A057937 -id:A057937 - cp's OEIS frontend

A054992 Number of prime factors of 2^n + 1 (counted with multiplicity).

1, 1, 2, 1, 2, 2, 2, 1, 4, 3, 2, 2, 2, 3, 4, 1, 2, 4, 2, 2, 4, 3, 2, 3, 4, 4, 6, 2, 3, 6, 2, 2, 5, 4, 5, 4, 3, 4, 4, 2, 3, 6, 2, 3, 7, 5, 3, 3, 3, 7, 6, 3, 3, 6, 6, 3, 5, 3, 4, 4, 2, 5, 7, 2, 6, 6, 3, 4, 5, 7, 3, 5, 3, 5, 7, 4, 6, 10, 2, 3, 10, 5, 6, 5, 4, 5, 5, 4, 4, 11, 6, 2, 5, 4, 5, 3, 5, 6, 9, 6, 2, 9, 3

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

nonn

The length of row n in A001269.

			a(3) = 2 because 2^3 + 1 = 9 = 3*3.

Max Alekseyev, Table of n, a(n) for n = 1..1128
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), this sequence (b=2).
Cf. A000051, A002586, A002587, A003260, A001222, A001269, A001348, A054988, A054989, A054990, A054991, A000978.
Cf. A046051 (number of prime factors of 2^n-1).
Cf. A086257 (number of primitive prime factors).

a[n_] := Module[{x=FactorInteger[2^n+1]}, Sum[x[[i]][[2]], {i, Length[x]}]]
A054992[n_Integer] := PrimeOmega[2^n + 1]; Table[A054992[n], {n,103}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

a(n)=bigomega(2^n+1) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = A046051(2n) - A046051(n). - T. D. Noe, Jun 18 2003

a(n) = A001222(A000051(n)). - Amiram Eldar, Oct 04 2019

Extended by Patrick De Geest, Oct 01 2000
Terms to a(500) in b-file from T. D. Noe, Nov 10 2007
Deleted duplicate (and broken) Wagstaff link. - N. J. A. Sloane, Jan 18 2019
a(500)-a(1062) in b-file from Amiram Eldar, Oct 04 2019
a(1063)-a(1128) in b-file from Max Alekseyev, Jul 15 2023, Mar 15 2025

A057934 Number of prime factors of 10^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 3, 2, 2, 2, 2, 2, 5, 3, 5, 3, 3, 4, 7, 5, 4, 3, 2, 4, 8, 4, 5, 3, 5, 3, 7, 4, 3, 7, 2, 4, 9, 4, 5, 6, 4, 3, 10, 4, 3, 7, 4, 4, 12, 4, 4, 9, 4, 7, 8, 4, 2, 6, 10, 5, 6, 5, 4, 6, 3, 3, 12, 3, 6, 8, 2, 4, 10, 11, 3, 5, 4, 7, 11, 6, 12, 7, 4, 9, 11, 3, 7, 8, 8, 3, 8, 4, 4, 11, 6, 4, 8, 4, 6, 8, 4, 5, 13

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

2^(a(2n)-1)-1 predicts the number of pair-solutions of even length L for AB = A^2 + B^2. For instance, with length 18 we have 10^18 + 1 = 101*9901*999999000001 or 3 divisors F which when put into the Mersenne formula 2^(F-1)-1 yields 3 pairs (see reference 'Puzzle 104' for details).

Max Alekseyev, Table of n, a(n) for n = 1..331 (first 310 terms from Ray Chandler)
Makoto Kamada, Factorizations of 100...001.
Carlos Rivera, Puzzle 104, The Prime Puzzles & Problems Connection.
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): this sequence (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A003021, A001271, A046053, A001562, A057951, A119704, A344897.

PrimeOmega[10^Range[100]+1] (* Harvey P. Dale, May 02 2021 *)

a(n)=bigomega(10^n+1) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A057951(2n) - A057951(n). - T. D. Noe, Jun 19 2003

A057935 Number of prime factors of 9^n + 1 (counted with multiplicity).

Original entry on oeis.org

2, 2, 3, 3, 4, 3, 4, 2, 4, 3, 4, 6, 4, 4, 6, 2, 4, 4, 4, 5, 7, 5, 4, 4, 8, 4, 5, 6, 4, 7, 5, 2, 6, 5, 9, 8, 5, 6, 7, 5, 5, 10, 7, 6, 9, 4, 4, 6, 9, 6, 8, 7, 6, 9, 8, 9, 9, 5, 3, 11, 6, 4, 11, 6, 8, 9, 9, 8, 6, 9, 5, 6, 6, 6, 13, 4, 8, 7, 5, 4, 7, 6, 5, 11, 8, 5, 8, 7, 4, 11, 7, 9, 9, 5, 9, 7, 5, 6, 10, 7, 6

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..345 (first 329 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), this sequence (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A002592, A046053, A001222, A057941, A057952, A062396.

f:=func; [f(9^n + 1):n in [1..100]]; // Marius A. Burtea, Feb 02 2020

PrimeOmega[Table[9^n + 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)

a(n) = A057952(2n) - A057952(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A062396(n)) = A057941(2*n). - Amiram Eldar, Feb 02 2020

A057941 Number of prime factors of 3^n + 1 (counted with multiplicity).

Original entry on oeis.org

2, 2, 3, 2, 3, 3, 3, 3, 5, 4, 4, 3, 3, 4, 6, 2, 5, 4, 4, 3, 7, 4, 3, 6, 5, 4, 7, 4, 5, 6, 4, 2, 7, 4, 5, 4, 5, 4, 8, 5, 4, 7, 3, 5, 10, 4, 5, 4, 5, 8, 9, 4, 4, 5, 7, 6, 8, 4, 4, 7, 4, 5, 13, 2, 5, 6, 4, 5, 9, 9, 7, 8, 4, 5, 12, 6, 6, 7, 5, 5, 12, 5, 6, 10, 9, 7, 11, 6, 5, 9, 8, 4, 9, 4, 8, 6, 5, 9, 14, 6, 4

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..691 (first 658 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), this sequence (b=3), A054992 (b=2).

Cf. A001222, A007658, A034472, A057958, A074476.

PrimeOmega[3^Range[110]+1] (* Harvey P. Dale, Jun 20 2015 *)

a(n)=bigomega(n^3+1) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A057958(2n) - A057958(n) - T. D. Noe, Jun 19 2003

a(n) = A001222(A034472(n)). - Amiram Eldar, Feb 01 2020

A057936 Number of prime factors of 8^n + 1 (counted with multiplicity).

Original entry on oeis.org

2, 2, 4, 2, 4, 4, 4, 3, 6, 6, 5, 4, 4, 6, 7, 3, 6, 6, 5, 4, 7, 6, 5, 5, 7, 10, 10, 5, 5, 11, 5, 3, 9, 9, 11, 6, 7, 8, 7, 6, 7, 10, 6, 7, 12, 8, 7, 7, 7, 14, 11, 5, 6, 10, 12, 8, 9, 8, 8, 8, 4, 9, 13, 4, 11, 12, 8, 9, 8, 15, 8, 8, 6, 10, 12, 8, 12, 17, 6, 7, 15, 10, 9, 12, 12, 10, 11, 8, 8, 18, 12

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..502 (first 354 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), this sequence (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A057953, A062395, A274905.

f:=func; [f(8^n + 1):n in [1..110]]; // Marius A. Burtea, Feb 02 2020

PrimeOmega[8^Range[100]+1] (* Harvey P. Dale, Dec 16 2014 *)

a(n) = A057953(2n) - A057953(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A062395(n)) = A054992(3*n). - Amiram Eldar, Feb 02 2020

A057939 Number of prime factors of 5^n + 1 (counted with multiplicity).

Original entry on oeis.org

2, 2, 4, 2, 3, 3, 4, 3, 6, 4, 5, 3, 4, 3, 7, 3, 4, 5, 5, 4, 10, 4, 4, 4, 5, 5, 10, 3, 4, 7, 5, 4, 9, 6, 7, 6, 5, 4, 8, 5, 6, 6, 6, 3, 10, 3, 5, 5, 7, 7, 10, 5, 5, 6, 7, 7, 9, 3, 6, 6, 6, 4, 16, 4, 8, 7, 3, 7, 9, 7, 5, 6, 5, 5, 13, 5, 9, 10, 6, 6, 14, 6, 5, 7, 9, 5, 9, 7, 5, 12, 8, 4, 10, 6, 9, 7, 7, 7, 12

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..471 (first 451 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), this sequence (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A034474, A057956, A074478.

PrimeOmega[5^Range[100]+1] (* Harvey P. Dale, Nov 27 2013 *)

a(n) = A057956(2n) - A057956(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A034474(n)). - Amiram Eldar, Feb 01 2020

A057940 Number of prime factors of 4^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 1, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 3, 5, 3, 7, 3, 6, 3, 3, 4, 5, 2, 6, 4, 7, 5, 5, 4, 10, 3, 5, 5, 5, 4, 11, 2, 4, 3, 6, 6, 9, 2, 4, 6, 7, 5, 8, 3, 7, 6, 6, 4, 10, 2, 10, 7, 6, 4, 8, 4, 6, 7, 5, 2, 14, 4, 9, 5, 4, 4, 10, 4, 6, 8, 11, 4, 8, 3, 4, 8, 11, 4, 9, 5, 10, 4, 9, 8, 12, 6

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..583 (first 531 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), this sequence (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A052539, A057957, A274903.

with(numtheory); A057940:=n->bigomega(4^n + 1); seq(A057940(n), n=1..100); # Wesley Ivan Hurt, Jan 28 2014

Table[PrimeOmega[4^n + 1], {n, 100}] (* Wesley Ivan Hurt, Jan 28 2014 *)

a(n) = A057957(2n) - A057957(n). - T. D. Noe, Jun 19 2003
a(n) = Omega(4^n + 1) = A001222(A052539(n)). - Wesley Ivan Hurt, Jan 28 2014
a(n) = A054992(2*n). - Amiram Eldar, Feb 01 2020

A057938 Number of prime factors of 6^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 4, 2, 3, 3, 2, 2, 4, 3, 5, 3, 3, 6, 3, 3, 6, 4, 3, 4, 4, 4, 5, 5, 4, 9, 2, 3, 6, 3, 11, 5, 4, 3, 9, 5, 4, 7, 2, 3, 7, 5, 2, 7, 7, 6, 8, 4, 5, 10, 8, 6, 7, 3, 2, 6, 3, 2, 10, 3, 8, 11, 5, 5, 6, 7, 4, 5, 6, 5, 10, 5, 6, 11, 6, 6, 8, 4, 5, 8, 8, 5, 11, 4, 3, 16, 12, 4, 5, 5, 10, 4, 4, 5

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..430 (first 408 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), this sequence (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A057955, A062394, A274904.

f:=func; [f(6^n + 1):n in [1..100]]; // Marius A. Burtea, Feb 02 2020

PrimeOmega[6^Range[100]+1] (* Harvey P. Dale, Mar 10 2013 *)

a(n) = A057955(2n) - A057955(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A062394(n)). - Amiram Eldar, Feb 02 2020

A366638 Sum of the divisors of 7^n+1.

Original entry on oeis.org

3, 15, 93, 660, 3606, 34560, 236964, 1559520, 9155916, 77423280, 530807472, 3868683120, 21224771760, 185094572580, 1261494915594, 9988783073280, 49990612274316, 436182213726030, 3279858902194056, 21372989348391720, 122709716651985624, 1082323574100172800

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

			a(4)=3606 because 7^4+1 has divisors {1, 2, 1201, 2402}.

Max Alekseyev, Table of n, a(n) for n = 0..387

Cf. A034491, A000203, A057937, A227575, A366634, A366636, A366637, A366639.

Cf. A069061, A366578, A366607, A366617, A366629, A366657, A366666, A366668, A366689, A366715.

a:=n->numtheory[sigma](7^n+1):
seq(a(n), n=0..100);

DivisorSigma[1, 7^Range[0, 21] + 1] (* Paul F. Marrero Romero, Oct 16 2023 *)

a(n) = sigma(7^n+1) = A000203(A034491(n)).

A366639 a(n) = phi(7^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 4, 20, 168, 1200, 7600, 43200, 407680, 2712832, 19707408, 112560000, 945677920, 6768230400, 47530457728, 271289229120, 2096760960000, 16569393144832, 116315256993600, 597938524646400, 5699431359135360, 38890647857280000, 270061302781670400

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..387

Cf. A034491, A000010, A057937, A227575, A366635, A366636, A366637, A366638.

Cf. A053285, A366579, A366608, A366618, A366630, A366658, A366667, A366669, A366690, A366716.

EulerPhi[7^Range[0,21] + 1] (* Paul F. Marrero Romero, Nov 05 2023 *)

PARI
```
{a(n) = eulerphi(7^n+1)}
```

a(n) = A000010(A034491(n)). - Paul F. Marrero Romero, Nov 06 2023

cp's OEIS Frontend

A054992 Number of prime factors of 2^n + 1 (counted with multiplicity).

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A057934 Number of prime factors of 10^n + 1 (counted with multiplicity).

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A057935 Number of prime factors of 9^n + 1 (counted with multiplicity).

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Magma

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A057941 Number of prime factors of 3^n + 1 (counted with multiplicity).

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A057936 Number of prime factors of 8^n + 1 (counted with multiplicity).

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Magma

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A057939 Number of prime factors of 5^n + 1 (counted with multiplicity).

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A057940 Number of prime factors of 4^n + 1 (counted with multiplicity).

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A057938 Number of prime factors of 6^n + 1 (counted with multiplicity).

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