A391317 - OEIS

%I #9 Dec 26 2025 19:38:07

%S 48,420,1680,2520,3120,3360,3780,4200,4200,4320,4620,4620,4680,5040,

%T 5040,5040,5040,5040,5040,5040,6120,6300,6300,6300,6720,6720,6720,

%U 6720,6720,7200,7560,7560,7560,7560,7560,7560,7560,8400,8400,8400,8400,9240,9240,9240,9240

%N a(n) is the smallest positive integer that has at least one subset of proper divisors that adds to the n-th weird number (A006037(n)).

%C This sequence is conjectured to be a subsequence of A005101.

%e a(1) = 48 because 48 is the smallest positive integer that has at least one subset of proper divisors that adds to A006037(1) = 70: {1, 2, 3, 4, 8, 12, 16, 24}, {1, 3, 6, 8, 12, 16, 24} and {4, 6, 8, 12, 16, 24} add to 70.

%p with(numtheory):

%p c := proc(n, k) local b, l; l:= sort([(divisors(n) minus {n})[]]):

%p     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))))

%p     end; forget(b):

%p     b(k, nops(l))

%p end:

%p A391317List := proc(); # To calculate the first 45 terms

%p     local a, i, k, l;

%p     a:=[];

%p     l := [70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20510, 21490, 21770, 21910, 22190, 23170, 23590, 24290, 24430];

%p     for i in l do

%p         for k from 2*floor((sqrt(1 + 8*i) - 1)/2) do

%p             if sigma(k) - k >= i and c(k, i) > 0 then

%p                 a := [op(a), k];

%p                 break

%p             fi

%p         od

%p     od;

%p     return op(a)

%p end proc;

%p A391317List();

%Y Cf. A005101, A005835, A006037.

%K nonn

%O 1,1

%A _Felix Huber_, Dec 16 2025