A007446 Exponentiation of e.g.f. for primes.

1, 2, 7, 31, 162, 973, 6539, 48410, 390097, 3389877, 31534538, 312151125, 3271508959, 36149187780, 419604275375, 5100408982825, 64743452239424, 856157851884881, 11768914560546973, 167841252874889898, 2479014206472819045, 37860543940437797897

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

From Tilman Neumann, Oct 05 2008: (Start)

a(n) is also given by

- substituting the primes (A000040) into (the simplest) Faa di Bruno's formula, or

- the complete Bell polynomial of the first n prime arguments, or

- computing n-th moments from the first n primes as cumulants

The examples show that the coefficients of the prime power products are just A036040/A080575 (these are just rearrangements of the same coefficients). Moreover, the prime products of the additional terms span the whole space of natural numbers, thus what we see here is a reordering of the natural numbers! (End)

Examples

			From _Tilman Neumann_, Oct 05 2008: (Start)
Let p_i denote the i-th prime A000040(i). Then
a(1)=2 = 1*p_1
a(2)=7 = 1*p_2 + 1*p_1^2
a(3)=31 = 1*p_3 + 3*p_2*p_1 + 1*p_1^3
a(4)=162= 1*p_4 + 4*p_3*p_1 + 3*p_2^2 + 6*p_2*p_1^2 + 1*p_1^4
a(5)=973= 1*p_5 + 5*p_4*p_1 + 10*p_3*p_2 + 10*p_3*p_1^2 + 15*p_2^2*p_1 + 10*p_2*p_1^3 + 1*p_1^5
(End)
		

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A036040, A080575. - Tilman Neumann, Oct 05 2008

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*ithprime(j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 18 2015

Mathematica

a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n-1, j-1]*Prime[j]*a[n-j], {j, 1, n}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)
Table[Sum[BellY[n, k, Prime[Range[n]]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

MuPAD

completeBellMatrix := proc(x,n)
// x - vector x[1]...x[m], m>=n
local i,j,M;
begin
M:=matrix(n,n): // zero-initialized
for i from 1 to n-1 do
M[i,i+1]:=-1:
end_for:
for i from 1 to n do
for j from 1 to i do
M[i,j] := binomial(i-1,j-1)*x[i-j+1]:
end_for:
end_for:
return (M):
end_proc:
completeBellPoly := proc(x, n)
begin
return (linalg::det(completeBellMatrix(x,n))):
end_proc:
x:=[2,3,5,7,11,13,17,19,23,29]:
for i from 1 to 10 do print(i,completeBellPoly(x,i)): end_for:
// Tilman Neumann, Oct 05 2008

Formula

E.g.f.: exp(Sum_{k>=1} prime(k)*x^k/k!). - Ilya Gutkovskiy, Nov 26 2017