A351318 a(n) is the least prime prime(k), k > n, such that A036689(k) or A036690(k) is s(n) + s(n+1) + ... + s(j), j < k, where each s(i) is either A036689(i) or A036690(i).

3, 7, 13, 31, 47, 47, 53, 53, 73, 137, 103, 131, 109, 137, 239, 257, 229, 349, 257, 269, 331, 347, 389, 409, 257, 389, 251, 229, 499, 487, 509, 491, 541, 487, 353, 739, 571, 743, 727, 307, 883, 743, 929, 827, 971, 911, 887, 569, 1063, 751, 1013, 883, 1451, 977, 1259, 853, 983, 947, 967, 1049

Offset: 1

Author

J. M. Bergot and Robert Israel, Mar 18 2022

Keywords

• nonn

Comments

a(n) is the least prime p such that p*(p-1) or p*(p+1) is the sum of a sequence where each term is either prime(i)*(prime(i)-1) or prime(i)*(prime(i)+1), for i from n to some j.

Examples

			a(3) = 13 because prime(3) = 5, the next two primes are 7 and 11, and 5*6 + 7*6 + 11*10 = 182 = 13*14.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2400

Crossrefs

Cf. A036889, A036890.

Programs

Maple

P:= select(isprime, [2,seq(i,i=3..10^6,2)]):
R:= convert(map(p -> (p*(p-1),p*(p+1)),P),set):
f:= proc(n) local S,T,SR,i,s;
  S:= {P[n]*(P[n]-1),P[n]*(P[n]+1)};
  for i from n+1 do
    T:= [P[i]*(P[i]-1),P[i]*(P[i]+1)];
    S:= map(s -> (s+T[1],s+T[2]),S);
    SR:= S intersect R;
    if SR <> {} then
        s:= (sqrt(1+4*min(SR))-1)/2;
      if isprime(s) then return s else return s+1 fi
    fi
  od
end proc:
map(f, [$1..100]);