A277583 Goldbach's problem extended to squares of prime gaps (>=2): smallest integer >= ((A078587(n) - A078496(n))^2)/n for n >= 4.

1, 4, 1, 10, 5, 2, 4, 14, 1, 12, 3, 2, 3, 9, 1, 31, 2, 1, 15, 7, 5, 6, 2, 3, 12, 20, 1, 19, 11, 2, 2, 5, 3, 4, 9, 1, 1, 15, 1, 54, 1, 1, 20, 4, 3, 12, 1, 6, 7, 3, 4, 11, 1, 2, 16, 10, 1, 22, 6, 2, 1, 3, 2, 3, 14, 1, 1, 9, 1, 2, 13, 1, 1, 2, 2, 17, 5, 1, 11, 28, 2, 7, 1, 10, 4, 15

Offset: 4

Author

Juri-Stepan Gerasimov, Oct 22 2016

Keywords

• nonn

Comments

Where A078587(n) + A078496(n) = 2n and A078587(n) < A078496(n).

Examples

			a(5) = 4 because ((A078587(5) - A078496(5))^2)/5 = ((3 - 7)^2)/5 < 4, where 3 (prime) = 7 (prime) = 2*5;
a(6) = 1 because ((A078587(6) - A078496(6))^2)/6 = ((5 - 7)^2)/6 < 1, where 5 (prime) + 7 (prime) = 2*6;
a(7) = 10 because ((A078587(7) - A078496(7))^2)/7 = ((3 - 11)^2)/7 < 10, where 3 (prime) + 11 (prime) = 2*7.
		

Crossrefs

Cf. A002375, A078496, A078587, A277581 (Goldbach's problem extended to squares of prime gaps >= 0).

Programs

Mathematica

Table[k = 1; While[k < ((Last@ # - First@ #)^2)/n, k++] &@ Block[{p = n + 1, q}, q = 2 n - p; While[q > 0 && Nand[PrimeQ@ p, PrimeQ@ q], p++; q--]; {p, q}]; k, {n, 4, 89}] (* or *)
Table[Ceiling[4 (n - #)^2/n] &@ Block[{p = n + 1, q}, q = 2 n - p; While[q > 0 && Nand[PrimeQ@ p, PrimeQ@ q], p++; q--]; p], {n, 4, 89}] (* Michael De Vlieger, Oct 26 2016, after T. D. Noe at A078587 and Michel Marcus PARI *)

PARI

maxp(n) = {my(p = precprime(n-1)); while(!isprime(2*n-p), p = precprime(p-1)); p;}
a(n) = ceil(4*(n - maxp(n))^2/n); \\ Michel Marcus, Oct 22 2016