0,2

Contribution from Tilman Neumann (Tilman.Neumann(AT)web.de), 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)

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

Contribution from Tilman Neumann (Tilman.Neumann(AT)web.de), Oct 05 2008: (Start)

Let p_i denote the i.th prime (A000040). 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)

Contribution from Tilman Neumann (Tilman.Neumann(AT)web.de), Oct 05 2008: (Start)

(Other) 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:

(End)

Cf. A036040, A080575, A000040 [From Tilman Neumann (Tilman.Neumann(AT)web.de), Oct 05 2008]

Sequence in context: A030966 A009132 A125275 this_sequence A002872 A105216 A005977

Adjacent sequences: A007443 A007444 A007445 this_sequence A007447 A007448 A007449

easy,nonn

N. J. A. Sloane (njas(AT)research.att.com).