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A189000 Bi-unitary multiperfect numbers.

%I A189000 #58 Jan 05 2025 19:51:39
%S A189000 1,6,60,90,120,672,2160,10080,22848,30240,342720,523776,1028160,
%T A189000 1528800,6168960,7856640,7983360,14443520,22932000,23569920,43330560,
%U A189000 44553600,51979200,57657600,68796000,133660800,172972800,779688000,1476304896,2339064000,6840038400
%N A189000 Bi-unitary multiperfect numbers.
%C A189000 All entries greater than 1 are even [Hagis].
%C A189000 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 A189000 Wall shows that 6, 60, and 90 are the only bi-unitary perfect numbers. - _Tomohiro Yamada_, Apr 15 2017
%C A189000 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 A189000 a(43) > 4.66*10^12. - _Giovanni Resta_, Sep 07 2018
%C A189000 We include 1 here, although this is not "multi"-perfect. - _R. J. Mathar_, Sep 08 2018
%H A189000 Giovanni Resta, <a href="/A189000/b189000.txt">Table of n, a(n) for n = 1..42</a>
%H A189000 Peter Hagis, <a href="https://web.archive.org/web/2024*/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 A189000 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 A189000 Michel Marcus, <a href="/A189000/a189000.txt">Unexhaustive list of terms</a>
%H A189000 C. R. Wall, <a href="http://dx.doi.org/10.1090/S0002-9939-1972-0289403-9">Bi-unitary perfect numbers</a>, Proc. Amer. Math. Soc. 33 (1) (1972) 39-42.
%H A189000 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 A189000 {n | A188999(n)}.
%e A189000 n=120 divides A188999(120)=360.
%e A189000 n=90 divides A188999(90)=180.
%e A189000 n=672 divides A188999(672)=2016.
%t A189000 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 A189000 (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 A189000 isok(n) = ! frac(a188999(n)/n); \\ _Michel Marcus_, Sep 03 2018
%Y A189000 Cf. A007691 (analog for sigma).
%Y A189000 Cf. A188999 (bi-unitary sigma), A318175, A318781 (the k coefficients).
%K A189000 nonn
%O A189000 1,2
%A A189000 _R. J. Mathar_, Apr 15 2011
%E A189000 a(18)-a(27) by _D. S. McNeil_, Apr 15 2011
%E A189000 a(28)-a(31) from _Giovanni Resta_, Apr 15 2017
%E A189000 a(1)=1 inserted by _Giovanni Resta_, Sep 07 2018