A259643 Numbers k such that sum of first k odd primes divides product of first k odd primes.

1, 3, 5, 11, 25, 29, 41, 49, 51, 59, 69, 81, 99, 103, 113, 131, 133, 135, 147, 149, 153, 181, 187, 193, 197, 199, 205, 211, 213, 217, 219, 229, 235, 239, 243, 255, 271, 277, 281, 287, 289, 303, 309, 313, 323, 333, 335, 343, 347, 357, 359, 365, 367, 381, 383, 389

Offset

1,2

Comments

Obviously, a(n) is always an odd number.

Links

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

Example

1 is a term because prime(2) mod prime(2) = 3 mod 3 = 0.
3 is a term because (prime(2) * prime(3) * prime(4)) mod (prime(2) + prime(3) + prime(4)) = 105 mod 15 = 0.
5 is a term because (prime(2) * prime(3) * prime(4) * prime(5) * prime(6)) mod (prime(2) + prime(3) + prime(4) + prime(5) + prime(6)) = 15015 mod 39 = 0.

Maple

N:= 1000: # for terms <= N
Pr:= [seq(ithprime(i), i=2..N+1)]:
S:= ListTools:-PartialSums(Pr):
filter:= proc(k)
  local x, j;
  x:= S[k];
  for j from 1 to k while x > 1 do
    if x mod Pr[j] = 0 then x:= x/Pr[j] fi
  od;
  x = 1
end proc:
select(filter, [seq(i, i=1..N, 2)]); # Robert Israel, Nov 23 2025

Mathematica

Module[{nn=400, op}, op=Prime[Range[2, nn+1]]; Select[Range[nn], Divisible[ Times@@ Take[op, #], Total[Take[op, #]]]&]] (* Harvey P. Dale, Nov 16 2022 *)

Prog

(PARI) for(n=1, 1e3, if( prod(k=1, n, prime(k+1)) % sum(k=1, n, prime(k+1)) == 0 , print1(n", ")))

Crossrefs

Cf. A051838, A065091, A070826, A071089, A071148, A262807.

Sequence in context: A214873 A330951 A018116 * A167796 A261896 A285184

Adjacent sequences: A259640 A259641 A259642 * A259644 A259645 A259646

Keyword

nonn,easy

Author

Altug Alkan, Oct 02 2015

Status

approved