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A318175 Numbers m such that A188999(A188999(m)) = k*m for some k where A188999 is the bi-unitary sigma function. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Like 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 like:

- 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).

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

EXAMPLE

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.

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-Fran├žois Alcover, Sep 22 2018 *)

PROG

(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;

CROSSREFS

Cf. A188999 (bi-unitary sigma).

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

Sequence in context: A175463 A167450 A050569 * A318182 A047469 A283774

Adjacent sequences:  A318172 A318173 A318174 * A318176 A318177 A318178

KEYWORD

nonn

AUTHOR

Michel Marcus, Aug 20 2018

STATUS

approved

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Last modified April 1 10:33 EDT 2020. Contains 333159 sequences. (Running on oeis4.)