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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 (list; graph; refs; listen; history; text; internal format)
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) = 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

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. Adams-Watters, Aug 29 2006

STATUS

approved

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Last modified October 15 10:15 EDT 2019. Contains 328026 sequences. (Running on oeis4.)