cp's OEIS Frontend

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).

%I A351318 #23 Jun 02 2025 15:25:04
%S A351318 3,7,13,31,47,47,53,53,73,137,103,131,109,137,239,257,229,349,257,269,
%T A351318 331,347,389,409,257,389,251,229,499,487,509,491,541,487,353,739,571,
%U A351318 743,727,307,883,743,929,827,971,911,887,569,1063,751,1013,883,1451,977,1259,853,983,947,967,1049
%N 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).
%C A351318 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.
%H A351318 Robert Israel, <a href="/A351318/b351318.txt">Table of n, a(n) for n = 1..2400</a>
%e A351318 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.
%p A351318 P:= select(isprime, [2,seq(i,i=3..10^6,2)]):
%p A351318 R:= convert(map(p -> (p*(p-1),p*(p+1)),P),set):
%p A351318 f:= proc(n) local S,T,SR,i,s;
%p A351318   S:= {P[n]*(P[n]-1),P[n]*(P[n]+1)};
%p A351318   for i from n+1 do
%p A351318     T:= [P[i]*(P[i]-1),P[i]*(P[i]+1)];
%p A351318     S:= map(s -> (s+T[1],s+T[2]),S);
%p A351318     SR:= S intersect R;
%p A351318     if SR <> {} then
%p A351318         s:= (sqrt(1+4*min(SR))-1)/2;
%p A351318       if isprime(s) then return s else return s+1 fi
%p A351318     fi
%p A351318   od
%p A351318 end proc:
%p A351318 map(f, [$1..100]);
%Y A351318 Cf. A036889, A036890.
%K A351318 nonn
%O A351318 1,1
%A A351318 _J. M. Bergot_ and _Robert Israel_, Mar 18 2022