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.

A366714 Number of divisors of 12^n+1.

Original entry on oeis.org

2, 2, 4, 8, 4, 4, 8, 8, 8, 32, 12, 4, 16, 24, 16, 128, 4, 8, 32, 16, 64, 384, 64, 16, 64, 64, 32, 1024, 8, 8, 48, 8, 4, 512, 16, 32, 128, 16, 32, 1536, 16, 32, 64, 32, 16, 4096, 8, 32, 32, 32, 512, 512, 32, 32, 1024, 128, 512, 1536, 192, 64, 1024, 32, 64
Offset: 0

Views

Author

Sean A. Irvine, Oct 17 2023

Keywords

Examples

			a(4)=4 because 12^4+1 has divisors {1, 89, 233, 20737}.

Links

Crossrefs

Cf. A178248, A000005, A046798, A344897, A366709, A366712, A366713, A366715, A366716, A366719, A366720, A366688.

Programs

  • Maple
    a:=n->numtheory[tau](12^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[0, 12^Range[0, 70] + 1] (* Paolo Xausa, Apr 20 2025 *)
  • PARI
    a(n) = numdiv(12^n+1);

Formula

a(n) = sigma0(12^n+1) = A000005(A178248(n)).

A366716 a(n) = phi(12^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 12, 112, 1296, 20416, 229680, 2306304, 32916240, 400515072, 3863116800, 47825825600, 685853880624, 8732596764672, 97509650382144, 990242755633152, 11148606564480000, 184883057981234176, 2047145911595946000, 20281543142263603200, 294779525244632305920
Offset: 0

Views

Author

Sean A. Irvine, Oct 17 2023

Keywords

Links

Crossrefs

Cf. A178248, A000010, A366711, A366712, A366713, A366714, A366715, A366719, A366720, A366690.

Programs

  • Mathematica
    EulerPhi[12^Range[0,19] + 1] (* Paul F. Marrero Romero, Oct 27 2023 *)
  • PARI
    {a(n) = eulerphi(12^n+1)}

Formula

a(n) = A000010(A178248(n)). - Paul F. Marrero Romero, Oct 27 2023

A366715 Sum of the divisors of 12^n+1.

Original entry on oeis.org

3, 14, 180, 2240, 21060, 267988, 3706920, 38773952, 459970056, 6692483840, 79425033660, 800162860756, 9101898907920, 117326869641600, 1596198064568400, 20655000929239040, 184885459808838660, 2390210102271311936, 33504016991491136160, 344201347103878781440
Offset: 0

Views

Author

Sean A. Irvine, Oct 17 2023

Keywords

Examples

			a(4)=21060 because 12^4+1 has divisors {1, 89, 233, 20737}.

Links

Crossrefs

Cf. A178248, A000203, A366710, A366712, A366713, A366714, A366716, A366719, A366720, A366689.

Programs

  • Maple
    a:=n->numtheory[sigma](12^n+1):
seq(a(n), n=0..100);

Formula

a(n) = sigma(12^n+1) = A000203(A178248(n)).

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.

A366713 Number of prime factors of 12^n + 1 (counted with multiplicity).

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Oct 17 2023

Keywords

Links

Crossrefs

Cf. A178248, A001222, A054992, A057934, A057935, A057936, A057937, A057938, A057939, A057940, A057941, A366712, A366714, A366715, A366716, A366719, A366720, A366708, A366687.

Programs

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

Formula

a(n) = bigomega(12^n+1) = A001222(A178248(n)).

A366720 Largest prime factor of 12^n+1.

Original entry on oeis.org

2, 13, 29, 19, 233, 19141, 20593, 13063, 260753, 1801, 85403261, 57154490053, 2227777, 222379, 13156924369, 35671, 1200913648289, 66900193189411, 122138321401, 905265296671, 67657441, 1885339, 68368660537, 49489630860836437, 592734049, 438472201
Offset: 0

Views

Author

Sean A. Irvine, Oct 17 2023

Keywords

Links

Crossrefs

Cf. A178248, A006530, A002587, A074476, A274903, A074478, A274904, A227575, A274905, A002592, A003021, A062308, A002590, A366712, A366713, A366714, A366715, A366716, A366717, A366718, A366719, A324941.

Programs

  • Mathematica
    Table[FactorInteger[12^n + 1][[-1, 1]], {n, 0, 20}]

Formula

a(n) = A006530(A178248(n)). - Paul F. Marrero Romero, Dec 07 2023

A366670 Smallest prime dividing 6^n + 1.

Original entry on oeis.org

2, 7, 37, 7, 1297, 7, 13, 7, 17, 7, 37, 7, 1297, 7, 37, 7, 353, 7, 13, 7, 41, 7, 37, 7, 17, 7, 37, 7, 281, 7, 13, 7, 2753, 7, 37, 7, 577, 7, 37, 7, 17, 7, 13, 7, 89, 7, 37, 7, 193, 7, 37, 7, 1297, 7, 13, 7, 17, 7, 37, 7, 41, 7, 37, 7, 4926056449, 7, 13, 7, 137
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A020639, A062394, A274904, A366630.
Cf. A002586, A366609, A366671, A038371, A366719.

Programs

Formula

a(n) = A020639(A062394(n)). - Paul F. Marrero Romero, Oct 17 2023

A366671 Smallest prime dividing 8^n + 1.

Original entry on oeis.org

2, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 641, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 769, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Comments

a(n) = 3 if n is odd. a(n) = 5 if n == 2 (mod 4). - Robert Israel, Nov 20 2023

Links

Crossrefs

Cf. A002586, A366609, A052536, A366655, A274905, A366670, A038371, A366719.

Programs

  • Maple
    P1000:= mul(ithprime(i),i= 4..1000):
f:= proc(n) local t;
  if n::odd then return 3 elif n mod 4 = 2 then return 5 fi;
  t:= igcd(8^n+1,P1000);
  if t <> 1 then min(numtheory:-factorset(t)) else min(numtheory:-factorset(8^n+1)) fi
end proc:
map(f, [$0..100]); # Robert Israel, Nov 20 2023
  • Mathematica
    Table[FactorInteger[8^n + 1][[1,1]], {n, 0, 78}] (* Paul F. Marrero Romero, Oct 20 2023 *)
  • Python
    from sympy import primefactors
def A366671(n): return min(primefactors((1<<3*n)+1)) # Chai Wah Wu, Oct 16 2023

Formula

a(n) = A020639(A062395(n)). - Paul F. Marrero Romero, Oct 20 2023
a(n) = A002586(3*n) for n >= 1. - Robert Israel, Nov 20 2023

A366717 Smallest prime dividing 12^n - 1.

Original entry on oeis.org

11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11
Offset: 1

Views

Author

Sean A. Irvine, Oct 17 2023

Keywords

Comments

Periodic with period 12, repeat of 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5.

Crossrefs

Cf. A024140, A366718, A366719.

Programs

Formula

a(n) = A020639(A024140(n)). - Paul F. Marrero Romero, Oct 25 2023
Showing 1-9 of 9 results.

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