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%I #13 Sep 11 2018 17:02:56
%S 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,
%T 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,
%U 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
%N Number of primes between smallest prime divisor of n and largest prime divisor of n.
%C This sequence first differs from sequence A117371 at the 30th term.
%C 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
%H Antti Karttunen, <a href="/A117370/b117370.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F If A001221(n)<=1, a(n) = 0, otherwise a(n) = A243055(n) - 1 = (A061395(n)-A055396(n))-1. - _Antti Karttunen_, Sep 10 2018
%e 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).
%o (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
%Y Cf. A117371.
%Y Cf. A000961, A006530, A020639, A055396, A061395, A243055.
%K nonn
%O 1,14
%A _Leroy Quet_, Mar 10 2006
%E More terms from _Jonathan Vos Post_, Mar 11 2006
%E More terms from _Franklin T. Adams-Watters_, Aug 29 2006
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