A117370 Number of primes between smallest prime divisor of n and largest prime divisor of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 0, 4, 0, 2, 0, 1, 0, 0, 2, 5, 0, 0, 0, 6, 3, 1, 0, 2, 0, 3, 0, 7, 0, 0, 0, 1, 4, 4, 0, 0, 1, 2, 5, 8, 0, 1, 0, 9, 1, 0, 2, 3, 0, 5, 6, 2, 0, 0, 0, 10, 0, 6, 0, 4, 0, 1, 0, 11, 0, 2, 3, 12, 7, 3, 0, 1, 1, 7, 8, 13, 4, 0, 0, 2, 2, 1, 0, 5, 0

Offset: 1

Author

Leroy Quet, Mar 10 2006

Keywords

• nonn

Comments

This sequence first differs from sequence A117371 at the 30th term.

Records in a(n) are for n = 2*prime(k), for which a(n) = k-2. Examples: a(14) = a(2*prime(4)) = 4-2 = 2; a(22) = a(2*prime(5)) = 5-2 = 3; a(26) = a(2*prime(6)) = 6-2 = 4; a(74) = a(2*prime(12)) = 12-2= 10. Those records are each repeated for n = 2*(prime(k)^e_1)*(prime(m)^e_2)*(prime(n)^e_3)...*(prime(x)^e_y) where e_i are positive integers and prime(m), ..., prime(x) are between 2 and prime(k). Minima a(n) = 0 iff least spf(n)=gpf(n) iff n is 1 or a prime power (A000961), or a product of powers of consecutive primes (prime(k)^e_1)*(prime(k+1)^e_2). Here gpf(n) = greatest prime factor = A006530(n) and spf(n) = smallest prime factor = A020639(n). - Jonathan Vos Post, Mar 11 2006

Examples

			a(30) is 1 because there is one prime (which is 3) between the smallest prime dividing 30 (which is 2) and the largest prime dividing 30 (which is 5).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A117371.

Cf. A000961, A006530, A020639, A055396, A061395, A243055.

Programs

PARI

A117370(n) = if(1>=omega(n),0,my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); -1+(primepi(gpf)-primepi(lpf))); \\ Antti Karttunen, Sep 10 2018

Formula

If A001221(n)<=1, a(n) = 0, otherwise a(n) = A243055(n) - 1 = (A061395(n)-A055396(n))-1. - Antti Karttunen, Sep 10 2018

Extensions

More terms from Jonathan Vos Post, Mar 11 2006

More terms from Franklin T. Adams-Watters, Aug 29 2006