VOOZH about

URL: https://oeis.org/A385462

⇱ A385462 - OEIS

login
A385462
Numbers t which have a proper divisor d_i(t) such that (d_i(t) + sigma(t))/t is an integer k.
1
2, 4, 8, 10, 16, 24, 32, 44, 60, 64, 84, 128, 136, 152, 168, 184, 252, 256, 270, 336, 512, 630, 752, 756, 792, 864, 884, 924, 936, 1024, 1140, 1170, 1488, 1638, 2048, 2144, 2268, 2272, 2528, 2808, 2970, 3672, 4096, 4320, 4464, 4680, 5148, 5472, 6804, 7308, 7644, 8192, 8384
OFFSET
1,1
COMMENTS
Consecutive elements of this sequence for which k = 2 are consecutive deficient-perfect numbers (A271816) > 1.
Consecutive elements of this sequence for which k = 3 are consecutive non-perfect elements of A364977.
Let b_k(m) be the number of elements of this sequence with the same k and <= m.
--------------------------------------------
m | b_2(m) | b_3(m) | b_4(m) | b_5(m) |
--------------------------------------------
10^3 | 16 | 13 | - | - |
10^4 | 24 | 31 | 2 | - |
10^5 | 37 | 62 | 5 | - |
10^6 | 54 | 107 | 19 | - |
10^7 | 73 | 175 | 43 | 1 |
10^8 | 98 | 254 | 80 | 3 |
10^9 | 128 | 357 | 141 | 13 |
--------------------------------------------
Are there any odd terms in this sequence for which k > 2? If they exist, they are > 10^9.
Contains 2^k * (2^(k+1) + 2^j - 1) if 0 <= j <= k and 2^(k+1) + 2^j - 1 is prime. - Robert Israel, Jun 30 2025
EXAMPLE
4 is in this sequence because sigma(4) + d_1(4) = 7 + 1 = 8 and 8/4 = 2.
24 is in this sequence because sigma(24) + d_7(24) = 60 + 12 = 72 and 72/24 = 3.
4320 is in this sequence because sigma(4320) + d_47(4320) = 15120 + 2160 = 17280 and 17280/4320 = 4.
MAPLE
filter:= proc(n) local s;
s:= - numtheory:-sigma(n) mod n;
ormap(d -> d mod n = s, numtheory:-divisors(n) minus {n})
end proc:
select(filter, [$1..10^4]); # Robert Israel, Jun 30 2025
MATHEMATICA
Select[Range[8384], AnyTrue[(Drop[Divisors[#], -1]+DivisorSigma[1, #])/#, IntegerQ]&] (* James C. McMahon, Jul 05 2025 *)
PROG
(Maxima)
(n:1, for t:1 thru 10000 do (s:divsum(t), (A:args(divisors(t)),
for i:1 thru length(A)-1 do (y:s+A[i],
if mod(y, t)=0 then (print(n, "", t), n:n+1)))));
(PARI) isok(t) = my(s=sigma(t)); fordiv(t, d, if ((d<t) && (denominator((d+s)/t) == 1), return(1))); \\ Michel Marcus, Jun 30 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Lechoslaw Ratajczak, Jun 29 2025
STATUS
approved