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.

A366605 Number of distinct prime divisors of 4^n + 1.

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A001221, A052539, A057940, A274903, A366604, A366606, A366607, A366608, A366609.
Cf. A046799, A366580, A366615, A366627, A366636, A366655, A366664, A119704, A366686, A366712.

Programs

  • Mathematica
    PrimeNu[4^Range[0,100]+1] (* Paolo Xausa, Oct 14 2023 *)
  • PARI
    for(n = 0, 100, print1(omega(4^n + 1), ", "))
  • Python
    from sympy import primenu
def A366605(n): return primenu((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

Formula

a(n) = omega(4^n+1) = A001221(A052539(n)).
a(n) = A046799(2*n). - Max Alekseyev, Jan 08 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

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)).

A366656 Number of divisors of 8^n+1.

Original entry on oeis.org

2, 3, 4, 8, 4, 12, 16, 12, 8, 20, 48, 24, 16, 12, 64, 64, 8, 48, 64, 24, 16, 64, 64, 24, 32, 96, 768, 192, 32, 24, 1536, 24, 8, 256, 512, 1536, 64, 96, 256, 64, 64, 96, 1024, 48, 128, 1280, 256, 96, 128, 96, 8192, 1024, 32, 48, 1024, 2304, 256, 192, 256, 192
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Examples

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

Links

Crossrefs

Cf. A062395, A000005, A057936, A274905, A366653, A366638, A366655, A366657, A366658.
Cf. A046798, A366577, A366606, A366616, A366628, A366637, A366665, A344897, A366688, A366714.

Programs

  • Maple
    a:=n->numtheory[tau](8^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[0, 8^Range[0,59] + 1] (* Paul F. Marrero Romero, Nov 12 2023 *)
  • PARI
    a(n) = numdiv(8^n+1);

Formula

a(n) = sigma0(8^n+1) = A000005(A062395(n)).
a(n) = A046798(3*n). - Max Alekseyev, Jan 09 2024

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

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

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
Showing 1-9 of 9 results.

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