A366219 Smallest positive integer whose smallest coprime divisor shift is n.

1, 3, 91, 325, 2093, 1001, 12025, 1045, 4945, 6391, 189, 455, 245, 11825, 128843, 368809, 273, 1295, 14495, 37961, 252263, 91375, 595, 13013, 46189, 104951, 63875, 136345, 42237, 22253, 192647, 18655, 8225, 194545, 200629, 192907, 27625, 1911, 464783, 27797

Offset: 0

Author

M. Farrokhi D. G., Oct 05 2023

Keywords

• nonn

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.

Examples

			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.
		

Links

• M. Farrokhi D. G., Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A366251, A366330.

Programs

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