A057755 Number of digits in n-th Fermat number (A000215).

1, 1, 2, 3, 5, 10, 20, 39, 78, 155, 309, 617, 1234, 2467, 4933, 9865, 19729, 39457, 78914, 157827, 315653, 631306, 1262612, 2525223, 5050446, 10100891, 20201782, 40403563, 80807125, 161614249, 323228497, 646456994, 1292913987, 2585827973

Offset: 0

Author

Robert G. Wilson v, Oct 30 2000

Keywords

• nonn
• base

Comments

Also number of digits of A001146(n) and A051179(n). - Michel Marcus, Dec 21 2018

Examples

			a(6) = 20 because 2^(2^6) + 1 = 18446744073709551617 which is a twenty-digit number.
		

References

• John H. Conway and R. K. Guy, The Book of Numbers, Copernicus, an imprint of Springer-Verlag, NY, 1995, page 139.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 100 terms from Jinyuan Wang)
• R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
• Eric Weisstein's World of Mathematics, Fermat Number

Crossrefs

Cf. A000215, A001146, A051179.

Programs

GAP

List([0..18],n->Size(ListOfDigits(2^(2^n)+1))); # Muniru A Asiru, Dec 20 2018

Magma

[Floor(2^n*Log(10,2)/Log(10,10))+1: n in [0..40]]; // Vincenzo Librandi, Nov 08 2018

Maple

seq(length(2^(2^n)),n=0..20); # Zerinvary Lajos, Apr 20 2008

Mathematica

Table[ Floor[ 2^n * N[ Log[ 10, 2 ], 24 ] + 1 ], {n, 0, 43} ]

PARI

for(n=0, 50, print(n, " ", floor(2^n*log(2)/log(10))+1); ) \\ Jinyuan Wang, Nov 07 2018

Formula

a(n) = floor(log_10(F_n)+1) (F_n is the n-th Fermat number). - Ivan Panchenko, Sep 06 2009