A199592 Generalized Fermat numbers: 11^(2^n) + 1, n >= 0.

12, 122, 14642, 214358882, 45949729863572162, 2111377674535255285545615254209922, 4457915684525902395869512133369841539490161434991526715513934826242

Offset: 0

Author

Arkadiusz Wesolowski, Nov 08 2011

Keywords

• easy
• nonn

Examples

			a(0) = 11^(2^0) + 1 = 11^1 + 1 = 12 = 10*(2^0) + 2;
a(1) = 11^(2^1) + 1 = 11^2 + 1 = 122 = 10*(2^1*6) + 2;
a(2) = 11^(2^2) + 1 = 11^4 + 1 = 14642 = 10*(2^2*6*61) + 2;
a(3) = 11^(2^3) + 1 = 11^8 + 1 = 214358882 = 10*(2^3*6*61*7321) + 2;
a(4) = 11^(2^4) + 1 = 11^16 + 1 = 45949729863572162 = 10*(2^4*6*61*7321*107179441) + 2;
a(5) = 11^(2^5) + 1 = 11^32 + 1 = 2111377674535255285545615254209922 = 10*(2^5*6*61*7321*107179441*22974864931786081) + 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.
• Eric Weisstein's World of Mathematics, Generalized Fermat Number.
• OEIS Wiki, Generalized Fermat numbers.

Crossrefs

Cf. A059919, A199591, A078303, A078304, A152581, A080176, A152585.

Programs

Magma

[11^2^n+1 : n in [0..6]]

Mathematica

Table[11^2^n + 1, {n, 0, 6}]

PARI

for(n=0, 6, print1(11^2^n+1, ", "))

Formula

a(0) = 12; a(n) = (a(n-1)-1)^2 + 1, n >= 1.

a(0) = 12, a(1) = 122; a(n) = a(n-1) + 10*11^(2^(n-1))*Product_{i=0..n-2} a(i), n >= 2.

a(0) = 12, a(1) = 122; a(n) = a(n-1)^2 - 2*(a(n-2)-1)^2, n >= 2.

a(0) = 12; a(n) = 10*(Product_{i=0..n-1} a(i)) + 2, n >= 1.

a(n) = A152583(n) - 1.

Sum_{n>=0} 2^n/a(n) = 1/10. - Amiram Eldar, Oct 03 2022