A137795 -id:A137795 - cp's OEIS frontend

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

Leroy Quet, Mar 10 2006

nonn

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

			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.

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

Cf. A117370, A137795.

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

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 *)

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

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

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

A178921 Product of distances between successive distinct prime divisors of n; zero if n has only 1 distinct prime factor.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 5, 2, 0, 0, 1, 0, 3, 4, 9, 0, 1, 0, 11, 0, 5, 0, 2, 0, 0, 8, 15, 2, 1, 0, 17, 10, 3, 0, 4, 0, 9, 2, 21, 0, 1, 0, 3, 14, 11, 0, 1, 6, 5, 16, 27, 0, 2, 0, 29, 4, 0, 8, 8, 0, 15, 20, 6, 0, 1, 0, 35, 2, 17, 4, 10, 0, 3, 0, 39, 0, 4, 12, 41, 26, 9, 0, 2, 6, 21, 28, 45, 14, 1, 0, 5, 8, 3, 0, 14, 0, 11

Offset: 1

Alex Ratushnyak, Aug 18 2012

For n <= 41, a(n) = A049087(n).

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

Cf. A081060, A049087.

Cf. also A137795.

f[n_] := Module[{ps}, If[n <= 1, 0, ps = Transpose[FactorInteger[n]][[1]]; Times @@ Differences[ps]]]; Table[f[n], {n, 100}] (* T. D. Noe, Aug 20 2012 *)
Array[Apply[Times, Differences@ FactorInteger[#][[All, 1]] /. {} -> 0] &, 105] (* Michael De Vlieger, Sep 10 2018 *)

A178921(n) = if(1>=omega(n), 0, my(ps = factor(n)[,1], m = 1); for(i=2, #ps, m *= (ps[i]-ps[i-1])); (m)); \\ Antti Karttunen, Sep 07 2018

from sympy import primerange
primes = list(primerange(2,500))
for n in range(1,100):
    d = n
    prev = 0
    product = 1
    for p in primes:
        if d%p==0:
            if prev:
                product *= p-prev
            while d%p==0:
                d//=p
            if d==1:
                break
            prev = p
    if prev==0:
        product = 0
    print(product, end=',')

More terms from Antti Karttunen, Sep 07 2018

A143265 a(n) = the smallest integer >= n such that all the distinct primes that divide n and a(n) together are members of a set of consecutive primes. In other words, a(n) is the smallest integer >= n such that n*a(n) is contained in sequence A073491.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 11, 12, 13, 15, 15, 16, 17, 18, 19, 21, 25, 105, 23, 24, 25, 1155, 27, 30, 29, 30, 31, 32, 35, 15015, 35, 36, 37, 255255, 385, 42, 41, 45, 43, 105, 45, 4849845, 47, 48, 49, 51, 5005, 1155, 53, 54, 56, 60, 85085, 111546435, 59, 60, 61

Offset: 1

Leroy Quet, Aug 03 2008

nonn

			20 is factored as 2^2 *5^1. Checking the integers >= 20: 20*20 is not factorable into consecutive primes, since 3 is missing. 21 is factored as 3^1 *7^1. Since the distinct primes that divide 20 and 21 (which are 2,3,5,7) form a set of consecutive primes, then a(20) = 21.

Ray Chandler, Table of n, a(n) for n = 1..4096 (computed from A137795 b-file)

Cf. A073491, A137795.

a(n) = A137795(n) * Ceiling(n/A137795(n)). - Ray Chandler, Nov 09 2008

Inserted a(15) and a(21) and extended by R. J. Mathar, Aug 14 2008

a(46)-a(61) from Ray Chandler, Nov 09 2008

cp's OEIS Frontend

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).

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A178921 Product of distances between successive distinct prime divisors of n; zero if n has only 1 distinct prime factor.

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Mathematica

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Python

Extensions

A143265 a(n) = the smallest integer >= n such that all the distinct primes that divide n and a(n) together are members of a set of consecutive primes. In other words, a(n) is the smallest integer >= n such that n*a(n) is contained in sequence A073491.

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