A189000 Bi-unitary multiperfect numbers.

1, 6, 60, 90, 120, 672, 2160, 10080, 22848, 30240, 342720, 523776, 1028160, 1528800, 6168960, 7856640, 7983360, 14443520, 22932000, 23569920, 43330560, 44553600, 51979200, 57657600, 68796000, 133660800, 172972800, 779688000, 1476304896, 2339064000, 6840038400

Offset: 1

Author

R. J. Mathar, Apr 15 2011

Keywords

• nonn

Comments

All entries greater than 1 are even [Hagis].

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

Wall shows that 6, 60, and 90 are the only bi-unitary perfect numbers. - Tomohiro Yamada, Apr 15 2017

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

a(43) > 4.66*10^12. - Giovanni Resta, Sep 07 2018

We include 1 here, although this is not "multi"-perfect. - R. J. Mathar, Sep 08 2018

Examples

			n=120 divides A188999(120)=360.
n=90 divides A188999(90)=180.
n=672 divides A188999(672)=2016.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..42
• Peter Hagis, Bi-Unitary amicable and multiperfect numbers, Fib. Quart. 25 (2) (1987) 144-151
• Pentti Haukkanen and V. Sitaramaiah, Bi-unitary multiperfect numbers, I, Notes Number Theory Discrete Math. 26 (1) (2020) 93-171.
• Michel Marcus, Unexhaustive list of terms
• C. R. Wall, Bi-unitary perfect numbers, Proc. Amer. Math. Soc. 33 (1) (1972) 39-42.
• Tomohiro Yamada, Determining all biunitary triperfect numbers of a certain form, arXiv:2406.19331 [math.NT], 2024.

Crossrefs

Cf. A007691 (analog for sigma).

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

Programs

Mathematica

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 *)

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); }
isok(n) = ! frac(a188999(n)/n); \\ Michel Marcus, Sep 03 2018

Formula

{n | A188999(n)}.

Extensions

a(18)-a(27) by D. S. McNeil, Apr 15 2011

a(28)-a(31) from Giovanni Resta, Apr 15 2017

a(1)=1 inserted by Giovanni Resta, Sep 07 2018