A389790 Number of ways to write 2*n as p + p' + q + q', where p and q are primes with p <= q, and r' is the first prime greater than r.

0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 1, 0, 2, 1, 0, 3, 1, 0, 3, 0, 0, 4, 0, 1, 2, 1, 1, 3, 0, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 0, 4, 2, 0, 4, 1, 2, 3, 0, 1, 6, 0, 2, 2, 2, 3, 2, 0, 5, 2, 1, 3, 2, 3, 1, 3, 4, 1, 4, 2, 2, 4, 3, 0, 5, 3, 2, 4, 1, 1, 8, 1, 2, 2, 3, 4, 1, 2, 4, 4, 1

Offset

1,18

Comments

Conjecture: a(n) > 0 for all n >= 474.

This is an analog of Goldbach's conjecture. It has been verified for n <= 2*10^5.

It seems that 683 is the largest value of n with a(n) = 1.

From Chai Wah Wu, Oct 15 2025: (Start)

Conjecture: for all k, there exists n_k such that a(m)>k for all m >= n_k.

k conjectured largest value of n for which a(n) = k

----------------

2 833

3 1487

4 1411

5 1523

6 1747

7 2621

8 2153

9 3091

10 3238

(End)

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(10) = 1 with prime(2) + prime(3) + prime(3) + prime(4) = 3 + 5 + 5 + 7 = 2*10.
a(70) = 1 with prime(3) + prime(4) + prime(18) + prime(19) = 5 + 7 + 61 + 67 = 2*70.
a(100) = 1 with prime(15) prime(16) + prime(15) + prime(16) = 47 + 53 + 47 + 53 = 2*100.
a(421) = 1 with prime(14) + prime(15) + prime(74) + prime(75) = 43 + 47 + 373 + 379 = 2*421.
a(511) = 1 with prime(37) + prime(38) + prime(70) + prime(71) = 157 + 163 + 349 + 353 = 2*511.
a(683) = 1 with prime(24) + prime(25) + prime(107) + prime(108) = 89 + 97 + 587 + 593 = 2*683.

Mathematica

p[n_]:=p[n]=Prime[n]; S[n_]:=S[n]=p[n]+p[n+1];
f[n_]:=f[n]=Sum[If[S[k]<=n&&S[k+1]>n, k, 0], {k, 1, PrimePi[n/2]}];
tab={}; Do[r=0; Do[If[S[f[2n-S[k]]]==2n-S[k], r=r+1], {k, 1, f[n]}];
tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000040, A001043, A002375, A389789.

Sequence in context: A283486 A389443 A386992 * A330759 A216607 A025672

Adjacent sequences: A389787 A389788 A389789 * A389791 A389792 A389793

Keyword

nonn

Author

Zhi-Wei Sun, Oct 15 2025

Status

approved