A007443 Binomial transform of primes.

2, 5, 13, 33, 83, 205, 495, 1169, 2707, 6169, 13889, 30993, 68701, 151469, 332349, 725837, 1577751, 3413221, 7349029, 15751187, 33616925, 71475193, 151466705, 320072415, 674721797, 1419327223, 2979993519, 6245693407, 13068049163

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Equals row sums of triangle A164738. Example: a(4) = 33 = sum of terms in row 4 of triangle A164738: (2, 3, 5, 3, 5, 7, 5, 3). - Gary W. Adamson, Aug 23 2009

It might have been more natural to define this sequence with offset 0, which would also make the formula simpler. Then a(n) would be the first term of the sequence obtained from the primes by applying n times the operation "take sums of successive terms", Ts(k) = s(k)+s(k+1). - M. F. Hasler, Jun 02 2017

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..3000 (terms 1..1000 from Vincenzo Librandi)
• Vaclav Kotesovec, Plot of a(n) / (2^n * n * log(n)) for n = 2..20000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A164738.

Cf. A001043, A096277, A096278, A096279. See A287915 for indices of primes.

First differences give A178167.

Programs

Maple

a:=n->add(binomial(n-1,k-1)*ithprime(k),k=1..n): seq(a(n),n=1..30); # Muniru A Asiru, Oct 23 2018

Mathematica

A007443[n_]:=Sum[Binomial[n-1,k-1]Prime[k],{k,n}];Array[A007443,50] (* or *)
Module[{nmax=50,b},b=Prime[Range[nmax]];Join[{2},Table[First[b=ListConvolve[{1,1},b]],nmax-1]]] (* Paolo Xausa, Oct 31 2023 *)

PARI

A007443(n)=sum(k=1,n,binomial(n-1,k-1)*prime(k)) \\ M. F. Hasler, Jun 02 2017

Formula

a(n) = Sum_{k=1..n} binomial(n-1,k-1)*prime(k). - M. F. Hasler, Jun 02 2017

G.f.: Sum_{k>=1} prime(k)*x^k/(1 - x)^k. - Ilya Gutkovskiy, Apr 21 2019

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, May 21 2010