A356178 Numbers k such that both Sum_{i=1..k} i*prime(i) and Sum_{i=1..k} (k+1-i)*prime(i) are prime.

1, 3, 199, 351, 1583, 1955, 2579, 2627, 3251, 3407, 3503, 5311, 6359, 6819, 7295, 7547, 8791, 9663, 10143, 10591, 11499, 11579, 12199, 12443, 14527, 15563, 15583, 16051, 16543, 16655, 18047, 18319, 20691, 20847, 23979, 24079, 24575, 25667, 26491, 28235, 30395, 30627, 32235, 32259, 33091, 33287, 33527

Offset: 1

Author

J. M. Bergot and Robert Israel, Jul 28 2022

Keywords

• nonn

Comments

Numbers k such that A014148(k) and A014285(k) are both prime.

a(n) == 3 (mod 4) for n > 1.

Examples

			a(2) = 3 is a term because Sum_{i=1..3} i*prime(i) = 1*2 + 2*3 + 3*5 = 23 and Sum_{i=1..3} (4-i)*prime(i) = 3*2 + 2*3 + 1*5 = 17 are prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A014148, A014285.

Programs

Maple

S1:= 2: S2:= 2: S3:= 2*S2-S1: R:= 1: count:= 1: p:= 2:
for n from 2 to 40000 do
  p:= nextprime(p);
  S1:= S1 + n*p;
  S2:= S2 + p;
  if n mod 4 = 3 and isprime(S1) then
    S3:= (n+1)*S2 - S1;
    if isprime(S3) then
       count:= count+1; R:= R, n;
    fi
  fi;
od:
R;

Mathematica

r = Range[35000]; p = Prime[r]; Intersection[Position[Accumulate[r*p], ?PrimeQ], Position[Accumulate[Accumulate[p]], ?PrimeQ]] // Flatten (* Amiram Eldar, Jul 28 2022 *)

PARI

isok(k) = my(vp=primes(k)); isprime(sum(i=1, k, i*vp[i])) && isprime(sum(i=1, k, (k+1-i)*vp[i])); \\ Michel Marcus, Jul 29 2022