A274649 a(n) is the smallest odd prime that divides n + the sum of all smaller primes, or 0 if no such prime exists.

5, 3, 30915397, 11339869, 3, 5, 859, 3, 41, 233, 3, 7, 4175194313, 3, 307, 5, 3, 1459, 7, 3, 5, 9907, 3, 647, 13, 3, 31, 11, 3, 193, 5, 3, 7, 2939, 3, 5, 3167, 3, 11, 7, 3, 1321, 86629, 3, 17, 5, 3

Offset: 0

Author

Neil Fernandez, Nov 10 2016

Keywords

• nonn
• more

Comments

From David A. Corneth, Nov 12 2016: (Start)

a(n) is the smallest odd prime p such that p|(n + A007504(primepi(p) - 1)) or zero if no such p exists.

If a(n) = p then a(n + p) <= p. (End)

If n is congruent to 1 (mod 3), then a(n)=3.

a(2), a(3) and a(12) were found by Jack Brennen.

From Robert G. Wilson v, Nov 13 2016: (Start)

If n == 1 (mod 3) then a(n) = 3;

If n == 0 (mod 5) then a(n) = 5;

If n == 4 (mod 7) then a(n) = 7;

if n == 5 (mod 11) then a(n) = 11;

if n == 11 (mod 13) then a(n) = 13;

if n == 10 (mod 17) then a(n) = 17;

if n == 18 (mod 19) then a(n) = 19;

if n == 23 (mod 23) then a(n) = 23;

in that order, i.e., from smallest to greatest prime modulus, etc.

First occurrence of p > 2: 1, 0, 11, 27, 24, 44, 56, 84, 161, ..., .

a(47) > 10^11. (End)

Examples

			a(6) = 859 because 859 is the smallest odd prime that divides the sum of 6 + (sum of all primes smaller than itself).
a(8) = 41 because 8+2+3+5+7+11+13+17+19+23+29+31+37+41 = 246 and 246/41 = 6.
		

Links

• Michael S. Branicky, Alternate Python program.
• Michael S. Branicky, n and a(n) for n = 0..10000 or 0 of no such value is known, search limit = 9*10^9.
• Robert G. Wilson v, n and a(n) for n = 0..10000 or 0 if no such value is known.

Crossrefs

Cf. A007504, A007506, A024011, A274995.

Cf. A016777 (n==1 (mod 3))

Programs

Mathematica

f[n_] := Block[{p = 3, s = n +2}, While[ Mod[s, p] != 0, s = s + p; p = NextPrime@ p]; p]; Array[f, 47, 0] (* Robert G. Wilson v, Nov 12 2016 *)

Python

# see link for an alternate that searches in parallel to a limit
from sympy import nextprime
def a(n):
  psum, p = 2, 3
  while (n + psum)%p: psum, p = psum + p, nextprime(p)
  return p
for n in range(12):
  print(a(n), end=", ") # Michael S. Branicky, May 03 2021