A087167 - OEIS

%I #53 Feb 08 2026 15:05:03

%S 8925,32445,442365,151115727449904501489664

%N Numbers k such that sigma(k) - 2k = 6.

%C If odd m is in this sequence and 5 doesn't divide m, then m is an odd Weird number.

%C Any term x of this sequence can be combined with any term y of A141548 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - _Timothy L. Tiffin_, Sep 13 2016

%C For k in A059609, 2^(k-1)*(2^k-7) is a term. - _Max Alekseyev_, Oct 11 2025

%D R. K. Guy, "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." B2 in Unsolved Problems in Number Theory, 2nd ed.New York:Springer- Verlag, pp. 45-53, 1994.

%H Max A. Alekseyev, <a href="https://arxiv.org/abs/2601.17832">Computing bounded solutions to linear Diophantine equations with the sum of divisors</a>, arXiv:2601.17832 [math.NT], 2026. See p. 9, Table 1.

%H Stan Benkoski, <a href="https://www.jstor.org/stable/2316276">Are All Weird Numbers Even?</a>, Problem E2308, Amer. Math. Monthly, 79 (7) (1972), 774.

%H Wenjie Fang, <a href="https://arxiv.org/abs/2207.12906">Searching on the boundary of abundance for odd weird numbers</a>, arXiv:2207.12906 [math.NT], 2022.

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_233.htm">Puzzle 233. A little twist</a>, The Prime Puzzles & Problems Connection.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WeirdNumber.html">Weird Number</a>.

%e a(1)=8925 because sigma(8925)=2*8925+6 and 8925 is the first odd number such that sigma(n)-2n=6.

%t Do[If[OddQ[n] && DivisorSigma[1, n] - 2n == 6, Print[n]], {n, 2*10^9}]

%o (PARI) is(n)=sigma(n)==2*n+6; \\ _Charles R Greathouse IV_, Mar 09 2014

%Y Subsequence of A088011, A088834, and A274551.

%Y Cf. A003380, A077374, A005101, A005835, A059609, A141548 (deficiency 6).

%K nonn,hard,more

%O 1,1

%A _Farideh Firoozbakht_, Oct 19 2003

%E "Odd" dropped from the definition and a(4) added by _Max Alekseyev_, Oct 11 2025