cp's OEIS Frontend

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.

%I A301852 #28 May 18 2023 08:33:01
%S A301852 2,7,12,83408,5290146169416
%N 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.
%C A301852 Integers k such that A071089(k) = k.
%C A301852 From _Robert Israel_, Mar 27 2018: (Start)
%C A301852 No more terms below 10^7.
%C A301852 Heuristically, the probability that k is a term is 1/prime(k) ~ 1/(k log k).
%C A301852 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)
%C A301852 No more terms below 10^9. - _Michel Marcus_, Mar 28 2018
%C A301852 No more terms below 1.44*10^12. - _Giovanni Resta_, Apr 06 2018
%C A301852 No mroe terms below 10^13. - _Lucas A. Brown_, May 18 2023
%H A301852 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A301852.py">Python program</a>.
%e A301852 2 is a term because prime(1)+prime(2) = 5 = 2 mod prime(2).
%p A301852 res:= NULL: p:= 1: s:= 0:
%p A301852 for m from 1 to 10^6 do
%p A301852   p:= nextprime(p);
%p A301852   s:= s+p;
%p A301852   if s mod p = m then res:= res, m fi
%p A301852 od:
%p A301852 res; # _Robert Israel_, Mar 27 2018
%o A301852 (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
%Y A301852 Cf. A000040, A071089.
%K A301852 nonn,more
%O A301852 1,1
%A A301852 _J. M. Bergot_, Mar 27 2018
%E A301852 a(4) from _Michel Marcus_, Mar 27 2018
%E A301852 a(5) from _Lucas A. Brown_, May 18 2023