A390352 Least positive integer m such that (2*n-1)*(prime(m)+prime(m+1)+prime(m+2)) = prime(k)+prime(k+1)+prime(k+2) for some k > 0.

1, 16, 28, 5, 37, 12, 5, 64, 8, 4, 180, 4, 9, 19, 11, 3, 27, 11, 9, 50, 4, 8, 322, 4, 7, 17, 11, 9, 11, 6, 4, 7, 27, 14, 41, 7, 16, 148, 12, 7, 8, 5, 9, 12, 4, 6, 26, 28, 11, 10, 7, 4, 127, 43, 5, 111, 3, 10, 28, 3, 69, 70, 42, 24, 5, 14, 7, 39, 15, 7, 10, 8, 6, 6, 6, 6, 75, 4, 4, 9, 7, 15, 106, 4, 7, 101, 8, 8, 56, 23

Offset

1,2

Comments

The existence of a(n) follows from the following general conjecture.

Conjecture: Let m > 1 and let P(m) be the set of all sums of m consecutive primes.

(i) If m is even, then each positive rational number can be written as p/q with p and q elements of P(m).

(ii) If m is odd, then for any positive odd integers a and b we can write a/b as p/q, where p and q belong to the set P(m).

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

Example

a(1) = 1 since (2*1-1)*(prime(1)+prime(2)+prime(3)) = 2 + 3 + 5 is a sum of three consecutive primes.
a(2) = 16 since (2*2-1)*(prime(16)+prime(17)+prime(18)) = 3*(53+59+61) = 519 = 167+173+179 = prime(41)+prime(42)+prime(43).

Mathematica

p[n_]:=p[n]=Prime[n]; S[n_]:=S[n]=p[n]+p[n+1]+p[n+2];
f[n_]:=f[n]=Sum[If[S[k]<=n&&S[k+1]>n, k, 0], {k, 1, PrimePi[n/3]}];
YQ[n_]:=YQ[n]=S[f[n]]==n;
tab={}; Do[m=1; Label[bb]; If[YQ[(2n-1)*S[m]], tab=Append[tab, m]; Goto[aa]]; m=m+1; Goto[bb]; Label[aa], {n, 1, 90}]; Print[tab]

Crossrefs

Cf. A000040, A034961, A389793, A386888.

Sequence in context: A302553 A300132 A101857 * A350302 A184039 A212051

Adjacent sequences: A390349 A390350 A390351 * A390353 A390354 A390355

Keyword

nonn

Author

Zhi-Wei Sun, Nov 02 2025

Status

approved