A267092 a(n) is the number of P-positions for n-modular Nim with 2 piles.

1, 3, 3, 8, 5, 9, 7, 20, 9, 15, 11, 24, 13, 21, 15, 48, 17, 27, 19, 40, 21, 33, 23, 60, 25, 39, 27, 56, 29, 45, 31, 112, 33, 51, 35, 72, 37, 57, 39, 100, 41, 63, 43, 88, 45, 69, 47, 144, 49, 75, 51, 104, 53, 81, 55, 140, 57, 87, 59, 120, 61, 93, 63, 256, 65, 99, 67, 136

Offset

1,2

Comments

The sequence is multiplicative.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Tanya Khovanova and Karan Sarkar, P-positions in modular extensions to Nim, International Journal of Game Theory, Vol. 46, No. 2 (2017), pp. 547-561, preprint, arXiv:1508.07054 [math.CO], 2015.

Formula

a(n) = n, if n is odd.

a(2*n) = n + 2*a(n).

a(n) = n(nu(n)/2+1), where nu(n) is the 2-adic order of n.

From Werner Schulte, Feb 07 2018: (Start)

Multiplicative with a(2^e)=(e+2)*2^(e-1) and a(p^e)=p^e for p>2 and e>0.

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

a(n) = Sum_{d|n} A006519(d)*A000010(n/d). (End)

Sum_{k=1..n} a(k) ~ 3*n^2/4. - Vaclav Kotesovec, Sep 10 2020

Example

The P-positions for 2-modular Nim with 2 piles are: (0,0), (1,2), (2,1). Thus a(2) = 3.

Mathematica

Table[n (IntegerExponent[n, 2]/2 + 1), {n, 100}]

Prog

(PARI) a(n) = n*(valuation(n, 2)/2 + 1); \\ Michel Marcus, Jan 13 2016

Crossrefs

Cf. A000010, A006519, A267093.

Sequence in context: A069462 A097469 A279727 * A272212 A319133 A266560

Adjacent sequences: A267089 A267090 A267091 * A267093 A267094 A267095

Keyword

nonn,mult,easy

Author

Tanya Khovanova and Karan Sarkar, Jan 10 2016

Status

approved