A119564 -id:A119564 - cp's OEIS frontend

A373580 Numbers k that divide 2^(2^k) - 2^k + 1.

1, 3, 5, 17, 257, 641, 1605, 4369, 32113, 65537, 94945, 114689, 159441, 164737, 274177, 319489, 974849, 2424833, 3862465, 6700417, 13631489, 13906833, 16843009, 26017793, 42009217, 45592577, 63766529, 70463489, 167772161, 175747457, 825753601, 1214251009, 1227890731, 1251711641

Offset: 1

Thomas Ordowski, Jun 10 2024

nonn

Numbers k that divide A119564(k).
A term is prime if and only if it is in A023394.
If k is in A307843, then it is a term of this sequence.
The terms that are not divisors of Fermat numbers are 1605, 4369, 32113, 94945, ... (they are all composite). Are there infinitely many of them?
Note that 2^(2^k) - 2^k + 1 = (2^(2^k) - 1) - (2^k - 2).

Cf. A023394 (primes in this sequence), A119564, A307843 (subsequence).

q[k_] := Mod[PowerMod[2, 2^k, k] - PowerMod[2, k, k] + 1, k] == 0; Select[Range[1, 10^5, 2], q] (* Amiram Eldar, Jun 10 2024 *)

isok(k) = Mod(Mod(2, k)^(2^k) - Mod(2,k)^k + 1, k) == 0; \\ Michel Marcus, Jun 12 2024

More terms from Amiram Eldar, Jun 10 2024

A119550 Prime numbers of the form 2^(2^k) + 2^k - 1.

Original entry on oeis.org

2, 5, 19, 263, 65551

Offset: 1

Cino Hilliard, May 31 2006

			F(2)= 2^(2^2)+1 = 17, M(2) = 2^2-1 = 3, F(2)+ M(2)-1 = 19 is prime, so 2 is a member.

Cf. A119563, A119564.

Select[Table[2^(2^k)+2^k-1,{k,0,10}],PrimeQ] (* James C. McMahon, Sep 15 2024 *)

fmp3(n)=for(x=0,n,y=2^(2^x)+2^x-1;if(ispseudoprime(y),print1(y",")))

Define F(n) = 2^(2^n)+1 = n-th Fermat number, M(n) = 2^n-1 = the n-th Mersenne number. Then we are considering the numbers f(n) = F(n)+M(n)-1 = 2^(2^n) + 2^n - 1 (cf. A119563).

Edited by N. J. A. Sloane, Jun 03 2006

Definition corrected by Stefan Steinerberger, Jun 10 2007

A119562 Let F(n) = 2^(2^n) + 1 = the n-th Fermat number, M(n) = 2^n - 1 = the n-th Mersenne number. Then a(n) = F(n) - M(n) + 1 = 2^(2^n) + 1 - (2^n - 1) + 1 = 2^(2^n) - 2^n + 3.

Original entry on oeis.org

4, 5, 15, 251, 65523, 4294967267, 18446744073709551555, 340282366920938463463374607431768211331, 115792089237316195423570985008687907853269984665640564039457584007913129639683

Offset: 0

Cino Hilliard, May 31 2006

nonn

			F(1) = 2^(2^1)+1 = 5
M(1) = 2^1-1 = 1
F(1) - M(2) + 1 = 5

fm2(n) = for(x=0,n,y=2^(2^x)-2^x+3;print1(y","))

a(n) = A001146(n)-A000079(n)+3 = A119564(n)+2. - R. J. Mathar, May 15 2007

Definition corrected by R. J. Mathar, May 15 2007

cp's OEIS Frontend

A373580 Numbers k that divide 2^(2^k) - 2^k + 1.

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A119550 Prime numbers of the form 2^(2^k) + 2^k - 1.

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A119562 Let F(n) = 2^(2^n) + 1 = the n-th Fermat number, M(n) = 2^n - 1 = the n-th Mersenne number. Then a(n) = F(n) - M(n) + 1 = 2^(2^n) + 1 - (2^n - 1) + 1 = 2^(2^n) - 2^n + 3.

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