A086685 Number of 1 <= i < n such that i*n+1 is prime.

0, 1, 1, 2, 1, 4, 2, 2, 3, 5, 3, 6, 4, 5, 5, 5, 3, 10, 2, 6, 6, 9, 4, 9, 5, 9, 7, 11, 4, 17, 3, 10, 9, 12, 9, 15, 4, 9, 11, 13, 5, 21, 7, 11, 10, 16, 8, 19, 6, 18, 13, 17, 5, 23, 10, 18, 9, 16, 8, 27, 7, 15, 13, 16, 13, 29, 9, 18, 13, 27, 9, 26, 10, 19, 18, 17, 11, 29, 11, 23, 18, 22, 11, 32

Offset: 1

Author

Jon Perry, Jul 28 2003

Keywords

• nonn

Comments

Number of primes p < n^2 such that p == 1 (mod n). The standard conjecture here is that a(n) ~ n^2/(2 phi(n)log n), where Euler's totient function phi(n) = A000010(n). - Thomas Ordowski, Oct 21 2014

Number of primes appearing in the 1st column of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

Examples

			For n=10, i=1,3,4,6,7 give primes, so a(10)=5.
		

Crossrefs

Cf. A000010 (phi), A000720 (pi).

Programs

Mathematica

f[n_] := Length[ Select[ Range[n - 1], PrimeQ[n# + 1] & ]]; Table[ f[n], {n, 1, 85}]
Table[Count[Range[n-1]n+1,?PrimeQ],{n,90}] (* _Harvey P. Dale, Oct 10 2013 *)

PARI

nphi(n)=local(c); c=0; for (i=1,n-1,if (isprime(i*n+1),c++)); c for(i=1,60,print1(","nphi(i)))

Formula

a(n) = Sum_{k=1..n} pi(1+n*(k-1)) - pi(n*(k-1)), where pi is the prime counting function. - Wesley Ivan Hurt, May 17 2021

Extensions

Extended by Robert G. Wilson v, Jul 31 2003