A055385 -id:A055385 - cp's OEIS frontend

A074966 a(n) = least k such that n^n + k is prime.

1, 1, 2, 1, 12, 7, 4, 43, 10, 19, 62, 35, 16, 27, 28, 13, 74, 107, 18, 91, 32, 87, 14, 95, 96, 43, 68, 135, 120, 19, 58, 7, 58, 63, 54, 31, 42, 115, 10, 157, 110, 13, 4, 403, 122, 457, 534, 37, 18, 31, 226, 253, 20, 193, 102, 177, 392, 45, 194, 257, 102, 79, 454, 231, 306

Offset: 1

Zak Seidov, Oct 03 2002

nonn

"Are there any n except 1, 2 and 4 that make n^n + 1 a prime? He [Sierpiński] has shown that if such a prime exists it is greater than 10^30000." - Ogilvy and Anderson

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

Hans Havermann and Seiichi Manyama, Table of n, a(n) for n = 1..2500 (Seiichi Manyama to 500)

Cf. A055385, A074967.

Array[Block[{k = 1}, While[! PrimeQ[k + #], k++]; k] &[#^#] &, 65] (* Michael De Vlieger, Jul 06 2018 *)

a(n)=(x->nextprime(x)-x)(n^n) \\ Charles R Greathouse IV, Nov 25 2014

More terms from Robert G. Wilson v, Oct 04 2002

Name taken from Comments section by Jon E. Schoenfield, Jan 14 2015

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

Original entry on oeis.org

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

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

A366820 a(n) is the sum of the divisors of n^n + 1.

Original entry on oeis.org

3, 3, 6, 56, 258, 6264, 52136, 1559520, 17041416, 706911048, 10102223208, 706019328000, 9101898907920, 519285252355776, 11672709747324912, 880565163670372352, 18446811354131136516, 1792353900753729655758, 54357680125881245248800, 4154723599066412190910560

Offset: 0

Sean A. Irvine, Oct 24 2023

nonn

Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind

Cf. A000203, A007571, A014566, A055385, A062727, A344859, A366819.

{3}~Join~Array[DivisorSigma[1, #^# + 1] &, 19] (* Michael De Vlieger, Oct 24 2023 *)

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

a(n) = A000203(A014566(n)).

cp's OEIS Frontend

A074966 a(n) = least k such that n^n + k is prime.

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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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Extensions

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

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Author

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Comments

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Crossrefs

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Mathematica

PARI

Formula

A366820 a(n) is the sum of the divisors of n^n + 1.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula