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

1,3

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

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.

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)

Example

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.

Prog

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

Crossrefs

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

Sequence in context: A125131 A003958 A326140 * A326069 A326068 A318878

Adjacent sequences: A082726 A082727 A082728 * A082730 A082731 A082732

Keyword

nonn

Author

Reinhard Zumkeller, Apr 13 2003

Status

approved