A287915 -id:A287915 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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

Cf. A164738.
Cf. A001043, A096277, A096278, A096279. See A287915 for indices of primes.
First differences give A178167.

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

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 *)

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

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

More terms from Vladimir Joseph Stephan Orlovsky, May 21 2010

A096280 Primes in A007443 (= binomial transform of primes).

Original entry on oeis.org

2, 5, 13, 83, 2707, 71475193, 674721797, 6245693407, 118543624847, 82736199371081, 72298621492552303967009812018997, 2454725173623452943975951834280921, 59966692897276736774965300014477948187539553

Offset: 1

Cino Hilliard, Jun 23 2004

nonn

Sum of reciprocals = 0.2893406979695919267175673140... Are these primes infinite?

The next term is too large to be displayed here. See A287915 for the indices k which yield these primes A007443(k). - M. F. Hasler, Jun 02 2017

Paolo Xausa, Table of n, a(n) for n = 1..20 (terms 1..17 from M. F. Hasler)

Cf. A007443, A287915, A001043, A096277, A096278, A096279.

See A287915 for the corresponding indices of A007443.

A007443[n_]:=Sum[Binomial[n-1,k-1]Prime[k],{k,n}];
With[{upto=500},Select[Array[A007443,upto],PrimeQ]] (* or *)
Module[{upto=500,b},b=Prime[Range[upto]];Join[{2},Select[Table[First[b=ListConvolve[{1,1},b]],upto-1],PrimeQ]]] (* Paolo Xausa, Oct 31 2023 *)

\\ n = terms to add, m = order.
sucsumspr(n,m) = { local(a,b,i,j,k,sr); sr=0; a = primes(1001); b = vector(1001); for(i=1,m, for(j=1,n+n, b[j] = a[j]+ a[j+1]; ); a=b; if(isprime(a[1]),print1(a[1]",");sr+=1.0/a[1]); ); print(); print(sr); }

for(n=1,999, ispseudoprime(A007443(n))&&print1(A007443(n)",")) \\ M. F. Hasler, Jun 02 2017

a(n) = A007443(A287915(n)). - M. F. Hasler, Jun 02 2017

Definition corrected, initial term 2 added, and edited by M. F. Hasler, Jun 02 2017

Name simplified by Paolo Xausa, Nov 05 2023

cp's OEIS Frontend

A007443 Binomial transform of primes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A096280 Primes in A007443 (= binomial transform of primes).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions