A085343 Number of primes between sigma(n) and phi(n) inclusive.

0, 2, 1, 3, 1, 4, 1, 4, 3, 5, 1, 7, 1, 6, 5, 7, 1, 9, 1, 9, 6, 7, 1, 13, 3, 8, 5, 11, 1, 16, 1, 12, 7, 10, 6, 19, 1, 10, 7, 18, 1, 19, 1, 15, 12, 12, 1, 24, 3, 16, 9, 16, 1, 23, 8, 21, 11, 15, 1, 33, 1, 14, 16, 20, 8, 26, 1, 19, 10, 25, 1, 35, 1, 19, 18, 23, 7, 30, 1, 31, 14, 18, 1, 39, 10, 19

Offset: 1

Author

Labos Elemer, Jul 10 2003

Keywords

• nonn

Comments

a(p) = 1 for prime p > 2. Since phi(p) = p - 1 and sigma(p) = p + 1, the largest prime q < p - 1 must be the prime previous to p, while p itself is the largest prime less than p + 1 for p > 2. - Michael De Vlieger, Jan 22 2020

Examples

			n=12: sigma(12)=28, phi(n)=4, Pi(28)-Pi(4)=9-2=7.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A000010, A000203, A070803, A070804, A085122, A085341-A085347.

Programs

Mathematica

Array[Subtract @@ PrimePi@{DivisorSigma[1, #], EulerPhi@ #} &, 86] (* Michael De Vlieger, Jan 22 2020 *)

PARI

a(n) = primepi(sigma(n)) - primepi(eulerphi(n)); \\ Michel Marcus, Aug 29 2019

Formula

a(n) = pi(sigma(n)) - pi(phi(n)) = A000720(A000203(n)) - A000720(A000010(n)).

a(n) = A070803(n) - A070804(n). - Antti Karttunen, Jan 22 2020