A007446 Exponentiation of e.g.f. for primes. (Formerly M1785)

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

Offset

0,2

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)

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

Formula

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

Example

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A030966 A009132 A125275 * A277396 A227119 A002872

Adjacent sequences: A007443 A007444 A007445 * A007447 A007448 A007449

Keyword

easy,nonn

Author

N. J. A. Sloane

Status

approved