A070592 -id:A070592 - cp's OEIS frontend

A093179 Smallest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

3, 5, 17, 257, 65537, 641, 274177, 59649589127497217, 1238926361552897, 2424833, 45592577, 319489, 114689, 2710954639361, 116928085873074369829035993834596371340386703423373313, 1214251009, 825753601, 31065037602817, 13631489, 70525124609

Offset: 0

Eric W. Weisstein, Mar 27 2004

a(14) might need to be corrected if F(14) turns out to have a smaller factor than 116928085873074369829035993834596371340386703423373313. F(20) is composite, but no explicit factor is known. - Jeppe Stig Nielsen, Feb 11 2010

			F(0) = 2^(2^0) + 1 = 3, prime.
F(5) = 2^(2^5) + 1 = 4294967297 = 641*6700417.
So 3 as the 0th entry and 641 is the 5th term.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 73.

Wilfrid Keller, Prime factors k·2^n + 1 of Fermat numbers F_m and complete factoring status
Ivars Peterson, Cracking Fermat Numbers
Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000051, A000215, A020639, A070592, A007117.

Leading entries in triangle A050922.

Table[With[{k = 2^n}, FactorInteger[2^k + 1]][[1, 1]], {n, 0, 15, 1}] (* Vincenzo Librandi, Jul 23 2013 *)

g(n)=for(x=9,n,y=Vec(ifactor(2^(2^x)+1));print1(y[1]",")) \\ Cino Hilliard, Jul 04 2007

a(n) = A007117(n)*2^(n+2) + 1 for n >= 2. - Jianing Song, Mar 02 2021

a(n) = A020639(A000215(n)). - Alois P. Heinz, Oct 25 2024

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar
a(14)-a(15) added by Jeppe Stig Nielsen, Feb 11 2010
a(16)-a(19) added based on terms of A007117 by Jianing Song, Mar 02 2021

A164307 Primes in A081175.

Original entry on oeis.org

3, 5, 17, 257, 65537

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

The 6th term is too large to include in the data section (see Example section or the b-file).
Primes of the form sum_{j=1..u} j^x for some x>0, u>1. (Since the case of x=1 leads to the triangular numbers with no additional primes, this is equivalent to the definition.)
Primes in A000330 (x=2), or in A000537 (x=3), or in A000538 (x=4), or in A000539 (x=5) etc. See A164312 for the corresponding x values.

			a(1) = 1^1 + 2^1 = 3.
a(2) = 1^2 + 2^2 = 5.
a(3) = 1^4 + 2^4 = 17.
a(4) = 1^8 + 2^8 = 257.
a(5) = 1^16 + 2^16 = 65537.
a(6) = 1^1440 + 2^1440 + 3^1440 + 4^1440 + 5^1440 = 3.287049497374559048967261852*10^1006 = 3287049497374559048967261852 ... 458593539025033893379.

N. J. A. Sloane, Table of n, a(n) for n = 1..6

Cf. A000215, A070592, A019434, A092506, A093179, A100270, A123599.

lst={};Do[s=0;Do[If[PrimeQ[s+=n^x],AppendTo[lst,s];Print[Date[],s]],{n, 4!}],{x,7!}];lst

Edited by R. J. Mathar, Aug 22 2009

Corrected by N. J. A. Sloane, Nov 23 2015 at the suggestion of Jaroslav Krizek.

A164312 Numbers n such that k^n + (k-1)^n + ... + 3^n + 2^n + 1 is prime for some k.

Original entry on oeis.org

1, 2, 4, 8, 16, 1440

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

These terms have k-values {2, 2, 2, 2, 2, 5} respectively. When k = 2, the prime mentioned in the definition is given in A164307. - Derek Orr, Jun 06 2014

			1^1 + 2^1 = 3 is prime (k = 2).
1^2 + 2^2 = 5 is prime (k = 2).
1^4 + 2^4 = 17 is prime (k = 2).
1^8 + 2^8 = 257 is prime (k = 2).
1^16 + 2^16 = 65537 is prime (k = 2).
1^1440 + 2^1440 + 3^1440 + 4^1440 + 5^1440 = 3.287049497374559048967261852*10^1006 = 3287049497374559048967261852 ... 458593539025033893379 is prime (k = 5).

Cf. A000215, A070592, A019434, A092506, A093179, A100270, A123599, A164307.

lst={};Do[s=0;Do[If[PrimeQ[s+=n^x],AppendTo[lst,x];Print[Date[],x]],{n,4!}],{x,7!}];lst

a(n)=for(k=1,10^3,if(ispseudoprime(sum(i=1,k,i^n)),return(k)))
n=1;while(n<5000,if(a(n),print1(n,", "));n++) \\ Derek Orr, Jun 06 2014

Definition improved by Derek Orr, Jun 06 2014

A176689 Prime factors of 2^128 - 1.

Original entry on oeis.org

3, 5, 17, 257, 641, 65537, 274177, 6700417, 67280421310721

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 23 2010

Cf. A000215, A002590, A019434, A023394, A050922, A070592.

Mathematica
```
First/@FactorInteger[2^128-1]
```

factor(2^128 - 1)[,1] \\ Charles R Greathouse IV, Sep 06 2016

Edited by T. D. Noe, May 06 2010

Typo in definition corrected by Arkadiusz Wesolowski, Feb 17 2011

A372867 Distinct terms in A242017, listed in the order of their appearance.

Original entry on oeis.org

3, 5, 17, 97, 641, 257, 193, 274177, 65537, 449, 59649589127497217, 769, 1238926361552897, 5441, 5953, 2424833, 7873, 2753, 3329, 10753, 45592577, 18433, 4673, 15937, 444929, 11777, 12161, 21698561, 6977

Offset: 1

Jean-Marc Rebert, May 15 2024

Conjecture: every term except 3 belongs to A366609. - Bill McEachen, Jun 12 2024

Cf. A023394, A070592, A093179, A242017.

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A093179 Smallest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

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A164307 Primes in A081175.

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A164312 Numbers n such that k^n + (k-1)^n + ... + 3^n + 2^n + 1 is prime for some k.

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A176689 Prime factors of 2^128 - 1.

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A372867 Distinct terms in A242017, listed in the order of their appearance.

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