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 A048298 a(n) = n if n=2^i with i=0,1,2,3,...; else a(n) = 0. 14
 0, 1, 2, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Expand x/(x-1) = Sum_{n >= 0} 1/x^n as Sum a(n) / (1+x^n). Nim-binomial transform of the natural numbers. If {t(n)} is the Nim-binomial transform of {a(n)}, then t(n)=(S^n)a(0), where Sf(n) denotes the Nim-sum of f(n) and f(n+1); and S^n=S(S^(n-1)). - John W. Layman, Mar 06 2001 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II. J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29. Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003. FORMULA Multiplicative with a(2^e)=2^e and a(p^e)=0 for p > 2. - Vladeta Jovovic, Jan 27 2002 Inverse mod 2 binomial transform of n. a(n) = sum{k=0..n, (-1)^A010060(n-k)*mod(C(n, k), 2)*k}. - Paul Barry, Jan 03 2005 If n=1 we have a(n)=1; if n=p is prime, then (-1)^(p+1)+a(p)=1, thus a(2)=2, and a(p)=0, if p>2. - Vladimir Shevelev, Jun 09 2009 Dirichlet g.f.: 2^s/(2^s-2). - Ralf Stephan, Jun 17 2007 Dirichlet g.f.: zeta(s)/eta(s). - Ralf Stephan, Mar 25 2015 For n>=1, we have a recursion Sum_{d|n}(-1)^(1+(n/d))a(d)=1. - Vladimir Shevelev, Jun 09 2009 For n>=1, there is the recurrence n=Sum_{k=1..n} a(k)*g(n/k) where g(x) = floor(x) - 2*floor(x/2). - Benoit Cloitre, Nov 11 2010 a(n) = A209229(n)*n. - Reinhard Zumkeller, Oct 17 2015 MAPLE 0, seq(op([2^n, 0\$(2^n-1)]), n=0..10); # Robert Israel, Mar 25 2015 MATHEMATICA Table[n*Boole[Or[n == 1, First /@ FactorInteger@ n == {2}]], {n, 0, 120}] (* Michael De Vlieger, Mar 25 2015 *) PROG (PARI) a(n)=direuler(p=1, n, if(p==2, 1/(1-2*X), 1))[n] /* Ralf Stephan, Mar 27 2015 */ (MAGMA) [n eq 2^Valuation(n, 2) select n else 0: n in [0..120]]; // Vincenzo Librandi, improved by Bruno Berselli, Mar 27 2015 (Haskell) a048298 n = a209229 n * n  -- Reinhard Zumkeller, Oct 17 2015 CROSSREFS A kind of inverse to A048272. Cf. A060147. This is Guy Steele's sequence GS(5, 1) (see A135416). Cf. A209229 (characteristic function of powers of 2). Sequence in context: A104774 A087263 A099894 * A123565 A258701 A246160 Adjacent sequences:  A048295 A048296 A048297 * A048299 A048300 A048301 KEYWORD easy,nonn,mult AUTHOR Adam Kertesz (adamkertesz(AT)worldnet.att.net) EXTENSIONS More terms from Keiko L. Noble (s1180624(AT)cedarville.edu) STATUS approved

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