A278693 Numbers n such that prime(n) + prime(n+1) is divisible by 2n + 1.

4, 181, 6458, 6460, 40083, 40121, 10553502, 69709942, 3140421718, 3140421737, 3140421740, 3140421768, 3140421805, 3140421845

Offset

1,1

Comments

The pairs of consecutive prime numbers corresponding to the terms of this sequence are 7,11; 1087,1091; 64579,64591; 64601,64609; 481001,481003; 481447,481469; 189963043,189963047; ...

The corresponding integers (prime(k) + prime(k+1))/(2*k+1) are 2, 6, 10, 10, 12, 12, 18, ...

Links

Table of n, a(n) for n=1..14.

Example

a(1) = 4 since prime(4) + prime(5) = 7 + 11 = 18 is divisible by 2*4 + 1 = 9.
a(2) = 181 since prime(181) + prime(182) = 1087 + 1091 = 2178 is divisible by 2*181 + 1 = 363.
a(3) = 6458 since prime(6458) + prime(6459) = 64579 + 64591 = 129170 is divisible by 2*6458 + 1 = 12917.
a(4) = 6460 since prime(6460) + prime(6461) = 64601 + 64609 = 129210 is divisible by 2*6460 + 1 = 12921.
a(5) = 40083 since prime(40083) + prime(40084) = 481001 + 481003 = 962004 is divisible by 2*40083 + 1 = 80167.
a(6) = 40121 since prime(40121) + prime(40122) = 481447 + 481469 = 962916 is divisible by 2*40121 + 1 = 80243.
a(7) = 10553502 since prime(10553502) + prime(10553503) = 189963043 + 189963047 = 379926090 is divisible by 2*10553502 + 1 = 21107005.
a(8) = 69709942 since prime(69709942) + prime(69709943) = 1394198837 + 1394198863 = 2788397700 is divisible by 2*69709942 + 1 = 139419885.
a(9) = 3140421718 since prime(3140421718) + prime(3140421719) = 75370121237 + 75370121251 = 150740242488 is divisible by 2*3140421718 + 1 = 6280843437.
a(10) = 3140421737 since prime(3140421737) + prime(3140421738) = 75370121689 + 75370121711 = 150740243400 is divisible by 2*3140421737 + 1 = 6280843475.
a(11) = 3140421740 since prime(3140421740) + prime(3140421741) = 75370121767 + 75370121777 = 150740243544 is divisible by 2*3140421740 + 1 = 6280843481.
a(12) = 3140421768 since prime(3140421768) + prime(3140421769) = 75370122439 + 75370122449 = 150740244888 is divisible by 2*3140421768 + 1 = 6280843537.
a(13) = 3140421805 since prime(3140421805) + prime(3140421806) = 75370123327 + 75370123337 = 150740246664 is divisible by 2*3140421805 + 1 = 6280843611.
a(14) = 3140421845 since prime(3140421845) + prime(3140421846) = 75370124273 + 75370124311 = 150740248584 is divisible by 2*3140421845 + 1 = 6280843691.

Prog

(PARI) isok(k) = denominator((prime(k)+prime(k+1))/(2*k+1)) == 1; \\ Michel Marcus, Nov 26 2016
(PARI) n=0; p=2; forprime(q=3, 1e9, n++; if((p+q)%(2*n+1)==0, print1(n", ")); p=q) \\ Charles R Greathouse IV, Nov 28 2016

Crossrefs

Cf. A000040, A001043.

Sequence in context: A269195 A269102 A240282 * A172018 A295285 A322915

Adjacent sequences: A278690 A278691 A278692 * A278694 A278695 A278696

Keyword

nonn,more

Author

Emre APARI, Nov 26 2016

Extensions

a(8)-a(14) from Charles R Greathouse IV, Nov 28 2016

Status

approved