A344897 -id:A344897 - cp's OEIS frontend

A046798 Number of divisors of 2^n + 1.

2, 2, 2, 3, 2, 4, 4, 4, 2, 8, 6, 4, 4, 4, 8, 12, 2, 4, 16, 4, 4, 12, 8, 4, 8, 16, 16, 20, 4, 8, 48, 4, 4, 24, 16, 32, 16, 8, 16, 12, 4, 8, 64, 4, 8, 64, 32, 8, 8, 8, 64, 48, 8, 8, 64, 48, 8, 24, 8, 16, 16, 4, 32, 64, 4, 64, 64, 8, 12, 24, 96, 8, 32, 8, 32, 96, 16, 64, 768, 4, 8, 192, 32, 64

Offset: 0

Labos Elemer

nonn

a(n) is odd iff n = 3, as a consequence of the Catalan-Mihăilescu theorem. - Bernard Schott, Oct 05 2021

			a(7)=4, because 2^7 + 1 = 129 has 4 divisors.

Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 0..500 from T. D. Noe, terms 501..1062 from Amiram Eldar, term 1108 from Tyler Busby)
Wikipedia, Catalan's conjecture.

Cf. A000005, A000051, A002587, A002589, A046801, A054992, A057957, A059886, A274906, A344897.

[NumberOfDivisors(2^n+1): n in [0..100]]; // Vincenzo Librandi, Feb 05 2018

a:= n-> numtheory[tau](2^n+1):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 23 2021

A046798[n_IntegerQ]:=DivisorSigma[0,1+2^n]; (* Enrique Pérez Herrero, Nov 09 2010 *)
DivisorSigma[0, 1 + 2^#] & /@ Range[0, 83] (* Jayanta Basu, Jun 29 2013 *)
Table[DivisorSigma[0, 2^n + 1], {n, 0, 100}] (* Vincenzo Librandi, Feb 05 2018 *)

a(n) = numdiv(2^n+1); \\ Michel Marcus, Mar 18 2017

from sympy.ntheory import divisor_count
def A046798(n): return divisor_count(2**n + 1) # Indranil Ghosh, Mar 18 2017

a(n) = A000005(A000051(n)). - Michel Marcus, Mar 18 2017

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

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

Sean A. Irvine, Oct 17 2023

nonn

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

Max Alekseyev, Table of n, a(n) for n = 0..306

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

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

DivisorSigma[0, 12^Range[0, 70] + 1] (* Paolo Xausa, Apr 20 2025 *)

PARI
```
a(n) = numdiv(12^n+1);
```

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

A366688 Number of divisors of 11^n+1.

Original entry on oeis.org

2, 6, 4, 18, 4, 12, 16, 12, 8, 48, 8, 96, 16, 48, 32, 144, 8, 48, 32, 96, 16, 72, 16, 96, 128, 48, 8, 240, 64, 48, 64, 96, 16, 4608, 64, 1152, 128, 24, 16, 1152, 32, 48, 512, 24, 64, 3072, 64, 96, 32, 192, 64, 1152, 8, 96, 512, 6144, 128, 2304, 64, 96, 256, 48

Offset: 0

Sean A. Irvine, Oct 16 2023

nonn

			a(4)=4 because 11^4+1 has divisors {1, 2, 7321, 14642}.

Max Alekseyev, Table of n, a(n) for n = 0..325

Cf. A034524, A000005, A062308, A366683, A366686, A366687, A366689, A366690.

Cf. A046798, A344897, A366714.

a:=n->numtheory[tau](11^n+1):
seq(a(n), n=0..100);

DivisorSigma[0,11^Range[0,70]+1] (* Harvey P. Dale, Mar 17 2025 *)

PARI
```
a(n) = numdiv(11^n+1);
```

a(n) = sigma0(11^n+1) = A000005(A034524(n)).

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

Sean A. Irvine, Oct 14 2023

nonn

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

Max Alekseyev, Table of n, a(n) for n = 0..583

Cf. A000005, A052539, A057940, A274903, A366602, A366605, A366607, A366608, A366609.

Cf. A046798, A366577, A366616, A366628, A366637, A366656, A366665, A344897, A366688, A366714.

a:=n->numtheory[tau](4^n+1):
seq(a(n), n=0..100);

DivisorSigma[0,4^Range[0,100]+1] (* Paolo Xausa, Oct 14 2023 *)

PARI
```
a(n) = numdiv(4^n+1);
```

from sympy import divisor_count
def A366606(n): return divisor_count((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

a(n) = sigma0(4^n+1) = A000005(A052539(n)).

a(n) = A046798(2*n). - Max Alekseyev, Jan 08 2024

A366616 Number of divisors of 5^n+1.

Original entry on oeis.org

2, 4, 4, 12, 4, 8, 8, 16, 8, 32, 16, 32, 8, 16, 8, 96, 8, 16, 32, 32, 16, 576, 16, 16, 16, 32, 24, 320, 8, 16, 128, 32, 16, 384, 64, 128, 64, 32, 16, 192, 32, 64, 64, 64, 8, 512, 8, 32, 32, 128, 128, 768, 32, 32, 64, 128, 128, 384, 8, 64, 64, 64, 16, 24576, 16

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=12 because 5^3+1 has divisors {1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126}.

Max Alekseyev, Table of n, a(n) for n = 0..471

Cf. A000005, A034474, A057939, A074478, A366612, A366607, A366615, A366617, A366618.

Cf. A046798, A366577, A366606, A366628, A366637, A366656, A366665, A344897, A366688, A366714.

a:=n->numtheory[tau](5^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 5^Range[0, 70] + 1] (* Paolo Xausa, Apr 20 2025 *)

PARI
```
a(n) = numdiv(5^n+1);
```

a(n) = sigma0(5^n+1) = A000005(A034474(n)).

A366628 Number of divisors of 6^n+1.

Original entry on oeis.org

2, 2, 2, 4, 2, 8, 8, 12, 4, 8, 8, 4, 4, 16, 8, 32, 8, 8, 64, 8, 8, 48, 16, 8, 16, 16, 16, 32, 32, 16, 512, 4, 8, 64, 8, 1536, 32, 16, 8, 512, 32, 16, 128, 4, 8, 128, 32, 4, 128, 64, 64, 256, 16, 32, 1024, 192, 64, 128, 8, 4, 64, 8, 4, 768, 8, 256, 2048, 32, 32

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=4 because 6^3+1 has divisors {1, 7, 31, 217}.

Max Alekseyev, Table of n, a(n) for n = 0..430

Cf. A062394, A000005, A057938, A366621, A366627, A366629, A366630, A366670.

Cf. A046798, A366577, A366606, A366616, A366637, A366656, A366665, A344897, A366688, A366714.

a:=n->numtheory[tau](6^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 6^Range[0, 70] + 1] (* Paolo Xausa, Apr 19 2025 *)

PARI
```
a(n) = numdiv(6^n+1);
```

a(n) = sigma0(6^n+1) = A000005(A062394(n)).

A366637 Number of divisors of 7^n+1.

Original entry on oeis.org

2, 4, 6, 8, 4, 16, 24, 16, 8, 16, 32, 16, 32, 16, 12, 64, 8, 8, 48, 16, 16, 128, 48, 8, 16, 32, 24, 32, 64, 8, 512, 32, 16, 128, 48, 1024, 256, 16, 12, 256, 64, 64, 96, 512, 32, 2048, 96, 8, 64, 2048, 640, 128, 32, 64, 384, 3072, 256, 256, 96, 64, 512, 8, 48

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

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

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

Cf. A034491, A000005, A057937, A227575, A366633, A366636, A366638, A366639.

Cf. A046798, A366577, A366606, A366616, A366628, A366656, A366665, A344897, A366688, A366714.

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

DivisorSigma[0, 7^Range[0, 62] + 1] (* Paul F. Marrero Romero, Oct 16 2023 *)

PARI
```
a(n) = numdiv(7^n+1);
```

a(n) = sigma0(7^n+1) = A000005(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

Sean A. Irvine, Oct 15 2023

nonn

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

Max Alekseyev, Table of n, a(n) for n = 0..502

Cf. A062395, A000005, A057936, A274905, A366653, A366638, A366655, A366657, A366658.

Cf. A046798, A366577, A366606, A366616, A366628, A366637, A366665, A344897, A366688, A366714.

a:=n->numtheory[tau](8^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 8^Range[0,59] + 1] (* Paul F. Marrero Romero, Nov 12 2023 *)

PARI
```
a(n) = numdiv(8^n+1);
```

a(n) = sigma0(8^n+1) = A000005(A062395(n)).

a(n) = A046798(3*n). - Max Alekseyev, Jan 09 2024

A366665 Number of divisors of 9^n+1.

Original entry on oeis.org

2, 4, 4, 8, 8, 12, 8, 16, 4, 16, 8, 16, 64, 16, 16, 48, 4, 16, 16, 16, 32, 128, 32, 16, 16, 128, 16, 32, 64, 16, 128, 32, 4, 64, 32, 384, 256, 32, 64, 128, 32, 32, 1024, 128, 64, 384, 16, 16, 64, 512, 64, 256, 128, 64, 512, 192, 512, 512, 32, 8, 2048, 64, 16

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

			a(2)=4 because 9^2+1 has divisors {1, 2, 41, 82}.

Max Alekseyev, Table of n, a(n) for n = 0..345

Cf. A062396, A000005, A057935, A366661, A366664, A366666, A366667.

Cf. A046798, A366577, A366606, A366616, A366628, A366637, A366656, A344897, A366688, A366714.

a:=n->numtheory[tau](9^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 9^Range[0,62] + 1] (* Paul F. Marrero Romero, Nov 13 2023 *)

PARI
```
a(n) = numdiv(9^n+1);
```

a(n) = sigma0(9^n+1) = A000005(A062396(n)).

a(n) = A366577(2*n). - Max Alekseyev, Jan 08 2024

cp's OEIS Frontend

A046798 Number of divisors of 2^n + 1.

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

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A366714 Number of divisors of 12^n+1.

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A366688 Number of divisors of 11^n+1.

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A366606 Number of divisors of 4^n+1.

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A366616 Number of divisors of 5^n+1.

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A366628 Number of divisors of 6^n+1.

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