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A082729 Least positive number that can be written using all divisors of n and the operations add and subtract.

%I A082729 #20 Jan 04 2025 16:11:07
%S A082729 1,1,2,1,4,2,6,1,5,2,10,2,12,4,6,1,16,1,18,2,10,8,22,2,19,10,14,2,28,
%T A082729 2,30,1,18,14,22,1,36,16,22,2,40,2,42,4,12,20,46,2,41,7,30,6,52,2,38,
%U A082729 2,34,26,58,2,60,28,22,1,46,2,66,10,42,2,70,1,72,34,26,12,58,2,78,2,41,38,82
%N A082729 Least positive number that can be written using all divisors of n and the operations add and subtract.
%C A082729 For all Zumkeller numbers, a(A083207(n)) = 2, which result is obtained when an equally partitioned set of divisors is modified by transferring the divisor 1 from the other set to the other set. - _Antti Karttunen_, Jan 04 2025
%H A082729 Antti Karttunen, <a href="/A082729/b082729.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from Donovan Johnson)
%H A082729 Jon Maiga, <a href="http://sequencedb.net/s/A082729">Computer-generated formulas for A082729</a>, Sequence Machine.
%F A082729 a(n) < A000203(n).
%F A082729 a(n) = A000010(n) = n-1 iff n is prime or n=6.
%F A082729 a(A000079(n)) = 1.
%F A082729 From _Antti Karttunen_, Jan 04 2025: (Start)
%F A082729 Apparently, a(n) <= A000010(n)
%F A082729 a(n) = A103977(n) + 2*A179527(n). [From Sequence Machine. For a proof, see comments]
%F A082729 (End)
%e A082729 Divisors of n=18: {1,2,3,6,9,18} -> a(18)=-1+2-3-6-9+18=1.
%e A082729 Divisors of n=15: {1,3,5,15} -> a(15)=-1-3-5+15=6, as
%e A082729 -1-3-5-15<1, 1-3-5-15<1, -1+3-5-15<1, 1+3-5-15<1, -1-3+5-15<1, 1-3+5-15<1,
%e A082729 -1+3+5-15<1, 1+3+5-15<1, 1-3-5+15>6, -1+3-5+15>6, 1+3-5+15>6, -1-3+5+15>6,
%e A082729 1-3+5+15>6, -1+3+5+15>6 and 1+3+5+15>6.
%o A082729 (PARI) A082729(n) = { my(nd=numdiv(n), d=divisors(n), nn=0); for(j=0, 2^nd-1, s=0; for(h=0, nd-1, if(bittest(j, h)==0, s=s-d[h+1], s=s+d[h+1])); if(s>0, if(!nn || s<nn, nn=s))); (nn); }; \\ _Donovan Johnson_, Sep 14 2013
%o A082729 (PARI) A082729(n) = ((d->if(d,d,2))(A103977(n))); \\ _Antti Karttunen_, Jan 04 2025
%Y A082729 Cf. A000005, A000010, A000079, A081806, A083207, A103977, A179527.
%K A082729 nonn
%O A082729 1,3
%A A082729 _Reinhard Zumkeller_, Apr 13 2003