A089748 Numbers k that divide (sum of proper divisors of k + product of proper divisors of k).

2, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240

Offset

1,1

Comments

All perfect numbers belong to this sequence.

Every term of A007691 is in this sequence. - T. D. Noe, Sep 29 2005

There are two sets of candidates of k: (i) k|A001065(k) and k|A007956(k) individually, or (ii) neither k|A001065(k) nor k|A007956(k) but the remainders of A001065(k)/k and A007956(k)/k sum up to k. If k has at least 4 divisors, the product of the second and penultimate divisor (in the sorted divisors list) is k, so k|A007956(k). This means for all k in A080257 we have k|A007956(k), and the k that do not divide A007956(k) are in A000430, which means k=p or k=p^2 for some prime p. If k=p, A001065(k)+A007956(k) = 1+1 =2, and the requirement here reduces to k|2 and only k=2 is left. If k=p^2, A001065(k) +A007956(k) = 1+p+p = 1+2*p, and the requirement here reduces to p^2 | (1+2*p), which has no solutions. This means case (ii) does not generate any solutions besides k=2. And this means all other solutions are from case (i), and therefore elements A007691 > 1 are the only remaining candidates. - R. J. Mathar, Oct 15 2021

Links

Table of n, a(n) for n=1..16.

Maple

isA087948 := proc(n)
    if modp( A001065(n)+A007956(n), n) = 0 then
        true;
    else
        false;
    end if;
end proc:
for n from 2 do
    if isA087948(n) then
        printf("%d\n", n) ;
    end if;
end do: # R. J. Mathar, Oct 15 2021

Mathematica

l = {}; Do[d = Drop[Divisors[n], -1]; p = Apply[Plus, d]; t = Apply[Times, d]; m = Mod[p + t, n]; If[m == 0, l = Append[l, n]], {n, 2, 10^6}]; l
Select[Range[2, 22*10^5], Mod[Total[Most[Divisors[#]]]+Times@@Most[Divisors[#]], #]==0&] (* The program generates the first 11 terms of the sequence. *) (* Harvey P. Dale, Jun 05 2024 *)

Prog

(Python)
from math import prod
from sympy import divisors
def ok(n): d = divisors(n)[:-1]; return n > 1 and (sum(d) + prod(d))%n == 0
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Oct 15 2021

Crossrefs

Cf. A001065, A007956, A007691, A080257 (k which divide A007691(k)).

Cf. A219544.

Sequence in context: A242511 A360035 A323268 * A047125 A189238 A226497

Adjacent sequences: A089745 A089746 A089747 * A089749 A089750 A089751

Keyword

nonn,more

Author

Joseph L. Pe, Jan 08 2004

Extensions

a(11)-a(16) from Michael S. Branicky, Oct 16 2021

Status

approved