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%I #44 Sep 08 2022 08:44:39
%S 4,12,29,58,111,188,305,462,679,968,1337,1806,2391,3104,3953,4978,
%T 6175,7568,9185,11030,13143,15516,18177,21150,24471,28152,32197,36678,
%U 41543,46828,52621,58874,65659,73000,80949,89462,98631,108396,118869,130102,142071
%N Convolution of primes with themselves.
%H Reinhard Zumkeller, <a href="/A014342/b014342.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{i=1..n} prime(i) * prime(n+1-i), where prime(i) is the i-th prime.
%F G.f.: (b(x)^2)/x, where b(x) is the g.f. of A000040. - _Mario C. Enriquez_, Dec 13 2016
%e a(2)=12 because a(2) = prime(1)*prime(2) + prime(2)*prime(1) = 2*3 + 3*2 = 12.
%p A014342:=n->add(ithprime(i)*ithprime(n+1-i), i=1..n): seq(A014342(n), n=1..50); # _Wesley Ivan Hurt_, Dec 14 2016
%t Table[Sum[Prime[i] Prime[n + 1 - i], {i, n}], {n, 40}] (* _Michael De Vlieger_, Dec 13 2016 *)
%t Table[With[{p=Prime[Range[n]]},ListConvolve[p,p]],{n,40}]//Flatten (* _Harvey P. Dale_, May 03 2018 *)
%o (PARI) {m=40;u=vector(m,x,prime(x));for(n=1,m,v=vecextract(u,concat("1..",n)); w=vector(n,x,u[n+1-x]);print1(v*w~,","))} \\ _Klaus Brockhaus_, Apr 28 2004
%o (Haskell)
%o a014342 n = a014342_list !! (n-1)
%o a014342_list= f (tail a000040_list) [head a000040_list] 1 where
%o f (p:ps) qs k = sum (zipWith (*) qs $ reverse qs) :
%o f ps (p : qs) (k + 1)
%o -- _Reinhard Zumkeller_, Apr 07 2014, Mar 08 2012
%o (Magma) [&+[NthPrime(n-i+1)*NthPrime(i): i in [1..n]]: n in [1..40]]; // _Bruno Berselli_, Apr 12 2016
%o (Python)
%o from numpy import convolve
%o from sympy import prime, primerange
%o def aupton(terms):
%o p = list(primerange(2, prime(terms)+1))
%o return list(convolve(p, p))[:terms]
%o print(aupton(41)) # _Michael S. Branicky_, Apr 12 2021
%Y Cf. A000040, A023626, A024697, A025129, A209403.
%Y Column k=2 of A340991.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms from Felix Goldberg (felixg(AT)tx.technion.ac.il), Feb 01 2001
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