A165285 Primes which can be expressed as the sum of 2*p-1 for two or more consecutive primes p.

17, 43, 59, 73, 79, 101, 109, 139, 163, 191, 197, 233, 239, 283, 317, 331, 379, 419, 433, 439, 443, 463, 467, 499, 521, 569, 571, 599, 617, 619, 641, 739, 743, 787, 811, 863, 911, 919, 941, 967, 971, 1021, 1039, 1061, 1063, 1087, 1097, 1109, 1117, 1229, 1289

Offset

1,1

Comments

The original description of this sequence expressed this in terms of pairs (p-1, p).

The pair (p-1,p) has sum 2*p-1 as given by A076274.

Thus, a prime is in this sequence if it can be represented as a sum of consecutive terms from A076274.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

(1+2)+(2+3)+(4+5)=17, (4+5)+(6+7)+(10+11)=43, (6+7)+(10+11)+(12+13)=59, ...

Maple

N:= 10^4: # for terms <= N
pmax:= nextprime(nextprime(ceil((N+3)/6))):
T:= map(p -> 2*p-1, select(isprime, [2, seq(i, i=3..pmax, 2)])):
S:= ListTools:-PartialSums([0, op(T)]):
R:= {}:
for m from 3 by 2 while S[m] <= N do
  for j from 1 do
    v:= S[m+j]-S[j];
    if v > N then break fi;
    if isprime(v) then R:= R union {v} fi
od od:
sort(convert(R, list)); # Robert Israel, Apr 17 2026

Mathematica

lst={}; Do[s=2*Prime[m]-1; Do[p=Prime[n]; s+=(2*p-1); If[PrimeQ[s], If[s<=6793, AppendTo[lst, s]]], {n, m+1, 3*5!}], {m, 1, 3*5!}]; lst=Take[Union@lst, 200]

Crossrefs

Cf. A076274, A130097, A062090, A050150, A015919.

Sequence in context: A215164 A123592 A260553 * A200321 A165981 A109998

Adjacent sequences: A165282 A165283 A165284 * A165286 A165287 A165288

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Sep 13 2009

Extensions

Revised by Sean A. Irvine, Feb 02 2026

Status

approved