A046798 -id:A046798 - cp's OEIS frontend

A062319 Number of divisors of n^n, or of A000312(n).

1, 1, 3, 4, 9, 6, 49, 8, 25, 19, 121, 12, 325, 14, 225, 256, 65, 18, 703, 20, 861, 484, 529, 24, 1825, 51, 729, 82, 1653, 30, 29791, 32, 161, 1156, 1225, 1296, 5329, 38, 1521, 1600, 4961, 42, 79507, 44, 4005, 4186, 2209, 48, 9457, 99, 5151, 2704, 5565, 54

Offset: 0

Jason Earls, Jul 05 2001

From Gus Wiseman, May 02 2021: (Start)
Conjecture: The number of divisors of n^n equals the number of pairwise coprime ordered n-tuples of divisors of n. Confirmed up to n = 30. For example, the a(1) = 1 through a(5) = 6 tuples are:
  (1)  (1,1)  (1,1,1)  (1,1,1,1)  (1,1,1,1,1)
       (1,2)  (1,1,3)  (1,1,1,2)  (1,1,1,1,5)
       (2,1)  (1,3,1)  (1,1,1,4)  (1,1,1,5,1)
              (3,1,1)  (1,1,2,1)  (1,1,5,1,1)
                       (1,1,4,1)  (1,5,1,1,1)
                       (1,2,1,1)  (5,1,1,1,1)
                       (1,4,1,1)
                       (2,1,1,1)
                       (4,1,1,1)
The unordered case (pairwise coprime n-multisets of divisors of n) is counted by A343654.
(End)

			From _Gus Wiseman_, May 02 2021: (Start)
The a(1) = 1 through a(5) = 6 divisors:
  1  1  1   1    1
     2  3   2    5
     4  9   4    25
        27  8    125
            16   625
            32   3125
            64
            128
            256
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)

Cf. A046798, A077592, A035116. - Enrique Pérez Herrero, Nov 09 2010
Number of divisors of A000312(n).
Taking Omega instead of sigma gives A066959.
Positions of squares are A173339.
Diagonal n = k of the array A343656.
A000005 counts divisors.
A059481 counts k-multisets of elements of {1..n}.
A334997 counts length-k strict chains of divisors of n.
A343658 counts k-multisets of divisors.
Pairwise coprimality:
- A018892 counts coprime pairs of divisors.
- A084422 counts pairwise coprime subsets of {1..n}.
- A100565 counts pairwise coprime triples of divisors.
- A225520 counts pairwise coprime sets of divisors.
- A343652 counts maximal pairwise coprime sets of divisors.
- A343653 counts pairwise coprime non-singleton sets of divisors > 1.
- A343654 counts pairwise coprime sets of divisors > 1.
Cf. A000169, A000272, A002064, A002109, A009998, A048691, A143773, A146291, A176029, A327527, A343657.

[NumberOfDivisors(n^n): n in  [0..60]]; // Vincenzo Librandi, Nov 09 2014

A062319[n_IntegerQ]:=DivisorSigma[0,n^n]; (* Enrique Pérez Herrero, Nov 09 2010 *)
Join[{1},DivisorSigma[0,#^#]&/@Range[60]] (* Harvey P. Dale, Jun 06 2024 *)

je=[]; for(n=0,200,je=concat(je,numdiv(n^n))); je

{ for (n=0, 1000, write("b062319.txt", n, " ", numdiv(n^n)); ) } \\ Harry J. Smith, Aug 04 2009

a(n)=local(fm);fm=factor(n);prod(k=1,matsize(fm)[1],fm[k,2]*n+1) \\ Franklin T. Adams-Watters, May 03 2011

a(n) = if(n==0, 1, sumdiv(n, d, n^omega(d))); \\ Seiichi Manyama, May 12 2021

from math import prod
from sympy import factorint
def A062319(n): return prod(n*d+1 for d in factorint(n).values()) # Chai Wah Wu, Jun 03 2021

a(n) = A000005(A000312(n)). - Enrique Pérez Herrero, Nov 09 2010
a(2^n) = A002064(n). - Gus Wiseman, May 02 2021
a(prime(n)) = prime(n) + 1. - Gus Wiseman, May 02 2021
a(n) = Product_{i=1..s} (1 + n * m_i) where (m_1,...,m_s) is the sequence of prime multiplicities (prime signature) of n. - Gus Wiseman, May 02 2021
a(n) = Sum_{d|n} n^omega(d) for n > 0. - Seiichi Manyama May 12 2021

A069061 Sum of divisors of 2^n+1.

Original entry on oeis.org

4, 6, 13, 18, 48, 84, 176, 258, 800, 1302, 2736, 4356, 10928, 20520, 51792, 65538, 174768, 351120, 699056, 1110276, 3100240, 5048232, 11184816, 17041416, 49012992, 82623888, 211053040, 284225796, 727960800, 1494039792, 2863311536, 4301668356, 12611914848, 20788904016

Offset: 1

Benoit Cloitre, Apr 04 2002

nonn

Max Alekseyev, Table of n, a(n) for n = 1..1128 (terms 1..1062 from Amiram Eldar)

Cf. A000051, A000203, A002185, A002587, A002589, A046798, A046799, A053285, A054992, A057957, A059886, A085029, A086257, A112092, A250291, A274906, A295501.

Row sums of A374237.

DivisorSigma[1, 2^Range[50] + 1] (* Paolo Xausa, Jul 05 2024 *)

a(n) = sigma(2^n+1); \\ Michel Marcus, Nov 24 2013

a(n) = sigma(2^n+1).

a(n) = A000203(A000051(n)). - Michel Marcus, Nov 24 2013

More terms from Amiram Eldar, Oct 04 2019

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)).

A053285 Totient of 2^n+1.

Original entry on oeis.org

1, 2, 4, 6, 16, 20, 48, 84, 256, 324, 800, 1364, 3840, 5460, 12544, 19800, 65536, 87380, 186624, 349524, 986880, 1365336, 3345408, 5592404, 16515072, 20250000, 52306176, 84768120, 252645120, 351847488, 760320000, 1431655764, 4288266240, 5632621632, 13628740608

Offset: 0

Labos Elemer, Mar 03 2000

nonn

			It is a power of 2 iff n is a Fermat prime.

Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 0..300 from Robert Israel; terms 301..1062 from Amiram Eldar)

Cf. A000010, A000225, A002587, A000051, A046798, A046799, A049479, A051953, A054992.

[EulerPhi(2^n+1) : n in [1..40]]; // Vincenzo Librandi, Aug 12 2015

seq(numtheory:-phi(2^n+1), n=0..50); # Robert Israel, Aug 12 2015

Table[EulerPhi[2^n + 1], {n, 35}] (* Vincenzo Librandi, Aug 12 2015 *)

vector(40, n, eulerphi(2^n+1)) \\ Michel Marcus, Aug 12 2015

a(n) = A000010(A000051(n)).

a(0)=1 prepended by Alois P. Heinz, Aug 12 2015

A344897 a(n) is the number of divisors of 10^n + 1.

Original entry on oeis.org

2, 2, 2, 8, 4, 4, 4, 4, 4, 32, 8, 24, 8, 8, 16, 128, 32, 16, 8, 4, 16, 192, 16, 32, 8, 32, 8, 128, 16, 8, 128, 4, 16, 384, 16, 32, 64, 16, 8, 768, 16, 8, 128, 16, 16, 4096, 16, 16, 512, 16, 128, 256, 16, 4, 64, 768, 32, 64, 32, 16, 64, 8, 8, 3072, 8, 64, 256, 4, 16, 1024, 2048, 8, 32, 16, 128, 2048, 64, 3072, 128, 16

Offset: 0

Seiichi Manyama, Jun 01 2021

nonn

a(n) is even because 10^n + 1 is not a square number.

Max Alekseyev, Table of n, a(n) for n = 0..331

Cf. A000005, A000533, A003021, A038371, A046798, A057934, A070528, A119704, A344859, A087021, A087022,

a[0] = 2; a[n_] := DivisorSigma[0, 10^n + 1]; Array[a, 60, 0] (* Amiram Eldar, Jun 01 2021 *)

PARI
```
a(n) = numdiv(10^n+1);
```

a(n) = A000005(A000533(n)).

A366577 Number of divisors of 3^n+1.

Original entry on oeis.org

2, 3, 4, 6, 4, 6, 8, 6, 8, 24, 12, 12, 8, 6, 16, 48, 4, 24, 16, 12, 8, 72, 16, 6, 64, 24, 16, 96, 16, 24, 48, 12, 4, 96, 16, 24, 16, 24, 16, 192, 32, 12, 128, 6, 32, 768, 16, 24, 16, 24, 128, 384, 16, 12, 32, 96, 64, 192, 16, 12, 128, 12, 32, 4608, 4, 24, 64

Offset: 0

Sean A. Irvine, Oct 13 2023

nonn

			a(4)=4 because 3^4+1 has divisors {1, 2, 41, 82}.

Max Alekseyev, Table of n, a(n) for n = 0..691

Cf. A000005, A002592, A034472, A046798, A057935, A057941, A074476, A274909, A366575, A366578, A366579, A366580

a:=n->numtheory[tau](3^n+1):
seq(a(n), n=0..100);

DivisorSigma[0,3^Range[0,100]+1] (* Paolo Xausa, Oct 15 2023 *)

a(n) = numdiv(3^n+1); \\ Michel Marcus, Oct 14 2023

a(n) = sigma0(3^n+1) = A000005(A034472(n)).

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

Sean A. Irvine, Oct 14 2023

nonn

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

Max Alekseyev, Table of n, a(n) for n = 1..1128

Cf. A024036, A000005, A057957, A295501, A366603, A366604.

Cf. A046798, A046801, A366575, A366612, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

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

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

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

a(n) = sigma0(4^n-1) = A000005(A024036(n)).

a(n) = A046801(2*n) = A046798(n) * A046801(n). - Max Alekseyev, Jan 07 2024

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)).

cp's OEIS Frontend

A062319 Number of divisors of n^n, or of A000312(n).

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A069061 Sum of divisors of 2^n+1.

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A366714 Number of divisors of 12^n+1.

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A053285 Totient of 2^n+1.

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A344897 a(n) is the number of divisors of 10^n + 1.

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A366577 Number of divisors of 3^n+1.

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Maple

Mathematica

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A366602 Number of divisors of 4^n-1.

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