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A121497
Binomial transform of the characteristic function of the prime numbers (A010051).
5
0, 0, 1, 4, 10, 21, 41, 78, 148, 282, 537, 1013, 1882, 3446, 6267, 11468, 21416, 41209, 81771, 166042, 340994, 700570, 1429375, 2886777, 5771828, 11453105, 22638215, 44742141, 88681674, 176545766, 352992931, 707922077, 1421120880, 2849433326
OFFSET
0,4
COMMENTS
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.
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
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
a(n) equals the number of subsets of [n] whose cardinalities are prime. - Ivan N. Ianakiev, Jul 14 2019
Upper and lower bounds are provided by Kim and Sinha (see links). - Jeffrey Shallit, Nov 14 2024
LINKS
T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 0..3324 (terms up to 1000 from Noe)
Sungjin Kim and Nilotpal Kanti Sinha, Binomial probability of prime number of successes, INTEGERS 20 (2020), #A99.
FORMULA
a(n) = Sum_{i=1..pi(n)} binomial(n,prime(i)), where pi(n) is A000720(n), the number of primes <= n.
E.g.f.: exp(x) * (x^2/2! + x^3/3! + x^5/5! + ...) - Eric M. Schmidt, Jul 14 2012
G.f.: Sum_{p prime} x^p/(1-x)^(p+1). - Robert Israel, Sep 27 2018
MAPLE
Primes:= select(isprime, [2, seq(i, i=3..100, 2)]):
G:= add((z/(1-z))^p/(1-z), p=Primes):
S:= series(G, z, 101):
seq(coeff(S, z, i), i=0..100); # Robert Israel, Sep 27 2018
MATHEMATICA
Table[Sum[Binomial[n, Prime[i]], {i, PrimePi[n]}], {n, 40}]
PROG
(PARI) a(n)=my(s); forprime(p=2, n, s+=binomial(n, p)); s \\ Charles R Greathouse IV, Mar 22 2013
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
T. D. Noe, Aug 03 2006
EXTENSIONS
a(0) inserted by Franklin T. Adams-Watters, Jul 13 2012
STATUS
approved