A121270 -id:A121270 - cp's OEIS frontend

A014566 Sierpiński numbers of the first kind: n^n + 1.

2, 2, 5, 28, 257, 3126, 46657, 823544, 16777217, 387420490, 10000000001, 285311670612, 8916100448257, 302875106592254, 11112006825558017, 437893890380859376, 18446744073709551617, 827240261886336764178, 39346408075296537575425, 1978419655660313589123980

Offset: 0

Eric W. Weisstein

Sierpiński primes of the form n^n + 1 are {2,5,257,...} = A121270. The prime p divides a((p-1)/2) for p = {5,7,13,23,29,31,37,47,53,61,71,...} = A003628 Primes congruent to {5, 7} mod 8. p^2 divides a((p-1)/2) for prime p = {29,37,3373,...}. - Alexander Adamchuk, Sep 11 2006

n divides a(n-1) for even n, or 2n divides a(2n-1). a(2n-1)/(2n) = A124899(n) = {1, 7, 521, 102943, 38742049, 23775972551, 21633936185161, 27368368148803711, 45957792327018709121, ...}. 2^n divides a(2^n-1). A014566[2^n - 1] / 2^n = A081216[2^n - 1] = A122000[n] = {1, 7, 102943, 27368368148803711, 533411691585101123706582594658103586126397951, ...}. p+1 divides a(p) for prime p. a(p)/(p+1) = A056852[n] = {7, 521, 102943, 23775972551, 21633936185161, ...}. p^2 divides a((p-1)/2) for prime p = {29, 37, 3373} = A121999(n). - Alexander Adamchuk, Nov 12 2006

Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Maohua Le, Primes in the sequences n^n+1 and n^n-1, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, pp. 156-157.
Paulo Ribenboim, The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 74, 1989.

M. F. Hasler, Table of n, a(n) for n = 0..100
Florentin Smarandache, Only Problems, Not Solutions!, Xiquan Publ. Hse., 1990, Problem 17.
Eric Weisstein's World of Mathematics, Sierpiński Number of the First Kind.

Cf. A000312, A048861, A121270, A003628, A122000, A081216, A056852, A121999, A124899, A134883.

a(0) = 2; for n>0 Table[n^n+1,{n,1,20}] (* Alexander Adamchuk, Sep 11 2006 *)

A014566[n]:=if n=0 then 2 else n^n+1$
makelist(A014566[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

A014566(n)=n^n+1 /* M. F. Hasler, Jan 21 2009 */

For n>0, resultant of x^n+1 and nx-1. - Ralf Stephan, Nov 20 2004
E.g.f.: exp(x) + 1/(1+LambertW(-x)). - Vaclav Kotesovec, Dec 20 2014
Sum_{n>=1} 1/a(n) = A134883. - Amiram Eldar, Nov 15 2020

More terms from Erich Friedman

A110932 Numbers k such that 2*k^k + 1 is prime.

Original entry on oeis.org

0, 1, 12, 18, 251, 82992

Offset: 0

Ray G. Opao, Sep 25 2005

As a "list of numbers such that ...", the sequence should have offset 1, but to preserve the validity of formulas referring to this sequence, the offset was set to 0 when the initial value a(0)=0 was added. - M. F. Hasler, Sep 02 2012

Cf. A110931, A121270 (= primes in A014566), A088790, A160360, A160600.

The primes 2n^n+1, for k<4, n=a(k)<251, are listed at A216148(k) = A216147(a(k)). - M. F. Hasler, Sep 02 2012

Join[{0}, Select[Range[1000], PrimeQ[2*#^# + 1] &]] (* Robert Price, Mar 27 2019 *)

is_A110932(n)=ispseudoprime(n^n*2+1) \\ M. F. Hasler, Sep 02 2012

a(5) from Serge Batalov, Apr 08 2018

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

A301641 Primes of form 4*k^k + 1.

Original entry on oeis.org

5, 17, 109, 3294173, 355271367880050092935562133789062501

Offset: 1

Seiichi Manyama, Mar 25 2018

nonn

No additional terms through k=1000. - Harvey P. Dale, Oct 06 2023

Primes of form b*k^k + 1: A121270 (b=1), A216148 (b=2), A301644 (b=3), this sequence (b=4), A301642 (b=16).

Cf. A301519, A301808.

a:=k->`if`(isprime(4*k^k+1),4*k^k+1,NULL): seq(a(k),k=1..1400); # Muniru A Asiru, Mar 25 2018

Select[Table[4n^n+1,{n,30}],PrimeQ] (* Harvey P. Dale, Oct 06 2023 *)

a(n) = 4*A301519(n+1)^A301519(n+1) + 1.

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,","))

A301642 Primes of form 16*k^k + 1.

Original entry on oeis.org

17, 433, 746497, 142657607172097

Offset: 1

Seiichi Manyama, Mar 25 2018

nonn

The next term is too large to include.

The next term a(5) has 108 digits; a(6) has 166 digits; a(7) has 170 digits. - Harvey P. Dale, Aug 23 2019

Primes of form b*k^k + 1: A121270 (b=1), A216148 (b=2), A301644 (b=3), A301641 (b=4), this sequence (b=16).

Cf. A301522.

Select[Table[16*k^k+1,{k,20}],PrimeQ] (* Harvey P. Dale, Aug 23 2019 *)

a(n) = 16*A301522(n+1)^A301522(n+1) + 1.

A301644 Primes of form 3*k^k + 1.

Original entry on oeis.org

13, 769, 139969, 30000000001

Offset: 1

Seiichi Manyama, Mar 25 2018

nonn

The next term is too large to include.

The next term (a(5)) has 703 digits. - Harvey P. Dale, Sep 04 2018

Primes of form b*k^k + 1: A121270 (b=1), A216148 (b=2), this sequence (b=3), A301641 (b=4), A301642 (b=16).

Cf. A160600.

Select[Table[3*k^k+1,{k,500}],PrimeQ] (* Harvey P. Dale, Sep 04 2018 *)

a(n) = 3*(2*A160600(n))^(2*A160600(n)) + 1.

A070855 Smallest prime of the form k*n^n + 1.

Original entry on oeis.org

2, 5, 109, 257, 37501, 139969, 3294173, 167772161, 3874204891, 30000000001, 24536803672547, 17832200896513, 12115004263690121, 344472211592298497, 12261028930664062501, 221360928884514619393, 6617922095090694113417

Offset: 1

Amarnath Murthy, May 15 2002

nonn

By Linnik's theorem, a(n) = O(n^(Ln)) for some effectively computable L.

For more terms, see A175763.

Charles R Greathouse IV, Table of n, a(n) for n = 1..385

Cf. A121270, A110932, A175763.

sp[n_]:=Module[{n2=n^n,k=1},While[!PrimeQ[k*n2+1],k++];k*n2+1]; Array[ sp,20] (* Harvey P. Dale, Jul 06 2014 *)

More terms from Don Reble, May 16 2002

Comments and b-file from Charles R Greathouse IV, Aug 30 2010

A269836 Primes p of the form k^(2*k) + 1.

Original entry on oeis.org

2, 17, 65537

Offset: 1

Jaroslav Krizek, Mar 06 2016

Prime terms from A167436.
Also primes p of the form (k^2)^k + 1.
Corresponding values of k: 1, 2, 4, ...; if a(4) exists, k must be larger than 1024.

			65537 = 4^8 + 1 = 16^4 + 1.

Cf. A121270 (primes of the form k^k + 1), A167436 (n^(2*n) + 1).

[n^(2*n) + 1: n in [1..700] | IsPrime(n^(2*n) + 1)];

Select[Table[n^(2n)+1,{n,20}],PrimeQ] (* Harvey P. Dale, Apr 01 2021 *)

cp's OEIS Frontend

A014566 Sierpiński numbers of the first kind: n^n + 1.

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A110932 Numbers k such that 2*k^k + 1 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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A301641 Primes of form 4*k^k + 1.

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

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A301642 Primes of form 16*k^k + 1.

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A301644 Primes of form 3*k^k + 1.

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