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

1,1

Links

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

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

Example

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

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 *)

Prog

(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
(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
(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

Crossrefs

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

Column k=2 of A340991.

Sequence in context: A350822 A050898 A009845 * A086274 A174121 A128563

Adjacent sequences: A014339 A014340 A014341 * A014343 A014344 A014345

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved