A284413 Exponent of 3 in 2^n + 1.

1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 5, 0, 1, 0, 1, 0, 2

Offset: 1

Author

Jon E. Schoenfield, Mar 26 2017

Keywords

• nonn
• mult
• easy

Comments

Records: a(3^(n-1)) = n and a(k) < n for k < 3^(n-1).

Multiplicative because A051064 is. - Andrew Howroyd, Jul 28 2018

Examples

			a(27) = 4 because 2^27 + 1 = 134217729 = 3^4 * 19 * 87211.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Indranil Ghosh)

Crossrefs

Cf. A051064, A168570 (exponent of 3 in 2^n - 1).

Programs

Magma

[IsEven(n) select 0 else Factorization(3*n)[1][2]: n in [1..87]];

Mathematica

Table[If[OddQ[n], IntegerExponent[3n, 3], 0], {n, 100}] (* Indranil Ghosh, Mar 27 2017 *)

PARI

a(n) = if(n%2, if(n<1, 0, 1 + valuation(n, 3)), 0); \\ Indranil Ghosh, Mar 27 2017

Formula

a(n) = A051064(n) if n is odd, 0 otherwise.

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Sep 11 2020

From Amiram Eldar, Dec 27 2022: (Start)

Multiplicative with a(2^e) = 0, a(3^e) = e+1, and a(p^e) = 1 if p >= 5.

Dirichlet g.f.: zeta(s)*(1-1/2^s)/(1-1/3^s). (End)