

A117370


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


2



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
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OFFSET

1,14


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) = k2. Examples: a(14) = a(2*prime(4)) = 42 = 2; a(22) = a(2*prime(5)) = 52 = 3; a(26) = a(2*prime(6)) = 62 = 4; a(74) = a(2*prime(12)) = 122= 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


LINKS

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


FORMULA

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


EXAMPLE

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).


PROG

(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


CROSSREFS

Cf. A117371.
Cf. A000961, A006530, A020639, A055396, A061395, A243055.
Sequence in context: A106671 A033776 A117371 * A309395 A260943 A316893
Adjacent sequences: A117367 A117368 A117369 * A117371 A117372 A117373


KEYWORD

nonn


AUTHOR

Leroy Quet, Mar 10 2006


EXTENSIONS

More terms from Jonathan Vos Post, Mar 11 2006
More terms from Franklin T. AdamsWatters, Aug 29 2006


STATUS

approved



