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 A121497 #37 Nov 14 2024 04:11:28
%S A121497 0,0,1,4,10,21,41,78,148,282,537,1013,1882,3446,6267,11468,21416,
%T A121497 41209,81771,166042,340994,700570,1429375,2886777,5771828,11453105,
%U A121497 22638215,44742141,88681674,176545766,352992931,707922077,1421120880,2849433326
%N A121497 Binomial transform of the characteristic function of the prime numbers (A010051).
%C A121497 This is the binomial transform of the sequence {0,0,1,1,0,1,0,1,...}. Sequence A052467, the binomial transform of the sequence {0,1,1,0,1,0,1,...} is very similar. In fact, the first differences of this sequence yields A052467.
%C A121497 The number of pernicious numbers (A052294) less than 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
%C A121497 a(n)/2^n is the probability that a series of Bernoulli trials with probability of success equal to 1/2 will result in a prime number of successes. Cf. A178851. - _Eric M. Schmidt_, Jul 13 2012
%C A121497 a(n) equals the number of subsets of [n] whose cardinalities are prime. - _Ivan N. Ianakiev_, Jul 14 2019
%C A121497 Upper and lower bounds are provided by Kim and Sinha (see links). - _Jeffrey Shallit_, Nov 14 2024
%H A121497 T. D. Noe and Charles R Greathouse IV, <a href="/A121497/b121497.txt">Table of n, a(n) for n = 0..3324</a> (terms up to 1000 from Noe)
%H A121497 Sungjin Kim and Nilotpal Kanti Sinha, <a href="https://math.colgate.edu/~integers/u99/u99.pdf">Binomial probability of prime number of successes</a>, INTEGERS 20 (2020), #A99.
%H A121497 Vaclav Kotesovec, <a href="/A121497/a121497_1.jpg">Plot of a(n) / (2^n/log(n/2)) for n = 2..10000</a>
%F A121497 a(n) = Sum_{i=1..pi(n)} binomial(n,prime(i)), where pi(n) is A000720(n), the number of primes <= n.
%F A121497 E.g.f.: exp(x) * (x^2/2! + x^3/3! + x^5/5! + ...) - _Eric M. Schmidt_, Jul 14 2012
%F A121497 G.f.: Sum_{p prime} x^p/(1-x)^(p+1). - _Robert Israel_, Sep 27 2018
%p A121497 Primes:= select(isprime, [2,seq(i,i=3..100,2)]):
%p A121497 G:= add((z/(1-z))^p/(1-z),p=Primes):
%p A121497 S:= series(G,z,101):
%p A121497 seq(coeff(S,z,i),i=0..100); # _Robert Israel_, Sep 27 2018
%t A121497 Table[Sum[Binomial[n,Prime[i]], {i,PrimePi[n]}], {n,40}]
%o A121497 (PARI) a(n)=my(s);forprime(p=2,n,s+=binomial(n,p));s \\ _Charles R Greathouse IV_, Mar 22 2013
%Y A121497 Cf. A000720, A010051, A052294, A052467, A178851
%K A121497 nonn
%O A121497 0,4
%A A121497 _T. D. Noe_, Aug 03 2006
%E A121497 a(0) inserted by _Franklin T. Adams-Watters_, Jul 13 2012