A361811 - OEIS

%I #9 Mar 25 2023 08:18:33

%S 1026,10098,10260,41800,45696,100980,241824,685440,4938136,13959680,

%T 14958944,25581600,28158165,32440716,36072320,55204500,74062944,

%U 81128632,149589440,178327008,192793770,209524210,283604220,319848642,498215416,581112000,740629440,1236402232

%N Smallest members of infinitary sociable quadruples.

%C The first 8 terms were found by Cohen (1990).

%H Graeme L. Cohen, <a href="http://dx.doi.org/10.1090/S0025-5718-1990-0993927-5">On an integer's infinitary divisors</a>, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.

%H Jan Munch Pedersen, <a href="http://62.198.248.44/aliquot/knwni4.htm">Known Infinitary Sociable Numbers of order four</a>.

%e 1026 is a term since the iterations of the sum of aliquot infinitary divisors function (A126168) that start with 1026 are cyclic with period 4: 1026, 1374, 1386, 1494, 1026, ..., and 1026 is the smallest member of the quadruple.

%e The first five quadruples are {1026, 1374, 1386, 1494}, {10098, 15822, 19458, 15102}, {10260, 13740, 13860, 14940}, {41800, 51800, 66760, 83540}, {45696, 101184, 94656, 88944}.

%t f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]]>0, 1 + p^(2^(m-j)), 1], {j, 1, m}]]; infs[n_] := Times @@ f @@@ FactorInteger[n] - n;  infs[1] = 0; seq[n_] := NestList[infs, n, 4][[2;; 5]] ; q[n_] := Module[{s = seq[n]}, n == Min[s] && Count[s, n] == 1]; Select[Range[10^6], q]

%o (PARI) infs(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) + 1, 1))) - n; }

%o is(n) = {my(m = n); for(k = 1, 4, m = infs(m); if(k < 4 && m <= n, return(0))); m == n; }

%Y Cf. A049417, A126168.

%Y Cf. A007357 (period 1), A126169 and A126170 (period 2).

%Y Subsequence of A004607 (all cycles of length > 2).

%Y Similar sequences: A090615 (all divisors), A319902 (unitary), A319915 (bi-unitary).

%K nonn

%O 1,1

%A _Amiram Eldar_, Mar 25 2023