A199591 Generalized Fermat numbers: 5^(2^n) + 1, n >= 0.

6, 26, 626, 390626, 152587890626, 23283064365386962890626, 542101086242752217003726400434970855712890626

Offset: 0

Author

Arkadiusz Wesolowski, Nov 08 2011

Keywords

• easy
• nonn

Examples

			a(0) = 5^(2^0) + 1 = 5^1 + 1 = 6 = 4*(2^0) + 2;
a(1) = 5^(2^1) + 1 = 5^2 + 1 = 26 = 4*(2^1*3) + 2;
a(2) = 5^(2^2) + 1 = 5^4 + 1 = 626 = 4*(2^2*3*13) + 2;
a(3) = 5^(2^3) + 1 = 5^8 + 1 = 390626 = 4*(2^3*3*13*313) + 2;
a(4) = 5^(2^4) + 1 = 5^16 + 1 = 152587890626 = 4*(2^4*3*13*313*195313) + 2;
a(5) = 5^(2^5) + 1 = 5^32 + 1 = 23283064365386962890626 = 4*(2^5*3*13*313*195313*76293945313) + 2;
		

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 0..11
• Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
• C. K. Caldwell, "Top Twenty" page, Generalized Fermat Divisors (base=5).
• Wilfrid Keller, GFN05 factoring status.
• Eric Weisstein's World of Mathematics, Generalized Fermat Number.
• OEIS Wiki, Generalized Fermat numbers.

Crossrefs

Cf. A059919, A078303, A078304, A152581, A080176, A199592, A152585.

Programs

Magma

[5^2^n+1 : n in [0..6]];

Mathematica

Table[5^2^n + 1, {n, 0, 6}]

PARI

for(n=0, 6, print1(5^2^n+1, ", "))

Formula

a(0) = 6; a(n) = (a(n-1)-1)^2 + 1, n >= 1.

a(0) = 6, a(1) = 26; a(n) = a(n-1) + 4*5^(2^(n-1))*Product_{i=0..n-2} a(i), n >= 2.

a(0) = 6, a(1) = 26; a(n) = a(n-1)^2 - 2*(a(n-2)-1)^2, n >= 2.

a(0) = 6; a(n) = 4*(Product_{i=0..n-1} a(i)) + 2, n >= 1.

a(n) = A152578(n) - 1.

Sum_{n>=0} 2^n/a(n) = 1/4. - Amiram Eldar, Oct 03 2022