A061853 Difference between smallest prime not dividing n and smallest nondivisor of n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Author

Henry Bottomley, May 10 2001

Keywords

• nonn

Comments

a(12m+6) is always positive since it involves subtracting 4 from a larger number; the first case where a term not of this form is positive is a(420).

Primorials from A002110(2)=6 onward seem to give the positions of records. - Antti Karttunen, Jul 28 2017

Difference between the smallest prime coprime to n and the smallest non-divisor of n. - Michael De Vlieger, Jul 28 2017

Examples

			a(29)=2-2=0; a(30)=7-4=3; a(420)=11-8=3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..30030

Crossrefs

Cf. A002110, A007978, A053669.

Programs

PARI

a(n) = {my(f = factor(n), d = divisors(f), res, p = 2, i = 1, j); while(i<=#f~ && f[i, 1]==p, i++; p = nextprime(p+1)); res = p; for(j=2, #d, if(d[j]!=j, return(res - d[j-1] - 1)))} \\ David A. Corneth, Jul 29 2017

Python

from sympy import nextprime
def a053669(n):
    p=2
    while n%p==0: p=nextprime(p)
    return p
def a007978(n):
    p=2
    while n%p==0: p+=1
    return p
def a(n): return a053669(n) - a007978(n)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 29 2017

Formula

a(n) = A053669(n) - A007978(n).

Extensions

Description corrected by Michael De Vlieger, Jul 28 2017