A361811 Smallest members of infinitary sociable quadruples.

1026, 10098, 10260, 41800, 45696, 100980, 241824, 685440, 4938136, 13959680, 14958944, 25581600, 28158165, 32440716, 36072320, 55204500, 74062944, 81128632, 149589440, 178327008, 192793770, 209524210, 283604220, 319848642, 498215416, 581112000, 740629440, 1236402232

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..28.

Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.

Jan Munch Pedersen, Known Infinitary Sociable Numbers of order four.

Example

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.
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}.

Mathematica

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]

Prog

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

Crossrefs

Cf. A049417, A126168.

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

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

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

Sequence in context: A371982 A031530 A004607 * A221008 A282254 A229332

Adjacent sequences: A361808 A361809 A361810 * A361812 A361813 A361814

Keyword

nonn

Author

Amiram Eldar, Mar 25 2023

Status

approved