A389789 - OEIS

A389789

Number of ways to write n as (2-(n mod 2))*p + q + q', where p and q are primes, and q' is the first prime after q.

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 0, 2, 3, 2, 1, 3, 1, 2, 2, 3, 2, 4, 1, 1, 4, 4, 1, 4, 2, 2, 3, 5, 2, 4, 2, 2, 5, 5, 1, 6, 3, 1, 3, 5, 1, 7, 4, 1, 4, 6, 0, 5, 3, 2, 4, 7, 3, 5, 2, 2, 5, 8, 1, 6, 5, 3, 3, 5, 2, 7, 2, 4, 6, 8, 1, 6, 4, 3, 4, 8, 4, 9, 3, 2, 7, 8, 1, 10, 6, 2, 4

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OFFSET

1,15

COMMENTS

Conjecture: (i) a(2*n+1) > 0 for all n > 4. In other words, any odd integer greater than 10 can be written as the sum of a prime and two consecutive primes.

(ii) a(2*n) > 0 for all n > 426. In other words, any even number greater than 853 can be written as 2*p + q + q', where p and q are primes.

This is a new variant of Goldbach's conjecture. It has been verified for positive integers up to 10^5.

It seems that the largest odd number n with a(n) = 1 is 57, and the largest even number n with a(n) = 1 is 2724.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310.

EXAMPLE

a(33) = 1 with 3 + prime(6) + prime(7) = 3 + 13 + 17 = 33.

a(57) = 1 with 5 + prime(9) + prime(10) = 5 + 23 + 29 = 57.

a(600) = 1 with 2*83 + prime(47) + prime(48) = 2*83 + 211 + 223 = 600.

a(1404) = 1 with 2*229 + prime(91) + prime(92) = 2*229 + 467 + 479 = 1404.

a(2724) = 1 with 2*359 + prime(168) + prime(169) = 2*359 + 997 + 1009 = 2724.

MATHEMATICA

S[n_]:=S[n]=Prime[n]+Prime[n+1];

tab={}; Do[r=0; k=1; Label[bb]; If[S[k]>=n, Goto[aa]]; If[PrimeQ[(n-S[k])/(2-Mod[n, 2])], r=r+1]; k=k+1; Goto[bb];

Label[aa]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

CROSSREFS

Cf. A000040, A001043, A002375, A054860, A389790.