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.

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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

A366602 Number of divisors of 4^n-1.

Original entry on oeis.org

2, 4, 6, 8, 8, 24, 8, 16, 32, 48, 16, 96, 8, 64, 96, 32, 8, 512, 8, 192, 144, 128, 16, 768, 128, 128, 160, 256, 64, 4608, 8, 128, 384, 128, 512, 8192, 32, 128, 192, 768, 32, 9216, 32, 1024, 4096, 512, 64, 6144, 32, 8192, 1536, 1024, 64, 10240, 3072, 2048, 384
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(4)=8 because 4^4-1 has divisors {1, 3, 5, 15, 17, 51, 85, 255}.

Links

Crossrefs

Cf. A024036, A000005, A057957, A295501, A366603, A366604.
Cf. A046798, A046801, A366575, A366612, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

Programs

  • Maple
    a:=n->numtheory[tau](4^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[0,4^Range[100]-1] (* Paolo Xausa, Oct 14 2023 *)
  • PARI
    a(n) = numdiv(4^n-1);

Formula

a(n) = sigma0(4^n-1) = A000005(A024036(n)).
a(n) = A046801(2*n) = A046798(n) * A046801(n). - Max Alekseyev, Jan 07 2024

A366603 Sum of the divisors of 4^n-1.

Original entry on oeis.org

4, 24, 104, 432, 1536, 8736, 22528, 111456, 473600, 1999872, 5909760, 38054016, 89522176, 462274560, 2015330304, 7304603328, 22907191296, 166290432000, 366506672128, 2220409884672, 7645340651520, 29833839544320, 95821839806976, 648494317126656
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(4)=432 because 4^4-1 has divisors {1, 3, 5, 15, 17, 51, 85, 255}.

Links

Crossrefs

Cf. A024036, A000203, A057957, A069061, A075708, A295501, A366602, A366604.
Cf. A075708, A366576, A366613, A366622, A366634, A366653, A366662, A102146, A366684, A366710.

Programs

  • Maple
    a:=n->numtheory[sigma](4^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[1,4^Range[30]-1] (* Paolo Xausa, Oct 14 2023 *)

Formula

a(n) = sigma(4^n-1) = A000203(A024036(n)).
a(n) = A069061(n) * A075708(n). - Robert Israel, Nov 22 2023

A027695 Number of primitive polynomials of degree n over GF(4).

Original entry on oeis.org

1, 2, 4, 12, 32, 120, 288, 1512, 4096, 15552, 48000, 240064, 552960, 3439800, 9483264, 35640000, 134217728, 673699800, 1451188224, 9644765256, 23685120000, 115605729792, 401556013056, 1996264531840, 4566087106560, 26244000000000, 89961392102400, 356237685227520
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=4 of A369291.
Cf. A027742, A295501.

Programs

  • Maple
    with(numtheory): seq(`if`(n=0, 1, phi(4^n-1)/n), n=0..27);
  • Mathematica
    Join[{1}, Array[EulerPhi[4^# - 1]/# &, 30]] (* Paolo Xausa, Jun 17 2024 *)
  • PARI
    a(n) = if(n==0, 1, eulerphi(4^n-1)/n) \\ Andrew Howroyd, Feb 01 2024

Formula

a(n) = A295501(n)/n = 2*A027742(n) for n >= 1. - Amiram Eldar, Nov 30 2024

Extensions

a(24) onwards from Andrew Howroyd, Feb 01 2024

A027742 a(n) = phi(4^n-1)/(2*n).

Original entry on oeis.org

1, 2, 6, 16, 60, 144, 756, 2048, 7776, 24000, 120032, 276480, 1719900, 4741632, 17820000, 67108864, 336849900, 725594112, 4822382628, 11842560000, 57802864896, 200778006528, 998132265920, 2283043553280, 13122000000000, 44980696051200, 178118842613760
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000010, A027695, A295501.

Programs

Formula

a(n) = A295501(n)/(2*n) = A027695(n)/2. - Amiram Eldar, Nov 30 2024

Extensions

Offset corrected by Sean A. Irvine, Dec 02 2019
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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