A318175 - OEIS

%I #33 Nov 05 2023 19:07:24

%S 1,2,8,9,10,15,18,21,24,30,42,60,144,160,168,240,270,288,324,480,512,

%T 630,648,960,1023,1200,1404,1428,1536,2046,2400,2808,2856,2880,3276,

%U 3570,4092,4320,4608,6552,8925,10080,10368,10752,11550,13824,14280,14976,15345,16368,17850

%N Numbers m such that A188999(A188999(m)) = k*m for some k where A188999 is the bi-unitary sigma function.

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

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

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

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

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

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

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

%H Giovanni Resta, <a href="/A318175/b318175.txt">Table of n, a(n) for n = 1..227</a> (terms < 10^12; first 185 terms from Tomohiro Yamada)

%H Tomohiro Yamada, <a href="https://arxiv.org/abs/1705.00189">2 and 9 are the only biunitary superperfect numbers</a>, arXiv:1705.00189 [math.NT], 2017. See Table 1.

%H Tomohiro Yamada, <a href="http://ac.inf.elte.hu/Vol_048_2018/247_48.pdf">2 and 9 are the only biunitary superperfect numbers</a>, Annales Univ. Sci. Budapest., Sec. Comp., Volume 48 (2018). See Table 1.

%H Michel Marcus, <a href="/A318175/a318175_1.log.txt">Unexhaustive list of terms</a>.

%e For m=2, A188999(2) = 3 and A188999(3) = 4, so 2 is a term with k=2.

%e For m=9, A188999(9) = 10 and A188999(10) = 18, so 9 is a term with k=2.

%t 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]}]];

%t Reap[For[m = 1, m < 20000, m++, If[Divisible[bsigma @ bsigma @ m, m], Sow[m]]]][[2, 1]] (* _Jean-François Alcover_, Sep 22 2018 *)

%o (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); }

%o isok(n) = frac(a188999(a188999(n))/n) == 0;

%Y Cf. A188999 (bi-unitary sigma).

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

%K nonn

%O 1,2

%A _Michel Marcus_, Aug 20 2018