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Binomial transform of {b(n)}, where b(n)=1 for prime n and b(n)=0 otherwise.
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%I #36 Jan 16 2022 23:29:03

%S 0,1,3,6,11,20,37,70,134,255,476,869,1564,2821,5201,9948,19793,40562,

%T 84271,174952,359576,728805,1457402,2885051,5681277,11185110,22103926,

%U 43939533,87864092,176447165,354929146,713198803,1428312446,2846268351

%N Binomial transform of {b(n)}, where b(n)=1 for prime n and b(n)=0 otherwise.

%C Number of compositions of n into a prime number of parts. - _Vladeta Jovovic_, Jan 31 2005

%C The number of pernicious numbers (A052294) between 2^(n-1) and 2^n. Although the graph looks almost like 2^n, the graph of a(n)/2^n has quite a bit of variation. - _T. D. Noe_, Mar 14 2009

%H T. D. Noe, <a href="/A052467/b052467.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialTransform.html">Binomial Transform</a>.

%F G.f.: Sum_{k>=1} (x/(1 - x))^prime(k). - _Ilya Gutkovskiy_, Dec 28 2016

%F a(n) = A121497(n+1) - A121497(n). - _Wesley Ivan Hurt_, Jan 14 2022

%t b[n_] := Boole[ PrimeQ[n]]; a[n_] := Sum[ Binomial[n, k]*b[k], {k, 0, n}]; Table[a[n], {n, 0, 34}] // Differences (* _Jean-François Alcover_, Oct 25 2012 *)

%Y Cf. A038499.

%Y Cf. A010051, A121497.

%K nonn

%O 1,3

%A _Eric W. Weisstein_

%E More terms from _David Wasserman_, Feb 25 2002

%E Description corrected by _T. D. Noe_, May 17 2003