%I #60 May 30 2026 16:40:35

%S 1,6,60,90,120,672,2160,10080,22848,30240,342720,523776,1028160,

%T 1528800,6168960,7856640,7983360,14443520,22932000,23569920,43330560,

%U 44553600,51979200,57657600,68796000,133660800,172972800,779688000,1476304896,2339064000,6840038400

%N Bi-unitary multiperfect numbers.

%C All entries greater than 1 are even [Hagis].

%C 14443520 is the first (only?) composite term not divisible by 3. Excluding the factor p=3, all composite terms <= 172972800 have nonincreasing exponents in the factorization (sorted by primes). - _D. S. McNeil_, Apr 15 2011

%C Wall shows that 6, 60, and 90 are the only bi-unitary perfect numbers. - _Tomohiro Yamada_, Apr 15 2017

%C McNeil's observation about exponents does not hold in general. Indeed, a(41) = 2^8 * 3^5 * 5^2 * 7 * 11 * 13^2 * 17. - _Giovanni Resta_, Apr 15 2017

%C a(43) > 4.66*10^12. - _Giovanni Resta_, Sep 07 2018

%C We include 1 here, although this is not "multi"-perfect. - _R. J. Mathar_, Sep 08 2018

%H Giovanni Resta, <a href="/A189000/b189000.txt">Table of n, a(n) for n = 1..42</a>

%H Peter Hagis, <a href="https://www.fq.math.ca/Scanned/25-2/hagis.pdf">Bi-Unitary amicable and multiperfect numbers</a>, Fib. Quart. 25 (2) (1987) 144-151

%H Pentti Haukkanen and V. Sitaramaiah, <a href="https://doi.org/10.7546/nntdm.2020.26.1.93-171">Bi-unitary multiperfect numbers, I</a>, Notes Number Theory Discrete Math. 26 (1) (2020) 93-171.

%H Michel Marcus, <a href="/A189000/a189000.txt">Unexhaustive list of terms</a>

%H C. R. Wall, <a href="https://doi.org/10.1090/S0002-9939-1972-0289403-9">Bi-unitary perfect numbers</a>, Proc. Amer. Math. Soc. 33 (1) (1972) 39-42.

%H Tomohiro Yamada, <a href="https://arxiv.org/abs/2406.19331">Determining all biunitary triperfect numbers of a certain form</a>, arXiv:2406.19331 [math.NT], 2024.

%F {n | A188999(n)}.

%e n=120 divides A188999(120)=360.

%e n=90 divides A188999(90)=180.

%e n=672 divides A188999(672)=2016.

%t bsig[n_] := If[n == 1, 1, Block[{p, e}, Product[{p, e} = pe; (p^(e + 1) - 1)/(p - 1) - If[EvenQ[e], p^(e/2), 0], {pe, FactorInteger[n]}]]]; Select[Range[10^5], Mod[bsig[#], #] == 0 &] (* _Giovanni Resta_, Apr 15 2017 *)

%o (PARI) a188999(n) = {my(f = factor(n)); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }

%o isok(n) = ! frac(a188999(n)/n); \\ _Michel Marcus_, Sep 03 2018

%Y Cf. A007691 (analog for sigma).

%Y Cf. A188999 (bi-unitary sigma), A318175, A318781 (the k coefficients).

%K nonn

%O 1,2

%A _R. J. Mathar_, Apr 15 2011

%E a(18)-a(27) by _D. S. McNeil_, Apr 15 2011

%E a(28)-a(31) from _Giovanni Resta_, Apr 15 2017

%E a(1)=1 inserted by _Giovanni Resta_, Sep 07 2018