A253633 -id:A253633 - cp's OEIS frontend

A080208 a(n) is the least k such that the generalized Fermat number (k+1)^(2^n) + k^(2^n) is prime.

1, 1, 1, 1, 1, 8, 95, 31, 85, 59, 1078, 754, 311, 3508, 1828, 49957, 22844

Offset: 0

T. D. Noe, Feb 10 2003

The first five terms correspond to the five known Fermat primes. The sequence A078902 lists some of the generalized Fermat primes. Bjorn and Riesel examined generalized Fermat numbers for k <= 11 and n <= 999. The sequence A080134 lists the conjectured number of primes for each k.

For n >= 10, a(n) yields a probable prime. a(13) was found by Henri Lifchitz. It is known that a(14) > 1000.

			a(5) = 8 because (k+1)^32 + k^32 is prime for k = 8 and composite for k < 8.

T. D. Noe, Table of generalized Fermat primes of the form (k+1)^2^m + k^2^m
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A019434, A078902, A080134, A153504, A152913, A194185, A253633.

a(n) = A253633(n) - 1.

a(14)-a(15) from Jeppe Stig Nielsen, Nov 27 2020

a(16) by Kellen Shenton communicated by Jeppe Stig Nielsen, May 19 2023

A291944 a(n) is the least A for which there exists B with 0 < B < A so that A^(2^n) + B^(2^n) is prime.

Original entry on oeis.org

2, 2, 2, 2, 2, 9, 11, 27, 14, 13, 47, 22, 53, 72, 216, 260, 124, 1196, 200

Offset: 0

Jeppe Stig Nielsen, Mar 09 2018

A^(2^n) + B^(2^n) is called an (extended) generalized Fermat prime, and often denoted F_n(A, B); or xGF(n, A, B).

If we require B=1, we get A056993. Therefore a(n) <= A056993(n).

			a(10)=47 corresponds to the prime number 47^1024 + 26^1024, the smallest prime number of the form A^1024 + B^1024 (or more precisely, it minimizes A).
a(14)=216 corresponds to the prime number 216^16384 + 109^16384, a 38248-decimal digit PRP, the smallest prime number of the form A^16384 + B^16384. - _Serge Batalov_, Mar 16 2018

Jeppe Stig Nielsen, List of [n, A, B] tuples for this sequence.
Chris K. Caldwell, The Prime Database: 72^8192 + 43^8192 (related to a(13)).
Henri Lifchitz & Renaud Lifchitz, 1196^131072+595^131072, a(17).
Henri Lifchitz & Renaud Lifchitz, 200^262144+119^262144, a(18).

Cf. A056993, A253633, A111635.

f[n_] := Monitor[ Block[{a = 2, b}, While[a < Infinity, b = 1 +Mod[a, 2]; While[b < a, If[ PrimeQ[a^2^n + b^2^n], Goto[fini]]; b+=2]; a++]; Label[fini]; {a, b}], {a, b}]; Array[f, 14, 0] (* Robert G. Wilson v, Mar 10 2018 *)

for(n=0,30,for(a=2,10^100,forstep(b=(a % 2)+1,a-1,2,if(ispseudoprime(a^(2^n)+b^(2^n)),print1(a,", ");next(3)))))

a(14) = 216 (and B = 109) from Serge Batalov, Mar 16 2018
a(15) = 260 (and B = 179) from Serge Batalov, Mar 16 2018
a(16) = 124 (and B = 57) from Serge Batalov, Mar 16 2018
a(17) = 1196 (and B = 595) from Kellen Shenton, Aug 10 2022
a(18) = 200 (and B = 119) from Kellen Shenton, Aug 27 2022

A250201 Least b such that Phi_n(b, b-1) is prime.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 3, 2, 3, 4, 2, 6, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 4, 5, 40, 2, 3, 2, 7, 2, 5, 3, 3, 2, 13, 3, 2, 14, 4, 22, 3, 3, 13, 2, 34, 5, 3, 5, 2, 2, 34, 9, 2, 17, 7, 3, 2, 3, 18, 9, 47, 4, 20, 3, 2, 2, 8, 2, 4, 17, 6, 14, 2, 2, 61, 18, 2, 2

Offset: 2

Eric Chen, Mar 09 2015

nonn

Phi_n(b, b-1) = (b-1)^EulerPhi(n) * Phi_n(b/(b-1)).
This sequence is not defined at n = 1 since Phi_1(b, b-1) = 1 for all b, and 1 is not prime. Conjecture: a(n) is defined for all n>1.
If b = 1, then Phi_n(b, b-1) = 1 for all n, and 1 is not prime, so all a(n) > 1.
a(n) = 2 if and only if n is in A072226.
n        Phi_n(a, b)
1        a-b
2        a+b
3        a^2+ab+b^2
4        a^2+b^2
5        a^4+a^3*b+a^2*b^2+a*b^3+b^4
6        a^2-ab+b^2
...      ...
n        b^EulerPhi(n)*Phi_n(a/b)

			a(11) = 6 because Phi_11(b, b-1) is composite for b = 2, 3, 4, 5 and prime for b = 6.
a(37) = 40 because Phi_37(b, b-1) is composite for b = 2, 3, 4, ..., 39 and prime for b = 40.

Eric Chen, Table of n, a(n) for n = 2..490

Cf. A103794, A253633, A085398, A058013.

Table[k = 2; While[!PrimeQ[(k-1)^EulerPhi(n)*Cyclotomic[n, k/(k-1)]], k++]; k, {n, 2, 300}]

a(n) = for(k = 2, 2^16, if(ispseudoprime((k-1)^eulerphi(n) * polcyclo(n, k/(k-1))), return(k)))

A302387 a(n) is least number k >= 3 such that (k^(2^n) + (k-2)^(2^n))/2 is prime.

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 3, 179, 169, 935, 663, 8723, 1481, 2035, 10199, 18203, 36395

Offset: 0

Jeppe Stig Nielsen, Apr 06 2018

			a(10)=663 corresponds to the prime (663^1024 + 661^1024)/2.

Henri Lifchitz & Renaud Lifchitz, (36395^65536+36393^65536)/2, a(16).

Cf. A253633, A291944, A301738.

lst = {};  For[n=0, n<=14, n++, k=3;  While[! PrimeQ[(k^(2^n) + (k-2)^(2^n))/2], k++];  AppendTo[lst, k]];  lst (* Robert Price, Apr 29 2018 *)

for(n=0,20,forstep(k=3,+oo,2,if(ispseudoprime((k^(2^n)+(k-2)^(2^n))/2),print1(k,", ");break())))

a(15) from Robert Price, May 28 2018

a(16) from Kellen Shenton, Apr 14 2022

cp's OEIS Frontend

A080208 a(n) is the least k such that the generalized Fermat number (k+1)^(2^n) + k^(2^n) is prime.

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A291944 a(n) is the least A for which there exists B with 0 < B < A so that A^(2^n) + B^(2^n) is prime.

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A250201 Least b such that Phi_n(b, b-1) is prime.

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A302387 a(n) is least number k >= 3 such that (k^(2^n) + (k-2)^(2^n))/2 is prime.

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