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

%I A203008 #23 Feb 06 2024 19:31:28
%S A203008 0,1,8,71,886,12673,230456,4633919,111429982,3343015913,106868339918,
%T A203008 4054408822031,169941130770676,7459593754902673,357142287146260646,
%U A203008 19235986110046059943,1151217759731312559002,71185663518687172418657
%N A203008 (n-1)-st elementary symmetric function of the first n odd primes; a(0) = 0.
%C A203008 Arithmetic derivative of the product of first n odd primes. - _Antti Karttunen_, Jan 31 2024
%C A203008 Primes occur at indices: 3, 19, 23, 117, 119, 127, 161, 209, ..., and they are: 71, 346723099672193960193396979, 15360643606799479140185671512081451, ... - _Antti Karttunen_, Feb 06 2024
%H A203008 Antti Karttunen, <a href="/A203008/b203008.txt">Table of n, a(n) for n = 0..349</a>
%F A203008 From _Antti Karttunen_, Jan 31 2024 and Feb 06 2024: (Start)
%F A203008 a(n) = A003415(A070826(1+n)) = (1/2)*(A024451(1+n)-A070826(1+n)).
%F A203008 For n >= 1, a(n) = A327860(A060389(n)).
%F A203008 A000035(a(n)) = A000035(n).
%F A203008 (End)
%t A203008 f[k_] := Prime[k + 1]; t[n_] := Table[f[k], {k, 1, n}]
%t A203008 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A203008 Table[a[n], {n, 1, 16}] (* A203008 *)
%o A203008 (PARI)
%o A203008 A002110(n) = prod(i=1,n,prime(i));
%o A203008 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A203008 A203008(n) = if(!n,n,A003415(A002110(1+n)/2)); \\ _Antti Karttunen_, Jan 31 2024
%Y A203008 Cf. A000035, A003415, A024451, A060389, A070826 (n-th. symm. function), A071148 (1st symm. func), A327860.
%K A203008 nonn
%O A203008 0,3
%A A203008 _Clark Kimberling_, Dec 29 2011
%E A203008 Term a(0) = 0 prepended by _Antti Karttunen_, Jan 31 2024