A318175 Numbers m such that A188999(A188999(m)) = k*m for some k where A188999 is the bi-unitary sigma function.
1, 2, 8, 9, 10, 15, 18, 21, 24, 30, 42, 60, 144, 160, 168, 240, 270, 288, 324, 480, 512, 630, 648, 960, 1023, 1200, 1404, 1428, 1536, 2046, 2400, 2808, 2856, 2880, 3276, 3570, 4092, 4320, 4608, 6552, 8925, 10080, 10368, 10752, 11550, 13824, 14280, 14976, 15345, 16368, 17850
Offset: 1
Keywords
Examples
For m=2, A188999(2) = 3 and A188999(3) = 4, so 2 is a term with k=2. For m=9, A188999(9) = 10 and A188999(10) = 18, so 9 is a term with k=2.
Links
- Giovanni Resta, Table of n, a(n) for n = 1..227 (terms < 10^12; first 185 terms from Tomohiro Yamada)
- Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, arXiv:1705.00189 [math.NT], 2017. See Table 1.
- Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, Annales Univ. Sci. Budapest., Sec. Comp., Volume 48 (2018). See Table 1.
- Michel Marcus, Unexhaustive list of terms.
Crossrefs
Programs
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Mathematica
bsigma[n_] := If[n==1, 1, Product[{p, e} = pe; If[OddQ[e], (p^(e+1)-1)/(p-1), ((p^(e+1)-1)/(p-1)-p^(e/2))], {pe, FactorInteger[n]}]]; Reap[For[m = 1, m < 20000, m++, If[Divisible[bsigma @ bsigma @ m, m], Sow[m]]]][[2, 1]] (* Jean-François Alcover, Sep 22 2018 *) -
PARI
a188999(n) = {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(a188999(n))/n) == 0;
Comments