A203008 (n-1)-st elementary symmetric function of the first n odd primes; a(0) = 0.

0, 1, 8, 71, 886, 12673, 230456, 4633919, 111429982, 3343015913, 106868339918, 4054408822031, 169941130770676, 7459593754902673, 357142287146260646, 19235986110046059943, 1151217759731312559002, 71185663518687172418657

Offset: 0

Author

Clark Kimberling, Dec 29 2011

Keywords

• nonn

Comments

Arithmetic derivative of the product of first n odd primes. - Antti Karttunen, Jan 31 2024

Primes occur at indices: 3, 19, 23, 117, 119, 127, 161, 209, ..., and they are: 71, 346723099672193960193396979, 15360643606799479140185671512081451, ... - Antti Karttunen, Feb 06 2024

Links

• Antti Karttunen, Table of n, a(n) for n = 0..349

Crossrefs

Cf. A000035, A003415, A024451, A060389, A070826 (n-th. symm. function), A071148 (1st symm. func), A327860.

Programs

Mathematica

f[k_] := Prime[k + 1]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203008 *)

PARI

A002110(n) = prod(i=1,n,prime(i));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A203008(n) = if(!n,n,A003415(A002110(1+n)/2)); \\ Antti Karttunen, Jan 31 2024

Formula

From Antti Karttunen, Jan 31 2024 and Feb 06 2024: (Start)

a(n) = A003415(A070826(1+n)) = (1/2)*(A024451(1+n)-A070826(1+n)).

For n >= 1, a(n) = A327860(A060389(n)).

A000035(a(n)) = A000035(n).

(End)

Extensions

Term a(0) = 0 prepended by Antti Karttunen, Jan 31 2024