This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A267092 #39 Apr 19 2025 19:36:35
%S A267092 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,
%T A267092 39,27,56,29,45,31,112,33,51,35,72,37,57,39,100,41,63,43,88,45,69,47,
%U A267092 144,49,75,51,104,53,81,55,140,57,87,59,120,61,93,63,256,65,99,67,136
%N A267092 a(n) is the number of P-positions for n-modular Nim with 2 piles.
%C A267092 The sequence is multiplicative.
%H A267092 Amiram Eldar, <a href="/A267092/b267092.txt">Table of n, a(n) for n = 1..10000</a>
%H A267092 Tanya Khovanova and Karan Sarkar, <a href="https://doi.org/10.1007/s00182-016-0545-7">P-positions in modular extensions to Nim</a>, International Journal of Game Theory, Vol. 46, No. 2 (2017), pp. 547-561, <a href="https://arxiv.org/abs/1508.07054">preprint</a>, arXiv:1508.07054 [math.CO], 2015.
%H A267092 Mircea Merca, <a href="https://doi.org/10.1007/s40590-024-00652-1">Euler's partition function in terms of 2-adic valuation</a>, Bol. Soc. Mat. Mex. 30, 76 (2024). See p. 11.
%H A267092 Mircea Merca, <a href="https://doi.org/10.1007/s00010-024-01117-6">Overpartitions in terms of 2-adic valuation</a>, Aequat. Math. (2024). See pp. 15-16.
%F A267092 a(n) = n, if n is odd.
%F A267092 a(2*n) = n + 2*a(n).
%F A267092 a(n) = n(nu(n)/2+1), where nu(n) is the 2-adic order of n.
%F A267092 From _Werner Schulte_, Feb 07 2018: (Start)
%F A267092 Multiplicative with a(2^e)=(e+2)*2^(e-1) and a(p^e)=p^e for p>2 and e>0.
%F A267092 Dirichlet g.f.: zeta(s-1)*(2^s-1)/(2^s-2).
%F A267092 a(n) = Sum_{d|n} A006519(d)*A000010(n/d). (End)
%F A267092 Sum_{k=1..n} a(k) ~ 3*n^2/4. - _Vaclav Kotesovec_, Sep 10 2020
%e A267092 The P-positions for 2-modular Nim with 2 piles are: (0,0), (1,2), (2,1). Thus a(2) = 3.
%t A267092 Table[n (IntegerExponent[n, 2]/2 + 1), {n, 100}]
%o A267092 (PARI) a(n) = n*(valuation(n, 2)/2 + 1); \\ _Michel Marcus_, Jan 13 2016
%Y A267092 Cf. A000010, A006519, A267093.
%K A267092 nonn,mult,easy
%O A267092 1,2
%A A267092 _Tanya Khovanova_ and Karan Sarkar, Jan 10 2016