A366219 - OEIS

%I #32 Nov 10 2023 10:42:44

%S 1,3,91,325,2093,1001,12025,1045,4945,6391,189,455,245,11825,128843,

%T 368809,273,1295,14495,37961,252263,91375,595,13013,46189,104951,

%U 63875,136345,42237,22253,192647,18655,8225,194545,200629,192907,27625,1911,464783,27797

%N Smallest positive integer whose smallest coprime divisor shift is n.

%C 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.

%C Conjecture. Every positive integer s is the coprime divisor shift of a positive integer.

%H M. Farrokhi D. G., <a href="/A366219/b366219.txt">Table of n, a(n) for n = 0..1000</a>

%e a(0) = 1 for GCD(1 + 0, 1) = 1.

%e a(1) = 3 for GCD(1 + 1, 3) = GCD(3 + 1, 3) = 1 but GCD(1 + 1, 2) > 1.

%e 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.

%o (PARI)

%o isds(k,s)={fordiv(k,d,if(gcd(d+s, k)<>1, return(0))); 1}

%o findds(k)={for(s=0, k-1, if(isds(k,s), return(s))); -1}

%o a(n)={for(k=1, oo, if(isds(k,n) && findds(k)==n, return(k)))} \\ _Andrew Howroyd_, Oct 05 2023

%Y Cf. A366251, A366330.

%K nonn

%O 0,2

%A _M. Farrokhi D. G._, Oct 05 2023