A046052 -id:A046052 - cp's OEIS frontend

A228846 Largest m such that (2k+1)*2^m + 1 is a prime factor of the Fermat number 2^(2^n) + 1.

1, 2, 4, 8, 16, 7, 8, 9, 11, 16, 14, 14

Offset: 0

Arkadiusz Wesolowski, Sep 05 2013

a(n) >= n + 2 for n >= 2.
a(n) = A228845(n) if 2^(2^n) + 1 is prime or semiprime.
a(n) = max (A007814(p_i-1)), where p_i are the prime factors of 2^(2^n)+1. - Ralf Stephan, Sep 09 2013
For n >= 2, a(n) >= n + 3 if A046052(n) is an odd number. - Arkadiusz Wesolowski, Aug 10 2021

			F(5) = 641*6700417 and max(A007814(640),A007814(6700416))=7, so a(5)=7.

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

Cf. A000215, A019434, A023394, A046052, A228845.

A229850 Number of prime factors congruent to 1 mod 3 that divide the Fermat number 2^(2^n) + 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 3, 2

Offset: 0

Arkadiusz Wesolowski, Oct 01 2013

a(n) < A046052(n) because all Fermat numbers greater than 3 are equal to 2 (mod 3).
a(n) = 1 if A046052(n) = 2.
If A046052(n) = 3, then a(n) = 0 or 2.
a(n) <= A228846(n) - n - 1 for n = 0 to 11.

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. 61-63, 65-66.

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

Cf. A000215, A002476, A023394, A046052, A229854.

A281576 Composite Fermat numbers.

Original entry on oeis.org

4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937

Offset: 1

Felix Fröhlich, Jan 24 2017

nonn

Complement of A019434 in A000215.
Intersection of A000215 and A002808.
Fermat numbers F_i such that A152155(i) != -1, where i is the index of F in A000215.
Is this sequence infinite?
All the terms are Fermat pseudoprimes to base 2 (A001567). For a proof see, e.g., Jaroma and Reddy (2007). - Amiram Eldar, Jul 24 2021

Giuseppe Coppoletta, Table of n, a(n) for n = 1..7
John H. Jaroma and Kamaliya N. Reddy, Classical and alternative approaches to the Mersenne and Fermat numbers, The American Mathematical Monthly, Vol. 114, No. 8 (2007), pp. 677-687.
Samuel S. Wagstaff, The Cunningham Project, Complete factoring status (compiled by Wilfrid Keller).

Cf. A000215, A001567, A002808, A019434, A046052.

a152155(n) = centerlift(Mod(3, 2^(2^n)+1)^(2^(2^n-1)))
terms(n) = my(i=0, k=1); while(1, if(a152155(k)!=-1, print1(2^(2^k)+1, ", "); i++); if(i==n, break); k++)
terms(4) \\ print initial 4 terms

A228845 Least m such that (2k+1)*2^m + 1 is a prime factor of the Fermat number 2^(2^n) + 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 7, 8, 9, 11, 11, 12, 13, 14

Offset: 0

Arkadiusz Wesolowski, Sep 05 2013

a(n) >= n + 2 for n >= 2.

a(n) = A228846(n) if 2^(2^n) + 1 is prime or semiprime.

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

Cf. A000215, A019434, A023394, A046052, A228846.

A023395 Only Fermat number divisible by A023394(n) is 2^2^a(n) + 1.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 12, 6, 11, 11, 9, 5, 18, 12, 10, 12, 23, 16, 15, 10, 19, 12, 19, 13, 36, 21, 38, 32, 25, 17, 39, 6, 26, 27, 30, 30, 8, 12, 15, 29, 38, 7, 25, 27, 36, 42, 25, 13, 13, 55

Offset: 1

David W. Wilson

From Jianing Song, Mar 02 2021: (Start)
2^(a(n)+1) is the multiplicative order of 2 modulo A023394(n).
Each k occurs A046052(k) times in this sequence provided that F(k) = 2^2^k + 1 is squarefree (no counterexamples are known). (End)
Alternatively, a(n) is the only k such that A023394(n) divides A000215(k). - Lorenzo Sauras Altuzarra, Feb 01 2023

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).
Index entries for sequences that are related to primes dividing Fermat numbers

Cf. A000215, A023394, A046052.

forprime(p=3,,r=znorder(Mod(2,p));hammingweight(r)==1&&print1(logint(r,2)-1,", ")) \\ Jeppe Stig Nielsen, Mar 04 2018

a(25)-a(41) computed using data from Wilfrid Keller by T. D. Noe, Feb 01 2009
Three more terms by T. D. Noe, Feb 03 2009
Six more terms from Wilfrid Keller by T. D. Noe, Jan 14 2013

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.

A063487 Number of distinct prime divisors of 2^(2^n)-1 (A051179).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 20, 25

Offset: 0

Jason Earls, Jul 28 2001

nonn

2^(2^n)-1 is the product of the first n Fermat numbers F(0),...,F(n-1) (A000215). Hence this sequence is just the summation of A046052, which gives the number of prime factors in each Fermat number. - T. D. Noe, Jan 07 2003

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, p. 238.

Eric Weisstein's World of Mathematics, Fermat Number

Cf. A051179, A000215, A046052.

PARI
```
for(n=0,22,print(omega(2^(2^n)-1)))
```

More terms from T. D. Noe, Jan 07 2003

A321213 a(n) is the number of divisors of n-th Fermat number (A000215).

Original entry on oeis.org

2, 2, 2, 2, 2, 4, 4, 4, 4, 8, 16, 32

Offset: 0

Jinyuan Wang, Oct 31 2018

			A000215(n) is prime for n=0 to 4, so a(n)= 2 for n=0 to 4.

Cf. A000005, A000215.

List(List([0..11],n->2^(2^n)+1),i->Number(DivisorsInt(i))); # Muniru A Asiru, Nov 03 2018

[DivisorSigma(0, 2^2^n + 1): n in [1..100]]

Table[DivisorSigma[0, 2^2^n + 1], {n, 120}]

PARI
```
a(n) = numdiv(2^2^n+1)
```

a(n) = A000005(A000215(n)). - Omar E. Pol, Oct 31 2018

a(n) = 2^A046052(n) for squarefree A000215(n). - Amiram Eldar, Oct 31 2018

a(10)-a(11) from Amiram Eldar, Oct 31 2018

cp's OEIS Frontend

A228846 Largest m such that (2k+1)*2^m + 1 is a prime factor of the Fermat number 2^(2^n) + 1.

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A229850 Number of prime factors congruent to 1 mod 3 that divide the Fermat number 2^(2^n) + 1.

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A281576 Composite Fermat numbers.

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A228845 Least m such that (2k+1)*2^m + 1 is a prime factor of the Fermat number 2^(2^n) + 1.

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A023395 Only Fermat number divisible by A023394(n) is 2^2^a(n) + 1.

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

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A063487 Number of distinct prime divisors of 2^(2^n)-1 (A051179).

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PARI

Extensions

A321213 a(n) is the number of divisors of n-th Fermat number (A000215).

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