A160804 - OEIS

%I #12 Jul 09 2023 03:18:14

%S 1,1,1,1,1,0,1,1,1,0,1,0,1,0,0,1,1,0,1,0,2,0,1,0,1,0,1,0,1,0,1,1,0,0,

%T 2,1,1,0,2,0,1,0,1,0,0,0,1,0,1,1,0,0,1,0,2,0,2,0,1,0,1,0,2,1,2,0,1,0,

%U 0,0,1,0,1,0,1,0,2,0,1,0,1,0,1,0,2,0,0,0,1,0,0,0,2,0,0

%N Consider a permutation K = (k(1),k(2),...k(A000005(n))) of the positive divisors of n. Consider the partial sums S = Sum_{j=1..m} k(j), 1 <= m <= A000005(n). Then, a(n) = the minimum number, for any permutation K, of partial sums S that are coprime to n.

%e The divisors of 12 are 1, 2, 3, 4, 6 and 12. If the permutation K is, for example, (12,6,3,1,2,4), then the partial sums are: 12 = 12, 12+6 = 18, 12+6+3 = 21, 12+6+3+1 = 22, 12+6+3+1+2 = 24, and 12+6+3+1+2+4 = 28. None of these sums are coprime to 12, proving that the minimum number of partial sums coprime to 12 = a(12) = 0.

%o (PARI) c(perm, n) = {my(s=0, k=0); for(i = 1, #perm, s += perm[i]; if(gcd(s, n) == 1, k++)); k;}

%o a(n) = {my(d = divisors(n), cmin = #d, c1); forperm(d, v, c1 = c(v, n); if(c1 < cmin, cmin = c1)); cmin;} \\ _Amiram Eldar_, Jul 09 2023

%Y Cf. A000005, A151613.

%K nonn

%O 1,21

%A _Leroy Quet_, May 26 2009

%E Extended thru a(59) by _Ray Chandler_, Jun 15 2009

%E a(60)-a(95) from _Amiram Eldar_, Jul 09 2023