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

Author

Michel Marcus, Aug 20 2018

Keywords

• nonn

Comments

As in A019278, here there are many instances where if x is a term, then A188999(x) is also a term.

Additionally, there exist longer chains of 3 or 4 elements; e.g.,

- 8 (3), 15 (4), 24 (5), 60 (6);

- 9 (2), 10 (3), 18 (4), 30 (5);

- 512 (3), 1023 (4), 1536 (5), 4092 (6);

- 8925 (4), 14976 (5), 35700 (6);

- 219969739395000 (16), 899826278400000 (17), 3519515830320000 (18).

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

Cf. A188999 (bi-unitary sigma).

Cf. A019278 (analog for sigma), A318182 (analog for infinitary sigma).

Programs

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;