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%I #10 Oct 28 2018 07:48:12
%S 162,1026,1620,10098,10260,41800,51282,100980,107920,512820,1479006,
%T 4612720,4938136,14790060,14800240,23168840,28158165,32440716,
%U 55204500,81128632,84392560,88886448,209524210,283604220,325903500,498215416,572062304,881697520
%N Smallest member of bi-unitary sociable quadruples.
%C The bi-unitary version of A090615.
%e 162 is in the sequence since the iterations of the sum of bi-unitary proper divisors function (A188999(n) - n) are cyclic with a period of 4: 162, 174, 186, 198, 162, ... and 162 is the smallest member of the quadruple.
%t fun[p_, e_]:=If[Mod[e, 2]==1, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)];
%t bs[n_] := If[n==1, 1, Times @@ (fun @@@ FactorInteger[n])]-n;seq[n_]:=NestList [bs, n,4][[2;;5]] ;aQ[n_] := Module[ {s=seq[n]}, n==Min[s] && Count[s,n]==1]; Do[If[aQ[n],Print[n]],{n,1,10^9}]
%o (PARI) fn(n) = {if (n==1, 1, 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) - n;);}
%o isok(n) = my(v = vector(5)); v[1] = n; for(k=2, 5, v[k] = fn(v[k-1])); (v[5] == n) && (vecmin(v) == n) && (#vecsort(v,,8)==4); \\ _Michel Marcus_, Oct 02 2018
%o (PARI) is(n) = my(c = n); for(i = 1, 3, c = fn(c); if(c <= n, return(0))); c = fn(c); c == n \\ uses Michel Marcus' fn _David A. Corneth_, Oct 02 2018
%Y Cf. A090615, A188999, A292980, A292981, A319902.
%K nonn
%O 1,1
%A _Amiram Eldar_, Oct 01 2018
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