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

1,2

Comments

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

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

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A014148, A014285.

Sequence in context: A157598 A023310 A054701 * A342578 A264674 A306630

Adjacent sequences: A356175 A356176 A356177 * A356179 A356180 A356181

Keyword

nonn

Author

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

Status

approved