OFFSET
1,3
COMMENTS
In other words, a(n) is the smallest number m such that m has n distinct divisors d_1, ..., d_n such that d_1+...+d_n = m. (The d_i do not need to be ALL the divisors of m.) For example, a(6) = m = 24, since the divisors of 24 are 1,2,3,4,6,8,12,24, and 1+2+3+4+6+8=24.
a(2) = 0. All other entries are nonzero.
In the following triangle the n-th row gives examples of the n divisors a(1), ..., a(7); a(n) = sum of the n-th row:
1
- -
1 2 3
1 2 3 6
1 2 3 6 12
1 2 3 4 6 8
1 2 3 6 8 12 16
For a given values of a(n) = m, however, there may be more than one way to choose d_1, ..., d_n so that d_1+...+d_n = m.
For n=10, a(10)=120, for example, there are the following equally valid solutions:
[1, 2, 3, 4, 5, 6, 15, 20, 24, 40]
[1, 2, 3, 4, 5, 8, 10, 12, 15, 60]
[1, 2, 3, 4, 5, 8, 12, 15, 30, 40]
[1, 2, 3, 4, 6, 8, 12, 20, 24, 40]
[1, 2, 3, 5, 6, 8, 10, 15, 30, 40]
[1, 2, 3, 5, 8, 10, 12, 15, 24, 40]
[1, 2, 3, 5, 8, 12, 15, 20, 24, 30]
[1, 2, 4, 5, 6, 8, 10, 20, 24, 40]
[1, 2, 4, 6, 8, 10, 15, 20, 24, 30]
[1, 3, 4, 5, 6, 10, 12, 15, 24, 40]
[1, 3, 4, 5, 6, 12, 15, 20, 24, 30]
[1, 3, 4, 5, 8, 10, 15, 20, 24, 30]
[1, 3, 5, 6, 8, 10, 12, 15, 20, 40]
[1, 4, 5, 6, 8, 10, 12, 20, 24, 30]
[2, 3, 4, 5, 6, 8, 10, 12, 30, 40]
[2, 3, 4, 6, 8, 10, 12, 15, 20, 40]
[2, 3, 5, 6, 8, 10, 12, 20, 24, 30]
(These solutions were provided by Jinyuan Wang.)
The lexicographically earliest solution is given as the n-th row of the triangle in A081514. The corresponding value d_n is given in A081513.
The lexicographically earliest solutions are:
..n....m: d_1 d_2 ... d_n
-------------------------
..1....1: 1
..2....0: - -
..3....6: 1, 2, 3
..4...12: 1, 2, 3, 6
..5...24: 1, 2, 3, 6, 12
..6...24: 1, 2, 3, 4, 6, 8
..7...48: 1, 2, 3, 4, 6, 8, 24
..8...60: 1, 2, 3, 4, 5, 10, 15, 20
..9...84: 1, 2, 3, 4, 6, 7, 12, 21, 28
.10..120: 1, 2, 3, 4, 5, 6, 15, 20, 24, 40
...
LINKS
David A. Corneth, Table of n, a(n) for n = 1..552 (first 144 terms from Robert Israel)
EXAMPLE
24 is a sum of 6 of its divisors. Namely, 1+2+3+4+6+8=24. Furthermore, 24 is the smallest natural number with at least 6 divisors (not including itself), so it must be the smallest natural number that is a sum of 6 of its divisors.
MAPLE
A081512 := proc(n) local a, dvs, dset, s, p; if n= 2 then RETURN(0) ; end if; for a from 1 do dvs := numtheory[divisors](a) ; dset := combinat[choose](dvs, n) ; for s in dset do if add(p, p=s) = a then RETURN(a) ; end if; end do; end do: end: for n from 2 do a := A081512(n) ; printf("%d, ", a) ; od: # R. J. Mathar, Nov 11 2008
MATHEMATICA
(* This partly empirical program is just a recomputation of existing data. *)
f[n_, k_] := Module[{c, cc, dd}, dd = Most@ Divisors@k; cc = c[#]& /@ Range@ Length@dd; FindInstance[AllTrue[cc, 0 <= # <= 1&] && cc.dd == k && Total[cc] == n, cc, Integers, 1]];
a[n_] := a[n] = Switch[n, 1, 1, 2, 0, 3, 6, _, For[k = a[n - 1], True, k = k + If[n < 25, 1, 60], If[f[n, k] != {}, Return[k]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 49}] (* Jean-François Alcover, Oct 21 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Mar 27 2003
EXTENSIONS
Corrected by Caleb M. Shor (cshor(AT)bates.edu), Sep 26 2007
Extended beyond a(7) by R. J. Mathar, Nov 11 2008
a(16)-a(49) from Max Alekseyev, Jul 27 2009
Edited by N. J. A. Sloane, May 24 2020, following advice from Jinyuan Wang.
STATUS
approved