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
Examples
The P-positions for 2-modular Nim with 2 piles are: (0,0), (1,2), (2,1). Thus a(2) = 3.
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.
- Mircea Merca, Euler's partition function in terms of 2-adic valuation, Bol. Soc. Mat. Mex. 30, 76 (2024). See p. 11.
- Mircea Merca, Overpartitions in terms of 2-adic valuation, Aequat. Math. (2024). See pp. 15-16.
Programs
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Mathematica
Table[n (IntegerExponent[n, 2]/2 + 1), {n, 100}] -
PARI
a(n) = n*(valuation(n, 2)/2 + 1); \\ Michel Marcus, Jan 13 2016
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).
Sum_{k=1..n} a(k) ~ 3*n^2/4. - Vaclav Kotesovec, Sep 10 2020
Comments