A277581 - OEIS

%I #40 Nov 19 2024 22:11:28

%S 0,0,1,0,1,0,5,2,4,0,1,0,3,2,3,0,1,0,2,1,15,0,5,6,2,3,12,0,1,0,11,2,2,

%T 5,3,0,9,1,1,0,1,0,1,1,20,0,3,12,1,6,7,0,4,11,1,2,16,0,1,0,6,2,1,3,2,

%U 0,14,1,1,0,1,0,13,1,1,2,2,0,5,1,11,0,2,7,1,10,4,0

%N Goldbach's problem extended to squares of nonnegative differences of primes: smallest integer >= ((A112823(n) - A234345(n))^2)/n for n >= 2.

%C Where A112823(n) + A234345(n) = 2n and A112823(n) <= A234345(n) (or nonnegative differences of primes). If n is prime, then a(n) = 0.

%C Conjecture: 1 <= a(n) <= m for all n, where m is largest value of a(n), i.e., the sequence of records in a(n) {1, 5, 15, 20, ..., m} is finite.

%e a(8) = 5 because ((A112823(8) - A234345(8))^2)/8 = ((5 - 11)^2)/8 < 5, where 5(prime) + 11(prime) = 2*8;

%e a(9) = 2 because ((A112823(9) - A234345(9))^2)/9 = ((7 - 11)^2)/9 < 2, where 7(prime) + 11(prime) = 2*9;

%e a(10) = 4 because ((A112823(10) - A234345(10))^2)/10 = ((7 - 13)^2)/10 < 4, where 7(prime) + 13(prime) = 2*10.

%Y Cf. A112823 (2 together with A002374), A234345, A277583 (Goldbach's problem extended to squares of prime gaps >= 2).

%K nonn,changed

%O 2,7

%A _Juri-Stepan Gerasimov_, Oct 21 2016