A387043 Number of ways to write n as (1 + (n mod 2))*p + prime(k) + prime(k+1) + prime(k+2), where p is a prime and k is a positive integer.

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

Offset

1,26

Comments

Conjecture: (i) a(2*n) > 0 for all n > 12. In other words, any even number greater than 25 is the sum of a prime and three consecutive primes.

(ii) a(2*n+1) > 0 for all n > 135. In other words, any odd number greater than 272 can be written as the sum of twice a prime and three consecutive primes.

It seems that 80 is the largest even number m with a(m) = 1 and 511 is the largest odd number n with a(n) = 1.

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(50) = 1 with 50 = 19 + (7 + 11 + 13), where 19 is prime, and 7, 11, 13 are consecutive primes.
a(80) = 1 with 80 = 31 + (13 + 17 + 19), where 31 is prime, and 13, 17, 19 are consecutive primes.
a(379) = 1 with 379 = 2*103 + (53 + 59 + 61), where 103 is prime, and 53, 59, 61 are consecutive primes.
a(511) = 1 with 511 = 2*43 + (137 + 139 + 149), where 43 is prime, and 137, 139, 149 are consecutive primes.

Mathematica

p[n_]:=p[n]=Prime[n];
S[n_]:=S[n]=p[n]+p[n+1]+p[n+2];
tab={}; Do[r=0; k=1; Label[bb]; If[S[k]>=n, Goto[aa]]; If[PrimeQ[(n-S[k])/(1+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, A034961, A389789, A389790.

Sequence in context: A127249 A127251 A063251 * A060208 A004570 A178064

Adjacent sequences: A387040 A387041 A387042 * A387044 A387045 A387046

Keyword

nonn

Author

Zhi-Wei Sun, Oct 15 2025

Status

approved