A048298 - OEIS

%I #82 Oct 10 2023 05:20:25

%S 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,

%T 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,

%U 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

%N a(n) = n if n=2^i for i >= 0, otherwise a(n) = 0.

%C Expand x/(x-1) = Sum_{n >= 0} 1/x^n as Sum a(n) / (1+x^n).

%C 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

%H Reinhard Zumkeller, <a href="/A048298/b048298.txt">Table of n, a(n) for n = 0..10000</a>

%H Jean-Paul Allouche and Jeffrey Shallit, <a href="http://webusers.imj-prg.fr/~jean-paul.allouche/kreg2.ps">The Ring of k-regular Sequences, II</a>.

%H Jean-Paul Allouche and Jeffrey Shallit, <a href="http://dx.doi.org/10.1016/S0304-3975(03)00090-2">The ring of k-regular sequences, II</a>, Theoret. Computer Sci., 307 (2003), 3-29.

%H Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seqs., Vol. 6, 2003.

%F Multiplicative with a(2^e)=2^e and a(p^e)=0 for p > 2. - _Vladeta Jovovic_, Jan 27 2002

%F 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

%F 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

%F Dirichlet g.f.: 2^s/(2^s-2). - _Ralf Stephan_, Jun 17 2007

%F Dirichlet g.f.: zeta(s)/eta(s). - _Ralf Stephan_, Mar 25 2015

%F For n>=1, we have a recursion Sum_{d|n}(-1)^(1+(n/d))a(d)=1. - _Vladimir Shevelev_, Jun 09 2009

%F 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

%F a(n) = A209229(n)*n. - _Reinhard Zumkeller_, Oct 17 2015

%F a(n) = n if 2^n mod n == 0 and a(n) = 0 otherwise. - _Chai Wah Wu_, Dec 01 2022

%p 0, seq(op([2^n,0$(2^n-1)]), n=0..10); # _Robert Israel_, Mar 25 2015

%p a := n -> if n = 2^ilog2(n) then n else 0 fi: # _Peter Luschny_, Oct 03 2022

%t Table[n*Boole[Or[n == 1, First /@ FactorInteger@ n == {2}]], {n, 0, 120}] (* _Michael De Vlieger_, Mar 25 2015 *)

%t a[n_] := If[n == 2^IntegerExponent[n, 2], n, 0]; Array[a, 100, 0] (* _Amiram Eldar_, Oct 10 2023 *)

%o (PARI) a(n)=direuler(p=1,n,if(p==2,1/(1-2*X),1))[n] /* _Ralf Stephan_, Mar 27 2015 */

%o (PARI) a(n) = if(n == 0, 0, if(n == 1 << valuation(n, 2), n, 0)); \\ _Amiram Eldar_, Oct 10 2023

%o (Magma) [n eq 2^Valuation(n,2) select n else 0: n in [0..120]]; // _Vincenzo Librandi_, improved by _Bruno Berselli_, Mar 27 2015

%o (Haskell)

%o a048298 n = a209229 n * n  -- _Reinhard Zumkeller_, Oct 17 2015

%o (Python)

%o def A048298(n): return n if n and not(n&-n)^n else 0 # _Chai Wah Wu_, Dec 01 2022

%Y A kind of inverse to A048272. Cf. A060147.

%Y This is Guy Steele's sequence GS(5, 1) (see A135416).

%Y Cf. A209229 (characteristic function of powers of 2).

%K easy,nonn,mult

%O 0,3

%A _Adam Kertesz_

%E More terms from Keiko L. Noble (s1180624(AT)cedarville.edu)