A117371 Number of primes between smallest prime divisor of n and largest prime divisor of n that are coprime to n (not factors 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, 0, 0, 0, 2, 5, 0, 0, 0, 6, 3, 1, 0, 1, 0, 3, 0, 7, 0, 0, 0, 1, 4, 4, 0, 0, 1, 2, 5, 8, 0, 0, 0, 9, 1, 0, 2, 2, 0, 5, 6, 1, 0, 0, 0, 10, 0, 6, 0, 3, 0, 1, 0, 11, 0, 1, 3, 12, 7, 3, 0, 0, 1, 7, 8, 13, 4, 0, 0, 2, 2, 1, 0, 4, 0

Offset: 1

Author

Leroy Quet, Mar 10 2006

Keywords

• nonn

Comments

This sequence first differs from sequence A117370 at the 30th term.

Examples

			a(30) is 0 because the one prime (which is 3) between the smallest prime dividing 30 (which is 2) and the largest prime dividing 30 (which is 5) is not coprime to 30. On the other hand, a(14) = 2 because there are two primes (3 and 5) that are between 14's least prime divisor (2) and greatest prime divisor (7) and 3 and 5 are both coprime to 14.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A117370, A137795.

Programs

Maple

A020639 := proc(n) local ifs; if n = 1 then 1 ; else ifs := ifactors(n)[2] ; min(seq(op(1,i),i=ifs)) ; fi ; end: A006530 := proc(n) local ifs; if n = 1 then 1 ; else ifs := ifactors(n)[2] ; max(seq(op(1,i),i=ifs)) ; fi ; end: A117371 := proc(n) local a,i ; a := 0 ; if n < 2 then 0 ; else for i from A020639(n)+1 to A006530(n)-1 do if isprime(i) and gcd(i,n) = 1 then a := a+1 ; fi ; od; fi ; RETURN(a) ; end: seq(A117371(n),n=1..140) ; # R. J. Mathar, Sep 05 2007

Mathematica

Table[Count[Prime[Range[PrimePi@ First@ # + 1, PrimePi@ Last@ # - 1]], ?(GCD[#, n] == 1 &)] &@ FactorInteger[n][[All, 1]], {n, 103}] (* _Michael De Vlieger, Sep 10 2018 *)

PARI

A117371(n) = if(1==n,0, my(f = factor(n), p = f[1, 1], gpf = f[#f~, 1], c = 0); while(pAntti Karttunen, Sep 10 2018

Formula

a(n) = A001221(A137795(n)). - Antti Karttunen, Sep 10 2018

Extensions

More terms from R. J. Mathar, Sep 05 2007