A023394 -id:A023394 - cp's OEIS frontend

A229851 Lucky Fermat factors.

641, 114689, 167772161, 6597069766657, 188894659314785808547841, 850705917302346158658436518579420528641, 2468256835981809063232453773836025757474103798450369795022913537

Offset: 1

Arkadiusz Wesolowski, Oct 01 2013

nonn

The prime k*2^(m+2) + 1 is a lucky Fermat factor if it divides 2^(2^m) + 1 and k = 3, 5, 6, 7, or 9 is the smallest value we can choose that is not excluded by congruence constraints modulo 12, which lead to divisibility of k*2^(m+2) + 1 by 3, 5, 7, or 13 (Krizek, Luca and Somer).

The m for which 2^(2^m) + 1 has a lucky factor are m = 5, 12, 23, 38, 73, 125, 207, 1945, 23471, 95328, 157167, 213319, 382447, 2145351, 2478782, ... From this it is trivial to write out a(1),...,a(15), but the numbers become too wide for a b-file. - Jeppe Stig Nielsen, Mar 13 2022

M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, pp. 77-79.

Wilfrid Keller, Fermat factoring status
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215. Subsequence of A023394.

r=vector(12,m,select(k->p=k*2^(m+2)+1;p%3!=0&&p%5!=0&&p%7!=0&&p%13!=0,[3,5,6,7])[1]);for(m=0,+oo,k=r[(m+11)%12+1];p=k*2^(m+2)+1;Mod(2,p)^(2^m)+1==0&&print1(p,", ")) \\ Jeppe Stig Nielsen, Mar 13 2022

A229857 Round(2^(m-n-2)/(m*log(8))), where m = 2^n - n - 2.

Original entry on oeis.org

5043, 2417158053779, 5245728941618725066052704993134, 215872416866954281715178071724040762825421437510476267629647193878371

Offset: 5

Arkadiusz Wesolowski, Oct 01 2013

a(9) has 145 digits and is too large to include.
Conjecture: a(n) < f(n) = number of primes of the form k*2^(n+2) + 1 with k odd that exist between a = 2^(n+2) + 1 and b = floor((2^(2^n) + 1)/(3*2^(n+2) + 1)).
For comparison, f(5) = 5746.
If the extended Riemann hypothesis is true, then for every fixed epsilon > 0, f(n) = Li(b)/(a - 1) + O(b^(1/2 + epsilon)), where Li(b) = integral(2..b, dt/log(t)).

P. Borwein, S. Choi, B. Rooney and A. Weirathmueller, The Riemann Hypothesis: A Resource for the Aficionado and Virtuoso Alike, Springer, Berlin, 2008, pp. 57-58.

Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A016631, A023394, A046052.

A343557 Indices of the prime factors of Fermat numbers in the sequence of primes.

Original entry on oeis.org

2, 3, 7, 55, 116, 6543, 10847, 23974, 27567, 76709, 177975, 457523, 887643, 1625567, 2751966, 3772007, 9385401, 42401669, 61136051, 301137372, 2946723445, 7632981296, 24728168164, 98261951745, 99582868271, 159657063059, 231641062432, 851793186025, 870658222248

Offset: 1

Lorenzo Sauras Altuzarra, Apr 28 2021

nonn

			A000040(a(5)) = A000040(116) = 641 = A023394(5).

Cf. A000040 (primes), A000720 (primepi), A023394 (prime factors of Fermat primes).

Supersequence of A159611.

q:=n->(irem(2^(2^padic:-ordp(ithprime(n)-1, 2))-1, ithprime(n)) = 0):
select(q, [$1..10^5])[]; # Lorenzo Sauras Altuzarra, Feb 20 2023

is_a023394(p)=p>2 && Mod(2,p)^lift(Mod(2,znorder(Mod(2,p)))^p)==1 && isprime(p) \\ after Charles R Greathouse IV in A023394
my(i=1); forprime(p=1, , if(is_a023394(p), print1(i, ", ")); i++) \\ Felix Fröhlich, Apr 30 2021

a(n) = A000720(A023394(n)).

A000040(a(n)) = A023394(n).

More terms from Michel Marcus, Apr 29 2021

More terms from Amiram Eldar, Apr 29 2021

A348062 Primes p such that the length of the (eventual) period of the sequence {2^(2^k) mod p: k >= 0} is odd.

Original entry on oeis.org

2, 3, 5, 17, 29, 43, 47, 113, 127, 179, 197, 257, 277, 283, 293, 317, 383, 439, 449, 467, 479, 509, 569, 641, 659, 719, 797, 863, 1013, 1069, 1289, 1373, 1399, 1427, 1439, 1487, 1579, 1627, 1657, 1753, 1823, 1913, 1933, 1949, 2063, 2203, 2207, 2213, 2273, 2339, 2351

Offset: 1

Arkadiusz Wesolowski, Sep 26 2021

nonn

Of these numbers only 3 and 5 are elite primes (A102742). (Aigner)

Every prime of the form A036259(n)*2^m + 1, with m, n >= 1, is in this sequence.

Alexander Aigner, Über Primzahlen, nach denen (fast) alle Fermatzahlen quadratische Nichtreste sind, Monatsh. Math., Vol. 101 (1986), pp. 85-93; alternative link.

Supersequence of A023394.

Cf. A102742 (elite primes), A256607.

L=List([2]); forprime(p=3, 2351, z=znorder(Mod(2, p)); if(znorder(Mod(2, z/2^valuation(z, 2)))%2, listput(L, p))); Vec(L)

A351865 Primes of the form x^2 + 64*y^2 that divide some Fermat number.

Original entry on oeis.org

257, 65537, 2424833, 26017793, 63766529, 825753601, 1214251009, 6487031809, 2710954639361, 2748779069441, 6597069766657, 25991531462657, 76861124116481, 151413703311361, 1095981164658689, 1238926361552897, 1529992420282859521, 2663848877152141313, 3603109844542291969

Offset: 1

Arkadiusz Wesolowski, Apr 10 2022

nonn

A prime p = k*2^j + 1 (with k odd) belongs to this sequence if and only if p is a factor of a Fermat number 2^(2^m) + 1 for some m <= j - 3.

			a(1) = 1^2 + 64*2^2 = 257 is a prime factor of 2^(2^3) + 1;
a(2) = 1^2 + 64*32^2 = 65537 is a prime factor of 2^(2^4) + 1;
a(3) = 127^2 + 64*194^2 = 2424833 is a prime factor of 2^(2^9) + 1;
a(4) = 2047^2 + 64*584^2 = 26017793 is a prime factor of 2^(2^12) + 1;
a(5) = 7295^2 + 64*406^2 = 63766529 is a prime factor of 2^(2^12) + 1;

Allan Cunningham, Haupt-exponents of 2, The Quarterly Journal of Pure and Applied Mathematics, Vol. 37 (1906), pp. 122-145.

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m.

Cf. A014754, A023394, A228846.

isok(p) = if(p%8==1 && isprime(p), my(d=Mod(2, p)); d^((p-1)/4)==1 && d^2^valuation(p-1, 2)==1, return(0));

A014754 INTERSECT A023394.

A354026 Primes that divide some k dividing 4^k + 3^k (A045584).

Original entry on oeis.org

7, 379, 14407, 689431, 4235659, 41647747, 137534083, 239900179, 242121643, 349909477, 1245283747, 1478065891, 1605314383, 2500276549, 2748751303, 5618210347, 7490947129, 11236420693, 11260421089, 16948514941, 29440659361, 74163546829, 75093609319, 82188727303

Offset: 1

Max Alekseyev, May 15 2022

nonn

Prime p > 3 is in this sequence iff all prime factors of the multiplicative order of -3/4 modulo p belong to this sequence.

Cf. A023394, A045584, A066364, A087807, A129729, A134360, A171980, A354027.

S=[]; forprime(p=5,oo, f=Set(factor(znorder(Mod(-3/4,p)))[,1]); if(#setintersect(S,f)==#f, S=setunion(S,[p]); print1(p,", ")));

a(18)-a(24) from Jinyuan Wang, Jan 29 2025

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

A188803 Primes that divide 2^(2^100) - 1 and 10^(10^100) - 1.

Original entry on oeis.org

3, 17, 257, 641, 65537, 167772161, 2748779069441, 46179488366593, 1328165573307087716353, 188894659314785808547841

Offset: 1

Arkadiusz Wesolowski, Apr 12 2011

Probably there are no more terms. [Arkadiusz Wesolowski, Jul 12 2011]

Dario A. Alpern, Known prime factors of Googolplex - 1
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 167772161
Eric Weisstein's World of Mathematics, Fermat Number
Eric Weisstein's World of Mathematics, Googolplex
Eric Weisstein's World of Mathematics, Greatest Common Divisor

Cf. A023394.

Select[Sort[Flatten[Table[z = k*2^n + 1; If[PowerMod[2, 2^100, z] == PowerMod[10, 10^100, z] == 1, z], {k, 1, 21, 2}, {n, 75}]]], PrimeQ] (* Arkadiusz Wesolowski, Sep 08 2011 *)

A215540 Least k such that (2n-1)2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m, or 0 if no such value exists.

Original entry on oeis.org

1, 41, 7, 14, 67, 18759, 20, 229, 147, 6838, 41

Offset: 1

Arkadiusz Wesolowski, Aug 15 2012

(2*n-1)*2^a(n) + 1 is in A023394.
a(n) >= 7 for n > 1.
a(39279) = 0. No n < 39279 with a(n)=0 is known.
a(12)>2500000, a(13)>2500000, a(14)=455, a(15)=57 (see Ballinger and Keller link).
No, a(13)=2141884, found in 2011. - Jeppe Stig Nielsen, Sep 07 2019

Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
Fermat factoring status, Prime factors of Fermat numbers
FermatSearch, Home page
PrimeGrid, Announcement of 25*2^2141884+1, related to a(13).
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A033809.

lst = {}; Do[k = 1; While[True, p = n*2^k + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, k]; Break[]]; k++], {n, 1, 9, 2}]; lst

A248802 Smallest prime factor of 2^(2^n+2) + 3.

Original entry on oeis.org

11, 19, 67, 13, 262147, 13, 1669, 13, 255127, 13, 2383, 13, 67, 13, 32544331, 13, 271, 13, 4057, 13

Offset: 0

Arkadiusz Wesolowski, Oct 14 2014

These numbers do not occur in A023394 (prime factors of Fermat numbers A000215).
From Chai Wah Wu, Oct 21 2019: (Start)
a(22) = 67, a(26) =  1399, a(28) = 10957, a(30) = 117127, a(32) = 67, a(36) = 12781849, a(38) = 262147, a(42) = 67, a(48) = 6391117, a(50) = 1265347, a(52) = 67, a(54) = 2383, a(58) = 26833, a(62) = 67, a(64) = 517261, a(68) = 2251, a(72) = 67, a(74) = 137077, a(78) = 562273, a(82) = 67, a(84) = 1399, a(86) = 3253, a(88) = 271, a(92) = 67, a(94) = 2203, a(96) = 329347, a(98) = 2383, a(100) = 5323, a(110) = 2759137, a(114) = 122653, a(116) = 659941, a(126) = 48337, a(130) = 2403229, a(134) = 2534659, a(140) = 41257.
Theorem: a(n) >= 13 for n > 0.
Proof. 2^(2^n+2) + 3 is odd and not a multiple of 3, so a(n) > 3. For all primes 3 < p < 14, p-3 is a power of 2. For p = 5, 2^4 == 1 mod 5, so for n = 1, 2^(2^n+2) + 3 == 4 mod 5 and for n > 1, 2^(2^n+2) + 3 == 7 == 2 mod 5. For p = 7, 2^3 == 1 mod 7. Since 2^n+2 <> 2 mod 3, 2^(2^n+2) <> 4 mod 7 and thus 2^(2^n+2) + 3 <> 0 mod 7.
For p = 11, 2^10 == 1 mod 11. Since 2^n+2 is even for n > 0, 2^n+2 <> 3 mod 10 and thus 2^(2^n+2) <> 2^3 mod 11 and 2^(2^n+2) + 3 <> 0 mod 11. End of proof.
Theorem: a(2n+1) = 13 for n >= 1.
Proof by induction. a(3) = 13 since 2^(2^3+2) + 3 = 1027 = 13*79.
Suppose a(2n+1) = 13, this implies that 2^(2^(2n+1)+2) == 10 mod 13.
Then 2^(2^(2n+3)+2) = 2^(3*2^(2n+1)) * 2^(2^(2n+1)+2). For n >= 1, 2^(2n+1) is a multiple of 4, and thus 2^(3*2^(2n+1)) == 2^12 == 1 mod 13.
This implies that 2^(2^(2n+3)+2) == 2^(2^(2n+1)+2) == 10 mod 13 and thus a(2n+3) <= 13. By the first result above, a(2n+3) = 13.
End of proof.
Conjecture 1: a(10n+2) = 67 for n >= 0.
Conjecture 2: a(36n+16) = 271 for n >= 0 and n <> 1 mod 5.
Conjecture 3: a(84n+22) = 523 for n >= 0 and n <> 0 mod 5.
Conjecture 4: a(58n+26) = 1399 for n >= 0 and when it is not covered by Conjectures 1-3.
Conjecture 5: a(138n+6) = 1669 for n >= 0 and n <> 2 mod 5.
Conjecture 6: a(44n+10) = 2383 for n >= 0 and when it is not covered by Conjectures 1-5.
(End)

Cf. A000215, A023394, A057733.

PrimeFactors[n_]:= Flatten[Table[#[[1]], {1}]&/@FactorInteger[n]]; Table[PrimeFactors[2^(2^n + 2) + 3] [[1]], {n, 0, 7}] (* Vincenzo Librandi, Oct 15 2014 *)

a(n) = factor(2^(2^n+2) + 3)[1, 1]; \\ Michel Marcus, Oct 15 2014

for(n=1,19,my(x=2^(2^n+2)+3);forprime(k=3,oo,if(x%k==0,print1(k,", ");break))) \\ Hugo Pfoertner, Aug 08 2019

Smallest prime factor of 4*A000215(n) - 1, with the Fermat numbers A000215. - Wolfdieter Lang, Nov 05 2014

cp's OEIS Frontend

A229851 Lucky Fermat factors.

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A229857 Round(2^(m-n-2)/(m*log(8))), where m = 2^n - n - 2.

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A343557 Indices of the prime factors of Fermat numbers in the sequence of primes.

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A348062 Primes p such that the length of the (eventual) period of the sequence {2^(2^k) mod p: k >= 0} is odd.

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A351865 Primes of the form x^2 + 64*y^2 that divide some Fermat number.

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PARI

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A354026 Primes that divide some k dividing 4^k + 3^k (A045584).

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A176689 Prime factors of 2^128 - 1.

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Mathematica

PARI

Extensions

A188803 Primes that divide 2^(2^100) - 1 and 10^(10^100) - 1.

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Mathematica

A215540 Least k such that (2*n-1)*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m, or 0 if no such value exists.

A215540 Least k such that (2n-1)2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m, or 0 if no such value exists.