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

1,2

Comments

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

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Felix Huber, Maple program to calculate the distinct subsets

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).

Example

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).

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);

Crossrefs

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

Sequence in context: A203953 A326933 A394734 * A123398 A277495 A188945

Adjacent sequences: A379753 A379754 A379755 * A379757 A379758 A379759

Keyword

nonn

Author

Felix Huber, Feb 07 2025

Status

approved