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

1,3

Links

Table of n, a(n) for n=1..59.

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).

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A019750 A308473 A362952 * A238396 A247671 A011125

Adjacent sequences: A237286 A237287 A237288 * A237290 A237291 A237292

Keyword

nonn

Author

Jaroslav Krizek, Mar 02 2014

Extensions

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

Status

approved