A078902 -id:A078902 - cp's OEIS frontend

A172521 Partial sums of A078902.

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.}

A077659 a(n) = smallest k>1 such that the sum n^k + (n+1)^k is prime, or -1 if no such k exists.

Original entry on oeis.org

2, 2, 4, 2, 2, 4, 2, 4, 2, 32

Offset: 1

T. D. Noe, Nov 14 2002

Checking k up through 1024 suggests that the sequence may continue -1, 2, 4, 2, -1, 4, 2, -1, 2, -1, 16, 2, 8, 2, 2, 4, 4, -1, 2, 2, 4, 2, 4, 2, 2, 4, 4, 4, 2, ...
For any a>1 and b>1, a^k + b^k is composite for all odd k>1. Hence if n^k + (n+1)^k is prime then k must be a power of 2.
It is known that a(11) > 2^22. Is it possible that 11^2^m + 12^2^m is composite for all m > 0?

			a(3)=4 because 3^2 + 4^2 = 25 is not prime, but 3^4 + 4^4 = 337 is prime.

T. D. Noe, Factorizations of Generalized Fermat Numbers 12^2^k + 11^2^k

Cf. A078902.

Cf. A080121.

lst={}; For[n=1, n<=100, n++, k=2; While[k<=2^10 && !PrimeQ[n^k+(n+1)^k], k=2*k]; If[k<=2^10, AppendTo[lst, k], AppendTo[lst, -1]]]; lst

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

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

Original entry on oeis.org

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

A080131 Conjectured number of generalized Fermat primes of the form (n+1)^2^k + n^2^k, with k>1.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Jan 30 2003

Primes that are the sum of consecutive integers (k=0) and consecutive squares (k=1) 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) = 3 because there are three Fermat primes (with k>1): 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, A080133, A080134.

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

A078901 Generalized Fermat numbers of the form (k+1)^2^m + k^2^m, with m>1.

Original entry on oeis.org

17, 97, 257, 337, 881, 1921, 3697, 6497, 6817, 10657, 16561, 24641, 35377, 49297, 65537, 66977, 72097, 89041, 116161, 149057, 188497, 235297, 290321, 354481, 428737, 456161, 514097, 611617, 722401, 847601, 988417, 1146097, 1321937

Offset: 1

T. D. Noe, Dec 12 2002

It can be shown that, like the Fermat numbers, two of these generalized Fermat numbers are coprime if they have the same base k. However, unlike the Fermat numbers (which are conjectured to be squarefree), these generalized Fermat numbers are not necessarily squarefree for k > 1. Riesel tabulates some prime factors of generalized Fermat numbers for k <= 5.

For k=1, these are the Fermat numbers A000215. See A078900 for the case m>0, which includes the sum of consecutive squares. By Legendre's theorem (Riesel, p. 165), the prime factors of a generalized Fermat number are of the form 1 + f 2^(m+1) for some integer f. See A078902 for generalized Fermat primes.

H. Riesel, "Prime numbers and computer methods for factorization," Second Edition, Progress in Mathematics, Vol. 126, Birkhauser, Boston, 1994, pp. 417-425.

T. D. Noe, Factorizations of Generalized Fermat Numbers
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A000215, A078900, A078902.

mx=10^7; maxK=Ceiling[Sqrt[mx/2]]; maxM=Ceiling[Log[2, Log[2, mx]]]; lst={}; Do[gf=(k+1)^2^m+k^2^m; If[gf

Offset corrected by Sean A. Irvine, Jul 23 2025

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

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

cp's OEIS Frontend

A172521 Partial sums of A078902.

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A077659 a(n) = smallest k>1 such that the sum n^k + (n+1)^k is prime, or -1 if no such k exists.

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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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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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A080131 Conjectured number of generalized Fermat primes of the form (n+1)^2^k + n^2^k, with k>1.

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A078901 Generalized Fermat numbers of the form (k+1)^2^m + k^2^m, with m>1.

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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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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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