A080208 -id:A080208 - cp's OEIS frontend

A078902 Generalized Fermat primes of the form (k+1)^2^m + k^2^m, with m>1.

17, 97, 257, 337, 881, 3697, 10657, 16561, 49297, 65537, 66977, 89041, 149057, 847601, 988417, 1146097, 1972097, 2070241, 2522257, 2836961, 3553777, 3959297, 4398577, 5385761, 7166897, 11073217, 17653681, 32530177, 41532497, 44048497

Offset: 1

T. D. Noe, Dec 12 2002

nonn

For k=1, these are the Fermat primes A019434. Is the set of generalized Fermat primes infinite? Conjecture that there are only a finite number of generalized Fermat primes for each value of k. See A077659, which shows that in cases such as k=11, there appear to be no primes. See A078901 for generalized Fermat numbers.

See A080131 for the conjectured number of primes for each k. See A080208 for the least k such that (k+1)^2^n + k^2^n is prime. The largest probable prime of this form discovered to date is the 10217-digit 312^2^12 + 311^2^12.

T. D. Noe, Table of n, a(n) for n = 1..525 (terms < 10^14)
T. D. Noe, Table of generalized Fermat primes of the form (k+1)^2^m + k^2^m
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A019434, A077659, A078900, A078901.

Cf. A080131, A080208, A019434, A078902, A080134, A153504, A152913, A194185.

lst3=Select[lst2, PrimeQ[ # ]&] (* lst2 is from A078901 *)

A253633 a(n) is the least positive integer b such that b^(2^n) + (b-1)^(2^n) is prime.

Original entry on oeis.org

2, 2, 2, 2, 2, 9, 96, 32, 86, 60, 1079, 755, 312, 3509, 1829, 49958, 22845

Offset: 0

Jeppe Stig Nielsen, Jan 07 2015

When a(n) is 2, the corresponding prime is a Fermat prime, otherwise it is a so-called extended generalized Fermat prime sometimes denoted xGF(n, b, b-1) or similar.

			For n = 5, 2^5 = 32 is the exponent.  The numbers 1^32 + 0^32, 2^32 + 1^32, ..., 8^32 + 7^32 are not prime, but 9^32 + 8^32 is prime, so a(5) = 9. - _Michael B. Porter_, Mar 28 2018

Henri Lifchitz & Renaud Lifchitz, PRP Top Records, search for x^16384+y^16384, related to a(14).
Henri Lifchitz & Renaud Lifchitz, 49958^32768+49957^32768, a(15).
Henri Lifchitz & Renaud Lifchitz, 22845^65536+22844^65536, a(16).

Cf. A056993, A080208.

a(n)=for(b=2,10^10,if(ispseudoprime(b^(2^n)+(b-1)^(2^n)),return(b)))

a(n) = A080208(n) + 1.

a(13) from Jeppe Stig Nielsen, Mar 27 2018
a(14) found by Henri Lifchitz in 2007, from Jeppe Stig Nielsen, Apr 17 2018
a(15) found by Kellen Shenton, from Jeppe Stig Nielsen, Nov 27 2020
a(16) found by Kellen Shenton, from Jeppe Stig Nielsen, Mar 31 2021

A122900 Minimum prime of the form n^k + (n+1)^k for k>1, or 0 if no such prime exists.

Original entry on oeis.org

5, 13, 337, 41, 61, 3697, 113, 10657, 181, 2211377674535255285545615254209921

Offset: 1

Alexander Adamchuk, Sep 18 2006

Currently a(n) is unknown for n = {11, 15, 18, 20, 28, 44, 46, 49, 51, 52, 53, 55, 57, 58, 61, 62, 64, 71, 73, 77, 81, 83, 91, 92, 94, ...}. All n < 100 and 1 < k < 2^10 have been checked.
All nonzero a(n) have a form n^(2^m) + (n+1)^(2^m).
The exponents m are listed in A080121. The first occurrence of each exponent m in A080121 is listed in A122902.

			a(1) = 5 because 1^2 + 2^2 = 5 is prime.
a(2) = 13 because 2^2 + 3^2 = 13 is prime.
a(3) = 337 because 3^4 + 4^4 = 337 is prime but 3^3 + 4^3 = 91 and 3^2 + 4^2 = 25 are composite.

Cf. A077659, A080121, A122902, A080208.

Edited by Max Alekseyev, Sep 09 2020

A122902 First occurrence of exponent n in A080121 corresponding to the minimum prime of the form (k^(2^n) + (k+1)^(2^n)) = A122900(k).

Original entry on oeis.org

1, 3, 23, 21, 10, 95, 255, 86, 59

Offset: 1

Alexander Adamchuk, Sep 18 2006, Oct 01 2006

Minimum primes of the form n^(2^m) + (n+1)^(2^m) are listed in A122900. The exponents m are listed in A080121.

a(10)-a(13)>1000, a(14)-a(16)>100.

			A080121 begins with 1,1,2,1,1,2,1,2,1,5,?,1,2,1,?,2,1,?,1,?,4,1,3,1,..., where the unknown terms (denoted with ?) are at least 10. So a(1) = 1, a(2) = 3, a(3) = 23, a(4) = 21, a(5) = 10.

T. D. Noe, Table of generalized Fermat primes of the form (k+1)^2^m + k^2^m
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A077659, A080121, A122900, A080208, A078902, A080134, A019434, A153504, A152913, A194185.

Edited by Max Alekseyev, Sep 09 2020

A172521 Partial sums of A078902.

Original entry on oeis.org

17, 114, 371, 708, 1589, 5286, 15943, 32504, 81801, 147338, 214315, 303356, 452413, 1300014, 2288431, 3434528, 5406625, 7476866, 9999123, 12836084, 16389861, 20349158, 24747735, 30133496, 37300393, 48373610, 66027291, 98557468

Offset: 1

Jonathan Vos Post, Feb 06 2010

nonn

It is unknown if this is a finite or infinite sequence. Can it ever have a prime value after a(1) = 17? It can be semiprime, as 371 = 7 * 53; 1589 = 7 * 227; 15943 = 107 * 149; 214315 = 5 * 42863; 2288431 = 23 * 99497; and 16389861 = 3 * 5463287.

			a(29) = 17 + 97 + 257 + 337 + 881 + 3697 + 10657 + 16561 + 49297 + 65537 + 66977 + 89041 + 149057 + 847601 + 988417 + 1146097 + 1972097 + 2070241 + 2522257 + 2836961 + 3553777 + 3959297 + 4398577 + 5385761 + 7166897 + 11073217 + 17653681 + 32530177 + 41532497 + 44048497.

Eric W. Weisstein, Generalized Fermat Number.

Cf. A000040, A001358, A019434, A077659, A078900, A078901, A078902, A080131, A080208.

SUM[i=1..n] {primes of the form (k+1)^2^m + k^2^m, with m>1.}

cp's OEIS Frontend

A078902 Generalized Fermat primes of the form (k+1)^2^m + k^2^m, with m>1.

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A253633 a(n) is the least positive integer b such that b^(2^n) + (b-1)^(2^n) is prime.

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A122900 Minimum prime of the form n^k + (n+1)^k for k>1, or 0 if no such prime exists.

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A122902 First occurrence of exponent n in A080121 corresponding to the minimum prime of the form (k^(2^n) + (k+1)^(2^n)) = A122900(k).

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A172521 Partial sums of A078902.

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