A065139 Numbers n such that the sum of prime(n) and pi(n) is divisible by n.

1, 2, 7, 9, 23, 57, 149, 368, 921, 5863, 14531, 36087, 36255, 36257, 233084, 1505984, 1151321194, 1151321361, 7826138122, 967823489175, 967823489458, 967823489477, 967823489490, 967823489491, 2545928465925, 123116092093107, 123116092093185, 123116092094024

Offset: 1

Author

Labos Elemer, Oct 16 2001

Keywords

• nonn

Examples

			p(233084) = 3242497, Pi(233084) = 20679; sum = 3263176 = 14*233084; order of quotient is log(n).
(prime(1505984) + pi(1505984))/1505984 = (23981141 + 114603)/1505984 = 16.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..33

Crossrefs

Cf. A000040, A000720, A006988.

Programs

Mathematica

Do[ If[ IntegerQ[ (Prime[n] + PrimePi[n]) /n ], Print[n]], {n, 1, 5*10^7} ]
Select[ Range[10^8], IntegerQ[(Prime[ # ] + PrimePi[ # ])/ # ] & ]

PARI

k=0;n=0;forprime(p=2,4e9,if(isprime(n++),k++);if((k+p)%n==0,print1(n", "))) \\ Charles R Greathouse IV, Aug 21 2011

Formula

Solutions to pi(x) + prime(x) = A000720(x) + A000040(x) = 0 (mod x).

Extensions

a(17)-a(19) from Donovan Johnson, Aug 21 2011

a(20)-a(28) from Giovanni Resta, Oct 15 2019