A080131 -id:A080131 - 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 *)

A080134 Conjectured number of generalized Fermat primes of the form (n+1)^2^k + n^2^k, with k>=0.

Original entry on oeis.org

5, 3, 3, 2, 4, 3, 2, 3, 3, 1, 1, 3, 1, 4, 1, 1

Offset: 1

T. D. Noe, Jan 30 2003

Values of k <= 16 were tested. The sequence A078902 lists some of the generalized Fermat primes. Bjorn and Riesel examined generalized Fermat numbers for n <= 11 and k <= 999. The next n>1 for which (n+1)^2^k + n^2^k is prime for k=0,1,2,3,4 is n=826284.

			a(1) = 5 because there are five known Fermat primes: 3, 5, 17, 257, 65537.

Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A019434, A078902, A080131, A080133.

lst={}; Do[prms=0; Do[If[PrimeQ[(n+1)^2^k+n^2^k], prms++ ], {k, 0, 16}]; AppendTo[lst, prms], {n, 16}]; lst

A080133 Conjectured number of generalized Fermat primes of the form (n+1)^2^k + n^2^k, with k>0.

Original entry on oeis.org

4, 2, 2, 2, 3, 2, 2, 2, 2, 1, 0, 3, 1, 3, 0, 1

Offset: 1

T. D. Noe, Jan 30 2003

Primes that are the sum of consecutive integers (k=0) are excluded. Values of k <= 16 were tested. The sequence A078902 lists some of the generalized Fermat primes. Bjorn and Riesel examined generalized Fermat numbers for n <= 11 and k <= 999.

			a(1) = 4 because there are four known Fermat primes (with k>0): 5, 17, 257, 65537.

Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A019434, A078902, A080131, A080134.

lst={}; Do[prms=0; Do[If[PrimeQ[(n+1)^2^k+n^2^k], prms++ ], {k, 1, 16}]; AppendTo[lst, prms], {n, 16}]; lst

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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A080134 Conjectured number of generalized Fermat primes of the form (n+1)^2^k + n^2^k, with k>=0.

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A080133 Conjectured number of generalized Fermat primes of the form (n+1)^2^k + n^2^k, with k>0.

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A172521 Partial sums of A078902.

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