] + 1] (* Paul F. Marrero Romero, Nov 11 2023 *)
  • PARI
    for(n = 0, 100, print1(omega(7^n + 1), ", "))

Formula

a(n) = omega(7^n+1) = A001221(A034491(n)).

A366664 Number of distinct prime divisors of 9^n + 1.

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A062396, A001221, A057935, A366660, A366665, A366666, A366667.
Cf. A046799, A366580, A366605, A366615, A366627, A366636, A366655, A119704, A366686, A366712.

Programs

  • Mathematica
    PrimeNu[9^Range[0,90]+1] (* Harvey P. Dale, Jul 04 2024 *)
  • PARI
    for(n = 0, 100, print1(omega(9^n + 1), ", "))

Formula

a(n) = omega(9^n+1) = A001221(A062396(n)).
a(n) = A366580(2*n). - Max Alekseyev, Jan 08 2024

A366609 Smallest prime dividing 4^n + 1.

Original entry on oeis.org

2, 5, 17, 5, 257, 5, 17, 5, 65537, 5, 17, 5, 97, 5, 17, 5, 641, 5, 17, 5, 257, 5, 17, 5, 193, 5, 17, 5, 257, 5, 17, 5, 274177, 5, 17, 5, 97, 5, 17, 5, 65537, 5, 17, 5, 257, 5, 17, 5, 641, 5, 17, 5, 257, 5, 17, 5, 449, 5, 17, 5, 97, 5, 17, 5, 59649589127497217
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Bisection of A002586.
Cf. A002586, A038371, A052539, A074476, A274903, A366605, A366606, A366607, A366608, A366670, A366671, A366719.
Showing 1-9 of 9 results.

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