A301852 Integers k such that the remainder of the sum of the first k primes divided by the k-th prime is equal to k.

2, 7, 12, 83408, 5290146169416

Offset: 1

Author

J. M. Bergot, Mar 27 2018

Keywords

• nonn
• more

Comments

Integers k such that A071089(k) = k.

From Robert Israel, Mar 27 2018: (Start)

No more terms below 10^7.

Heuristically, the probability that k is a term is 1/prime(k) ~ 1/(k log k).

Since Sum_{k>=2} 1/(k log(k)) diverges, there should be infinitely many terms. However, the sum diverges very slowly, so terms may be very sparse: approximately log(log(k)) terms <= k. (End)

No more terms below 10^9. - Michel Marcus, Mar 28 2018

No more terms below 1.44*10^12. - Giovanni Resta, Apr 06 2018

No mroe terms below 10^13. - Lucas A. Brown, May 18 2023

Examples

			2 is a term because prime(1)+prime(2) = 5 = 2 mod prime(2).
		

Links

• Lucas A. Brown, Python program.

Crossrefs

Cf. A000040, A071089.

Programs

Maple

res:= NULL: p:= 1: s:= 0:
for m from 1 to 10^6 do
  p:= nextprime(p);
  s:= s+p;
  if s mod p = m then res:= res, m fi
od:
res; # Robert Israel, Mar 27 2018

PARI

lista(nn)= my(p = 2, s = 2); for (n=1, nn, if ((s % p) == n, print1(n, ", ")); q = nextprime(p+1); s += q; p = q;); \\ Michel Marcus, Mar 27 2018

Extensions

a(4) from Michel Marcus, Mar 27 2018

a(5) from Lucas A. Brown, May 18 2023