A361259 a(n) is the least semiprime that is the sum of n consecutive primes.

10, 26, 39, 358, 58, 77, 155, 129, 583, 562, 323, 326, 551, 381, 629, 501, 707, 1294, 789, 791, 961, 1354, 1159, 1262, 1369, 1371, 1591, 1718, 1849, 1851, 2271, 2127, 3561, 2427, 3077, 2747, 3085, 3442, 4811, 3826, 3829, 3831, 5089, 4227, 4659, 4661, 5345, 7318, 5587, 8146, 6333, 6081, 6338

Offset

3,1

Comments

No sum of two consecutive primes is a semiprime. Proof: if prime(k) and prime(k+1) are odd primes, prime(k)+prime(k+1) is even, so the only way it could be a semiprime is for (prime(k)+prime(k+1))/2 to be prime. But this is impossible because prime(k) and prime(k+1) are consecutive primes and (prime(k)+prime(k+1))/2 is between them.

Links

Robert Israel, Table of n, a(n) for n = 3..10000

Example

a(3) = 10 = 2+3+5, a(4) = 26 = 3+5+7+11, a(5) = 39 = 3+5+7+11+13, a(6) = 358 = 47+53+59+61+67+71.

Maple

P:= select(isprime, [2, seq(i, i=3..10^6, 2)]):
S:= ListTools:-PartialSums(P):
f:= proc(n) local k, s;
  if numtheory:-bigomega(S[n])=2 then return S[n] fi;
  if n::even then
    for k from 1 do
      if isprime((S[n+k]-S[k])/2) then return S[n+k]-S[k] fi
    od
  else
    for k from 1 do
      if numtheory:-bigomega(S[n+k]-S[k]) = 2 then return S[n+k]-S[k] fi
    od
  fi
end proc:
map(f, [$3..100]);

Mathematica

pr = Prime[Range[10^6]]; Do[to = Total /@ Partition[pr, n, 1]; se =
Select[to, 2 == PrimeOmega[#] &, 1][[1]]; AppendTo[s, se], {n, 3, 30}]; s

Crossrefs

Cf. A001358.

Sequence in context: A322972 A080059 A071348 * A055042 A044071 A186279

Adjacent sequences: A361256 A361257 A361258 * A361260 A361261 A361262

Keyword

nonn

Author

Zak Seidov and Robert Israel, Mar 06 2023

Status

approved