OFFSET
1,2
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
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.
FORMULA
{n | A188999(n)}.
EXAMPLE
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
R. J. Mathar, Apr 15 2011
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
STATUS
approved