A274995 a(n) is the smallest odd prime that divides (-n) + the sum of all smaller primes, or 0 if no such prime exists.

5, 19, 3, 7, 82811, 3, 11, 17, 3, 191, 5, 3, 37, 29, 3, 5, 69431799799, 3, 1105589, 28463, 3, 431, 2947308589, 3, 7, 5, 3, 59, 11, 3, 5, 7, 3, 41

Offset: 0

Author

Neil Fernandez, Nov 11 2016

Keywords

• nonn
• more

Comments

From Robert G. Wilson v, Nov 15 2016: (Start)

If n == 2 (mod 3) then a(n) = 3;

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

If n == 3 (mod 7) then a(n) = 7;

If n == 6 (mod 11) then a(n) = 11;

If n == 2 (mod 13) then a(n) = 13;

If n == 7 (mod 17) then a(n) = 17;

If n == 1 (mod 19) then a(n) = 19;

If n == 8 (mod 23) then a(n) = 23;

in that order; i.e., from smaller to greater prime modulus, etc.

First occurrence of p>2: 2, 0, 3, 6, 54, 7, 1, 123, 13, 36, 12, 33, 453, 46, ..., .

(End)

The congruence classes in the above list, modulo the prime bases, namely 2, 0, 3, 6, 2, ..., are given by A071089, in which each term is the remainder when the sum of the first n primes is divided by the n-th prime. - Neil Fernandez, Nov 23 2016

Examples

			a(1) = 19 because 19 is the smallest odd prime that divides the sum of (-1) + (sum of all primes smaller than itself), that is, -1 + 58 = 57.
a(7) = 17 because -7 + 2 + 3 + 5 + 7 + 11 + 13 + 17 = 49 and 49/7 = 7.
		

Links

• Robert G. Wilson v, n and a(n), or 0 if no such value is known, for n=0..10000

Crossrefs

Cf. A007504, A007506, A071089, A274649.

Programs

Mathematica

f[n_] := Block[{p = 3, s = 2 - n}, While[ Mod[s, p] != 0, s = s + p; p = NextPrime@ p]; p]; Array[f, 16, 0] (* Robert G. Wilson v, Nov 15 2016 *)

PARI

sump(n) = s = 0; forprime(p=2, n-1, s+=p); s;
a(n) = {my(p=3); while ((sump(p)-n) % p, p = nextprime(p+1)); p;} \\ Michel Marcus, Nov 12 2016

PARI

a(n)=my(s=2); forprime(p=3,, if((s-n)%p==0, return(p)); s+=p) \\ Charles R Greathouse IV, Nov 15 2016

Extensions

a(16)-a(33) from Charles R Greathouse IV, Nov 15 2016