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

0,1

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

Links

Table of n, a(n) for n=0..33.

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A007504, A007506, A071089, A274649.

Sequence in context: A342175 A258080 A318181 * A370491 A089082 A217011

Adjacent sequences: A274992 A274993 A274994 * A274996 A274997 A274998

Keyword

nonn,more

Author

Neil Fernandez, Nov 11 2016

Extensions

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

Status

approved