A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.

1, 6, 8, 28, 12, 78, 16, 120, 52, 144, 24, 406, 28, 192, 192, 496, 36, 780, 40, 903, 256, 288, 48, 1830, 124, 336, 320, 1596, 60, 2628, 64, 2016, 384, 432, 384, 4186, 76, 480, 448, 4095, 84, 4656, 88, 2688, 2184, 576, 96, 7750, 228, 2976, 576, 3136, 108, 7260

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Formula

a(n) = A184387(n) - A237289(n).

a(p) = 2(p+2) for odd primes p.

a(n) = A184387(n) for practical numbers n (A005153), a(n) < A184387(n) for numbers n that are not practical (A237287).

Example

For n = 5, a(5) = 1 + 5 + 6 = 12 (each of the numbers 1, 5 and 6 is the sum of a subset of distinct divisors of 5).
The numbers n = 14 and 15 is an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237289(14) = A237289(15).
a(14) = 1+2+3+7+8+9+10+14+15+16+17+21+22+23+24 = a(15) = 1+3+4+5+6+8+9+15+16+18+19+20+21+23+24 = 192.

Maple

isSumDist := proc(n, k)
    local dvs, s ;
    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:
A237290 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if isSumDist(n, k) then
            a := a+k;
        end if;
    end do:
end proc:
seq(A237290(n), n=1..20) ; # R. J. Mathar, Mar 13 2014

Mathematica

a[n_] := Plus @@ Union[Plus @@@ Subsets@ Divisors@ n]; Array[a, 54] (* Giovanni Resta, Mar 13 2014 *)

Prog

(PARI) padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b); ); b; }
a(n) = {vks = []; d = divisors(n); nbd = #d; for (i=1, 2^nbd-1, b = padbin(i, nbd); onek = sum(j=1, nbd, d[j]*b[j]); vks = Set(concat(vks, onek)); ); sum(i=1, #vks, vks[i]); } \\ Michel Marcus, Mar 09 2014
(Python)
from sympy import divisors
def A237290(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

Crossrefs

Cf. A119348, A005153, A184387, A237287, A237289.

Sequence in context: A066231 A374120 A267477 * A229335 A007829 A345003

Adjacent sequences: A237287 A237288 A237289 * A237291 A237292 A237293

Keyword

nonn

Author

Jaroslav Krizek, Mar 02 2014

Status

approved