A014342 Convolution of primes with themselves.

4, 12, 29, 58, 111, 188, 305, 462, 679, 968, 1337, 1806, 2391, 3104, 3953, 4978, 6175, 7568, 9185, 11030, 13143, 15516, 18177, 21150, 24471, 28152, 32197, 36678, 41543, 46828, 52621, 58874, 65659, 73000, 80949, 89462, 98631, 108396, 118869, 130102, 142071

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			a(2)=12 because a(2) = prime(1)*prime(2) + prime(2)*prime(1) = 2*3 + 3*2 = 12.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A023626, A024697, A025129, A209403.

Column k=2 of A340991.

Programs

Haskell

a014342 n = a014342_list !! (n-1)
a014342_list= f (tail a000040_list) [head a000040_list] 1 where
   f (p:ps) qs k = sum (zipWith (*) qs $ reverse qs) :
                   f ps (p : qs) (k + 1)
-- Reinhard Zumkeller, Apr 07 2014, Mar 08 2012

Magma

[&+[NthPrime(n-i+1)*NthPrime(i): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 12 2016

Maple

A014342:=n->add(ithprime(i)*ithprime(n+1-i), i=1..n): seq(A014342(n), n=1..50); # Wesley Ivan Hurt, Dec 14 2016

Mathematica

Table[Sum[Prime[i] Prime[n + 1 - i], {i, n}], {n, 40}] (* Michael De Vlieger, Dec 13 2016 *)
Table[With[{p=Prime[Range[n]]},ListConvolve[p,p]],{n,40}]//Flatten (* Harvey P. Dale, May 03 2018 *)

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

Python

from numpy import convolve
from sympy import prime, primerange
def aupton(terms):
  p = list(primerange(2, prime(terms)+1))
  return list(convolve(p, p))[:terms]
print(aupton(41)) # Michael S. Branicky, Apr 12 2021

Formula

a(n) = Sum_{i=1..n} prime(i) * prime(n+1-i), where prime(i) is the i-th prime.

G.f.: (b(x)^2)/x, where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 13 2016

Extensions

More terms from Felix Goldberg (felixg(AT)tx.technion.ac.il), Feb 01 2001