A014566 -id:A014566 - cp's OEIS frontend

A344859 a(n) is the number of divisors of n^n + 1.

2, 2, 2, 6, 2, 8, 8, 16, 8, 16, 8, 96, 16, 32, 48, 160, 4, 12, 288, 48, 8, 64, 16, 512, 64, 128, 32, 3072, 64, 128, 1024, 384, 16, 2048, 64, 18432, 32, 128, 192, 512, 768, 64, 1024, 384, 256, 16384, 256, 2560, 64, 192, 1024, 3072, 32, 512, 16384, 4096, 128, 8192, 8192, 768, 4096, 256, 128, 1376256, 16

Offset: 0

Seiichi Manyama, May 31 2021

nonn

Tyler Busby, Table of n, a(n) for n = 0..148 (terms 0..123 from Max Alekseyev)
Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind

Cf. A000005, A014566, A062319, A064144, A121270, A334167.

Cf. A007571 (Lpf), A055385 (lpf).

a[0] = 2; a[n_] := DivisorSigma[0, n^n + 1]; Array[a, 45, 0] (* Amiram Eldar, May 31 2021 *)

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

a(n) = A000005(A014566(n)).

A085723 Number of prime divisors of n^n+1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 5, 3, 4, 3, 7, 4, 5, 6, 9, 2, 4, 9, 6, 3, 6, 4, 10, 6, 7, 5, 12, 6, 7, 10, 11, 4, 11, 6, 15, 5, 7, 8, 10, 10, 6, 10, 9, 8, 14, 8, 13, 6, 8, 10, 12, 5, 10, 14, 13, 7, 13, 13, 10, 12, 8, 7, 24, 4, 12, 8, 8, 7, 17, 10, 11, 12, 4, 8, 25, 7, 9, 14, 10, 5, 12, 7, 13, 8

Offset: 1

Jason Earls, Jul 20 2003

16^16+1 = 274177 * 67280421310721 is a semiprime. Where is the next?
a(73) >= 4. - Donovan Johnson, Sep 27 2010
According to factordb there are currently no other known candidates for semiprimes, with 781^781+1 being the largest fully factored number of this form. - Hugo Pfoertner, Aug 24 2019

			a(3) = 3: 3^3 + 1 = 28 = 2^2 * 7.
a(4) = 1: 4^4 + 1 = 257 is prime.
a(5) = 3: 5^5 + 1 = 3126 = 2 * 3 * 521.

Amiram Eldar, Table of n, a(n) for n = 1..148 (terms 1..123 from Chai Wah Wu)
factordb, Factors of n^n+1.

Cf. A001222, A007571, A014566, A055385, A121270, A309941, A344869.

for(k=1, 60, print1(bigomega(k^k+1),", ")) \\ Hugo Pfoertner, Aug 24 2019

a(n) = A001222(A014566(n)). - Amiram Eldar, Sep 27 2024

More terms from Ray G. Opao, Aug 25 2004
Corrected 8 existing terms and a(46)-a(72) from Donovan Johnson, Sep 27 2010
a(73)-a(84) added by Hugo Pfoertner, Aug 24 2019

A116893 Numbers k such that gcd(k!+1, k^k+1) > 1.

Original entry on oeis.org

1, 3, 23, 39, 51, 63, 95, 99, 131, 183, 191, 215, 239, 251, 299, 303, 315, 363, 371, 411, 419, 431, 443, 495, 543, 575, 659, 683, 711, 743, 755, 791, 831, 891, 911, 935, 975, 1019, 1031, 1055, 1071, 1143, 1155, 1191, 1211, 1223, 1251, 1275, 1295, 1355

Offset: 1

Giovanni Resta, Mar 01 2006

See A116892 for the corresponding values of the GCD. See also comments in A116891.

			gcd(1!+1, 1^1+1) = 2, gcd(2!+1, 2^2+1) = 1 and gcd(3!+1, 3^3+1) = 7, so 1 and 3 are the first two terms of the sequence.

Nick Hobson, Table of n, a(n) for n = 1..10000 (first 1832 terms from Antti Karttunen)
Nick Hobson, C program.

Cf. A014566, A038507, A067658, A116891, A116892, A116894.

C
```
See Links section.
```

Select[Range[1500], (GCD[ #!+1, #^#+1] > 1)&]

isok(n) = gcd(n! + 1, n^n + 1) != 1; \\ Michel Marcus, Jul 22 2018

A344869 Number of distinct prime factors of n^n+1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 3, 3, 4, 3, 6, 4, 5, 5, 6, 2, 3, 7, 5, 3, 6, 4, 8, 6, 7, 5, 11, 6, 7, 10, 7, 4, 11, 6, 13, 5, 7, 7, 8, 9, 6, 10, 8, 8, 14, 8, 10, 6, 7, 10, 11, 5, 8, 14, 11, 7, 13, 13, 9, 12, 8, 7, 18, 4, 12, 8, 7, 7, 16, 9, 8, 12, 4, 8, 24, 7, 9, 14, 7, 5, 12, 6, 12, 8, 13, 10, 12, 10, 6, 23, 15, 6, 9, 11, 16, 3, 8, 17, 23, 7

Offset: 0

Seiichi Manyama, May 31 2021

nonn

Amiram Eldar, Table of n, a(n) for n = 0..148
Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind.

Cf. A001221, A014566, A085723, A128428, A344859, A344870.

Magma
```
[#PrimeDivisors(n^n+1): n in [0..100]];
```

a[0] = 1; a[n_] := PrimeNu[n^n + 1]; Array[a, 45, 0] (* Amiram Eldar, May 31 2021 *)

PARI
```
a(n) = omega(n^n+1);
```

a(n) = A001221(A014566(n)).

a(67)-a(79) from Jon E. Schoenfield, May 31 2021

a(80)-a(100) from Seiichi Manyama, May 31 2021

A116891 a(n) = gcd(n! + 1, n^n + 1).

Original entry on oeis.org

2, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 79, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 103, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 127, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 191, 1, 1, 1, 199, 1, 1

Offset: 1

Giovanni Resta, Mar 01 2006

Apparently all the values greater than 1 (cf. A116892) are prime numbers and are equal to 2n+1 with only 4 exceptions for n<82000 (cf. A116894).
From Antti Karttunen, Jul 22 2018: (Start)
The first duplicated value > 1 is 157519 = a(43755) = a(78759). Note that 43755 = 15*2917, while 78759 = 27*2917.
It seems that for the long time after a(1) = 2, all other terms > 1 occur only at such positions k that k+1 is not squarefree. However, this turns out to be false as a(208161) = 555097, and 208162 is a squarefree number.
(End)

			a(3) = gcd(3! + 1, 3^3 + 1) = gcd(7,28) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..80001

Cf. A014566, A038507, A067658, A116892, A116893, A116894.

Table[GCD[n! + 1, n^n + 1], {n, 101}] (* Robert G. Wilson v, Mar 09 2006 *)

A116891(n) = gcd(n!+1,(n^n)+1); \\ Antti Karttunen, Jul 22 2018

A116892 Values of gcd(k!+1, k^k+1), when greater than 1.

Original entry on oeis.org

2, 7, 47, 79, 103, 127, 191, 199, 263, 367, 383, 431, 479, 503, 599, 607, 631, 727, 743, 823, 839, 863, 887, 991, 1087, 1151, 1319, 1367, 1423, 1487, 1511, 1583, 1663, 1783, 1823, 1871, 1951, 2039, 2063, 2111, 2143, 2287, 2311, 2383, 2423, 2447, 2503, 2551

Offset: 1

Giovanni Resta, Mar 01 2006

Apart from the initial term (2) and few exceptional values (A116894) this sequence seems to coincide with A067658. The values of k for which the terms of this sequence are obtained are in A116893.

			gcd(1!+1,1^1+1) = 2 gives the first term;
gcd(3!+1,3^3+1) = gcd(7,28) = 7 gives the second, and so on.

Nick Hobson, Table of n, a(n) for n = 1..10000 (first 1832 terms from Antti Karttunen)

Cf. A014566, A038507, A067658, A116891, A116893, A116894.

C
```
See Links section in A116893.
```

f[n_] := GCD[n! + 1, n^n + 1]; t = Array[f, 1295]; Rest@ Union@ t (* Robert G. Wilson v, Mar 09 2006 *)

lista(nn) = for (n=1, nn, if ((g=gcd(n! + 1, n^n + 1)) != 1, print1(g, ", "))); \\ Michel Marcus, Jul 22 2018

Entries checked by Robert G. Wilson v, Mar 09 2006

A116894 Numbers k such that gcd(k! + 1, k^k + 1) is neither 1 nor 2k+1.

Original entry on oeis.org

1, 5427, 41255, 43755, 208161, 496175, 497135

Offset: 1

Giovanni Resta, Mar 01 2006

g(n) = gcd(n! + 1, n^n + 1) is almost always equal to 1 or to 2n+1. These are the known exceptions: g(1) = 2, g(5427) = 10453, g(41255) = 129341, g(43755) = 157519, g(208161) = 555097. - Hans Havermann, Mar 28 2006

a(8) > 1000000. - Nick Hobson, Feb 20 2024

			gcd(1! + 1, 1^1 + 1) = 2 and 2 != 2*1 + 1, so 1 belongs to the sequence.

Cf. A014566, A038507, A067658, A116891, A116892, A116893.

C
```
// See Links section in A116893.
```

a(5) from Hans Havermann, Mar 28 2006

a(6)-a(7) from Nick Hobson, Feb 20 2024

A160600 Numbers k such that 3*(2k)^(2k)+1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 143, 225

Offset: 1

M. F. Hasler, Jul 10 2009

This corresponds to the numbers such that 3m^m+1 is prime, but these must all be even, m=2k, and therefore it is more natural to record the sequence of k=m/2.

Next term > 15000. - Matevz Markovic, Oct 09 2012

			a(1) = 1, because 2^2*3+1 = 13 is the smallest prime of this form.
a(2) = 2, because 4^4*3+1 = 769 is the next smallest prime of this form. a(3) = 3, because 6^6*3+1 = 139969 is again a prime.

Cf. A160360 (3n^n+2 is prime), A121270 = primes among Sierpinski numbers A014566(n)=n^n+1; A216148 = A216147(A110932): primes 2n^n+1; A088790, A065798.

q:= k-> isprime(3*(2*k)^(2*k)+1):
select(q, [$1..225])[];  # Alois P. Heinz, Aug 04 2025

for(i=1,9999,ispseudoprime(i^i*3+1)&print1(i/2,","))

A359700 a(n) = Sum_{d|n} d^(d + n/d - 1).

Original entry on oeis.org

1, 5, 28, 265, 3126, 46754, 823544, 16778273, 387420733, 10000015690, 285311670612, 8916100733146, 302875106592254, 11112006831323074, 437893890380939688, 18446744073843786241, 827240261886336764178, 39346408075300026047027

Offset: 1

Seiichi Manyama, Jan 11 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A014566, A055225, A087909, A294956, A353013, A353014.

a[n_] := DivisorSum[n, #^(# + n/# - 1) &]; Array[a, 20] (* Amiram Eldar, Aug 14 2023 *)

PARI
```
a(n) = sumdiv(n, d, d^(d+n/d-1));
```

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x)^k/(1-k*x^k)))

G.f.: Sum_{k>0} (k * x)^k / (1 - k * x^k).

If p is prime, a(p) = 1 + p^p.

A055386 Smallest factor of (2n)^(2n) + 1.

Original entry on oeis.org

5, 257, 13, 97, 101, 89, 29, 274177, 5, 148721, 5, 17, 53, 449, 17, 641, 13, 17, 5, 17, 5, 41, 29, 769, 41, 89, 13, 17, 5, 17, 5, 59649589127497217, 37, 41, 13, 97, 149, 17, 5, 15361, 5, 1753, 13, 17, 41, 449, 1129, 1153, 5, 17, 5, 1201, 17, 1777, 89, 4993, 41

Offset: 1

Walter Nissen, Jun 24 2000

nonn

If we use the commonly accepted convention that 0^0 = 1, then a(0) = 2. - Chai Wah Wu, Jul 22 2019

			8^8 + 1 = 97 * 257 * 673, so a(4) = 97.

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory. Dover. New York: 1988. Page 82.

Cf. A014566, A055385.

Table[With[{k = 2 n}, FactorInteger[k^k + 1]][[1, 1]], {n, 1, 60, 1}] (* Vincenzo Librandi, Jul 23 2013 *)

a(n) = factor((2*n)^(2*n) + 1)[1, 1] \\ Michel Marcus, Jul 23 2013; corrected by Jason Yuen, Jun 01 2025

a(n) = A055385(2*n). - Michel Marcus, Jul 23 2013

cp's OEIS Frontend

A344859 a(n) is the number of divisors of n^n + 1.

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A085723 Number of prime divisors of n^n+1 (counted with multiplicity).

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A116893 Numbers k such that gcd(k!+1, k^k+1) > 1.

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C

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PARI

A344869 Number of distinct prime factors of n^n+1.

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Magma

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Extensions

A116891 a(n) = gcd(n! + 1, n^n + 1).

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Mathematica

PARI

A116892 Values of gcd(k!+1, k^k+1), when greater than 1.

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Examples

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Programs

C

Mathematica

PARI

Extensions

A116894 Numbers k such that gcd(k! + 1, k^k + 1) is neither 1 nor 2k+1.

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C