A379756 a(n) is the number of subsets of S(n) that sum to A023196(n), where S(n) is the set of the proper divisors (or aliquot parts) of A023196(n).

1, 2, 2, 1, 5, 1, 3, 7, 3, 2, 10, 3, 2, 34, 2, 0, 31, 1, 6, 25, 1, 23, 21, 2, 1, 1, 20, 4, 1, 279, 13, 15, 1, 15, 116, 9, 11, 12, 4, 197, 1, 2, 755, 1, 42, 2, 9, 12, 6, 2, 151, 169, 7, 1, 9, 8, 6, 2190, 1, 516, 1, 6, 121, 130, 1, 6, 119, 1, 469, 4, 446, 1, 4, 6

Offset: 1

Author

Felix Huber, Feb 07 2025

Keywords

• nonn

Comments

This sequence is A065205 without the terms A065205(k) where k > sigma(k)/2.

Examples

			a(8) = 7 because exactly the 7 subsets {6, 12, 18}, {3, 6, 9, 18}, {2, 4, 12, 18}, {2, 3, 4, 9, 18}, {2, 3, 4, 6, 9, 12}, {1, 2, 6, 9, 18}, {1, 2, 3, 12, 18} of S(8) = {1, 2, 3, 4, 6, 9, 12, 18} sum to A023196(8) = 36.
a(16) = 0 because no subset of S(16) = {1, 2, 5, 7, 10, 14, 35} sums to A023196(16) = 70 (weird number).
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000
• Felix Huber, Maple program to calculate the distinct subsets

Crossrefs

Cf. A000203, A000396, A001065, A005101, A023196, A026793, A027750, A033630, A033882, A064771, A065205.

Programs

Maple

with(NumberTheory):
A023196:=proc(n)
    local a;
    option remember;
    if n=1 then
        6
    else
        for a from procname(n-1)+1 do
            if sigma(a)>=2*a then
                return a
            fi
        od
    fi;
end proc;
A379756:=proc(n)
    local b,d,l;
    d:=sigma(A023196(n))-2*A023196(n);
    l:= [select(x->x<=d,Divisors(A023196(n)))[]];
    b:= proc(m,i)
        option remember;
        `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
    end proc;
    forget(b);
    b(d,nops(l))
end proc;
seq(A379756(n),n=1..74);

Formula

Iff a(k) = 0, A023196(k) is a weird number (A006037).

Iff a(k) = 1, A023196(k) is a term of A064771.

a(A000396(k)) = 1 (A000396: perfect numbers).