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

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, 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, 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

Offset: 1

Author

Reinhard Zumkeller, Apr 13 2003

Keywords

• nonn

Comments

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

Examples

			Divisors of n=18: {1,2,3,6,9,18} -> a(18)=-1+2-3-6-9+18=1.
Divisors of n=15: {1,3,5,15} -> a(15)=-1-3-5+15=6, as
-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,
-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,
1-3+5+15>6, -1+3+5+15>6 and 1+3+5+15>6.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from Donovan Johnson)
• Jon Maiga, Computer-generated formulas for A082729, Sequence Machine.

Crossrefs

Cf. A000005, A000010, A000079, A081806, A083207, A103977, A179527.

Programs

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 || sDonovan Johnson, Sep 14 2013

PARI

A082729(n) = ((d->if(d,d,2))(A103977(n))); \\ Antti Karttunen, Jan 04 2025

Formula

a(n) < A000203(n).

a(n) = A000010(n) = n-1 iff n is prime or n=6.

a(A000079(n)) = 1.

From Antti Karttunen, Jan 04 2025: (Start)

Apparently, a(n) <= A000010(n)

a(n) = A103977(n) + 2*A179527(n). [From Sequence Machine. For a proof, see comments]

(End)