A136188 Digital roots of the Fermat numbers in A000215(n).

3, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8

Offset: 0

Author

Ant King, Dec 24 2007

Keywords

• easy
• base
• nonn

Comments

As 2^(2^n)+1=5 (mod 9) for odd values of n and 2^(2^n)+1=8 (mod 9) for even values of n>0, it follows that the digital roots of the Fermat numbers form a cyclic sequence, with the 5's corresponding to odd values of n and the 8's to even values of n.

Decimal expansion of 71/198. - Enrique Pérez Herrero, Nov 13 2021

Examples

			2^(2^3) + 1 = 257. This has digital root 5 and hence a(3) = 5.
		

Links

• I. Izmirli, On Some Properties of Digital Roots, Advances in Pure Mathematics, Vol. 4 No. 6 (2014), Article ID:47285.
• Eric Weisstein's World of Mathematics, Digital Root.
• Eric Weisstein's World of Mathematics, Fermat Number.
• Index entries for linear recurrences with constant coefficients, signature (0,1).

Crossrefs

Cf. A000215, A010888, A135928.

Essentially the same as A010719.

Programs

Mathematica

FermatNumber[n_]:=2^(2^n)+1;DigitalRoot[n_]:=FixedPoint[Plus@@IntegerDigits[ # ]&,n];DigitalRoot/@(FermatNumber[ # ] &/@Range[0,25])

PARI

a(n)=if(n,if(n%2,5,8),3) \\ Charles R Greathouse IV, May 01 2016

Formula

a(n) = A010888(A000215(n)).