A237289 Sum of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

0, 0, 2, 0, 9, 0, 20, 0, 39, 27, 54, 0, 77, 108, 108, 0, 135, 0, 170, 0, 272, 378, 252, 0, 372, 567, 500, 0, 405, 0, 464, 0, 792, 1053, 792, 0, 665, 1350, 1148, 0, 819, 0, 902, 882, 897, 2052, 1080, 0, 1425, 1395, 2052, 1715, 1377, 0, 2052, 0, 2600, 3375, 1710

Offset: 1

Author

Jaroslav Krizek, Mar 02 2014

Keywords

• nonn

Examples

			For n = 5, a(5) = 2 + 3 + 4 = 9 (numbers 2, 3 and 4 are not a sum of any subset of distinct divisors of 5).
Numbers n = 14 and 15 are an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237290(14) = A237290(15).
a(14) = 4+5+6+11+12+13+18+19+20 = a(15) = 2+7+10+11+12+13+14+17+22 = 108.
a(6) = 0 as 6 is practical; the sums into distinct divisors from 1 through 12 are 1 = 1, 2 = 2, 3 = 3, 4 = 1 + 3, 5 = 2 + 3, 6 = 1 + 2 + 3, 7 through 12 are (1 through 6) + 6. So none are not a sum distinct divisors of 6. - _David A. Corneth_, Jul 22 2025
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A119348, A005153, A184387, A237287, A237290.

Programs

Maple

isSumDist := proc(n,k)
    local dvs ;
    dvs := numtheory[divisors](n) ;
    for s in combinat[powerset](dvs) do
        add(m,m=op(s)) ;
        if % = k then
            return true;
        end if;
    end do:
    false ;
end proc:
A237289 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if not isSumDist(n,k) then
            a := a+k;
        end if;
    end do:
    a ;
end proc:
seq(A237289(n),n=1..20) ; # R. J. Mathar, Mar 13 2014

Mathematica

a[n_] := Block[{d = Divisors@n, s}, s = Plus @@ d; s*(s + 1)/2 - Plus @@ Union[Plus @@@ Subsets@d]]; m = Array[a, 59] (* Giovanni Resta, Mar 13 2014 *)

Python

from sympy import divisors
def A237289(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return (s*(s+1)>>1)-sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

Formula

a(n) = A184387(n) - A237290(n).

a(p) = p(p - 1) / 2 - 1 for p = prime > 2.

a(n) = 0 for practical numbers (A005153), a(n) > 0 for numbers that are not practical (A237287).

a(n) = A184387(n) - A229335(n) for numbers n such that A119347(n) = A100587(n).

Extensions

a(55) and a(57)-a(59) corrected by Giovanni Resta, Mar 13 2014