A309753 Approximation of the 2-adic integer arctanh(2) up to 2^n.

0, 0, 2, 2, 10, 26, 58, 122, 122, 122, 122, 1146, 1146, 5242, 13434, 29818, 29818, 95354, 95354, 95354, 619642, 619642, 619642, 4813946, 4813946, 21591162, 21591162, 21591162, 21591162, 290026618, 290026618, 290026618, 2437510266, 6732477562, 6732477562

Offset: 0

Author

Jianing Song, Aug 15 2019

Keywords

• nonn

Comments

arctanh(x) = x + x^3/3 + x^5/5 + x^7/7 + ...

Examples

			a(2) = 2^1 mod 2^2 = 2;
a(3) = 2^1 mod 2^3 = 2;
a(4) = (2^1 + 2^3/3) mod 2^4 = 2;
a(5) = (2^1 + 2^3/3) mod 2^5 = 26;
a(6) = (2^1 + 2^3/3 + 2^5/5) mod 2^6 = 58;
a(7) = (2^1 + 2^3/3 + 2^5/5) mod 2^7 = 122.
		

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A309751, A309754.

Programs

PARI

a(n) = lift(sum(i=0, n/2-1, Mod(2^(2*i+1)/(2*i+1), 2^n)))

Formula

a(n) = (Sum_{i=0..floor(n/2)-1} 2^(2*i+1)/(2*i+1)) mod 2^n.

Extensions

Offset corrected by Georg Fischer, Jun 22 2022