A387991 Number of ways to write n as i*prime(i) + j*prime(j) + k*prime(k) + m*prime(m) with 1 <= i <= j <= k <= m.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 3, 1, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 3, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 2, 2, 1, 0, 1, 2, 1, 0, 0, 1, 1, 1, 0, 1, 2

Offset

1,34

Comments

Conjecture: a(n) > 0 for all n >= 3070. In other words, any integer n >= 3070 is a sum of four terms of A033286. This has been verified for n <= 2*10^5.

It seems that the only values of n > 3000 with a(n) = 1 are 3029, 3065, 3093, 3382, 3729, 3926, 4100, 4118, 6683.

Links

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

Zhi-Wei Sun, Weighted sums of four primes, Question 501365 at MathOverflow, Oct. 8, 2025.

Example

a(55) = 1 with 2*prime(2) + 2*prime(2) + 3*prime(3) + 4*prime(4) = 2*3 + 2*3 + 3*5 + 4*7 = 55.
a(4100) = 1 with 4*prime(4) + 11*prime(11) + 11*prime(11) + 30*prime(30) = 4*7 + 11*31 + 11*31 + 30*113 = 4100.
a(4118) = 1 with 8*prime(8) + 19*prime(19) + 19*prime(19) + 20*prime(20) = 8*19 + 19*67 + 19*67 + 20*71 = 4118.
a(6683) = 1 with 4*prime(4) + 22*prime(22) + 24*prime(24) + 27*prime(27) = 4*7 + 22*79 + 24*89 + 27*103 = 6683.

Mathematica

S[n_]:=S[n]=n*Prime[n];
f[n_]:=f[n]=Sum[If[S[k]<=n&&S[k+1]>n, k, 0], {k, 1, PrimePi[n]}];
tab={}; Do[r=0; Do[If[n-S[k]-S[j]-S[i]==S[f[n-S[k]-S[j]-S[i]]], r=r+1], {k, 1, f[Floor[n/4]]}, {j, k, f[Floor[(n-S[k])/3]]}, {i, j, f[Floor[(n-S[k]-S[j])/2]]}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000040, A033286.

Sequence in context: A083891 A363860 A087623 * A265260 A363888 A338268

Adjacent sequences: A387988 A387989 A387990 * A387992 A387993 A387994

Keyword

nonn

Author

Zhi-Wei Sun, Oct 13 2025

Status

approved