OFFSET
0,2
COMMENTS
A nonnegative number s is a coprime divisor shift of n if GCD(d + s, n) = 1 for all divisors d of n. The coprime divisor shift of n is the infimum of the set of all nonnegative coprime divisor shifts of n.
Conjecture. Every positive integer s is the coprime divisor shift of a positive integer.
LINKS
M. Farrokhi D. G., Table of n, a(n) for n = 0..1000
EXAMPLE
a(0) = 1 for GCD(1 + 0, 1) = 1.
a(1) = 3 for GCD(1 + 1, 3) = GCD(3 + 1, 3) = 1 but GCD(1 + 1, 2) > 1.
a(2) = 91 for GCD(d + 2, 91) = 1 for all divisors d = 1, 7, 13, 91 of 91, GCD(13 + 1, 91) > 1, and 91 is the smallest number with this property.
PROG
(PARI)
isds(k, s)={fordiv(k, d, if(gcd(d+s, k)<>1, return(0))); 1}
findds(k)={for(s=0, k-1, if(isds(k, s), return(s))); -1}
a(n)={for(k=1, oo, if(isds(k, n) && findds(k)==n, return(k)))} \\ Andrew Howroyd, Oct 05 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
M. Farrokhi D. G., Oct 05 2023
STATUS
approved