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A336219 a(n) is the smallest member of the n-th purely periodic unitary sigma aliquot cycle listed in A336216.

%I A336219 #11 Jul 14 2020 23:20:04
%S A336219 6,30,60,90,114,1140,1482,2418,18018,24180,32130,35238,44772,56430,
%T A336219 67158,87360,142310,180180,197340,241110,263820,296010,308220,395730,
%U A336219 462330,473298,591030,669900,671580,698130,763620,785148,815100,1004850,1077890,1080150,1156870,1177722
%N A336219 a(n) is the smallest member of the n-th purely periodic unitary sigma aliquot cycle listed in A336216.
%C A336219 This is the first column of the irregular triangle in A336216.
%C A336219 From the formula of _Vladeta Jovovic_ in A034448 we get for an even number n not divisible by 4 and odd prime p: usigma(2^m * p * n) = (2^(m+1) + 1) * (p + 1) * usigma(n) / 3  so that usigma(2^m * p * n) = (2^m * p * n) * usigma(n) when 3* 2^m * p = (2^(m+1) + 1) * (p + 1), and consequently, p = (2^(m+1) + 1) / (2^m - 1), i.e. p = 5 for m = 1, and p = 3 for m = 2.
%C A336219 Therefore, if all members a_1, a_2, ... , a_k, a_1 of a cycle are even and not divisible by 4 and 5 then 10*a_1, 10*a_2, ... , 10*a_k, 10*a_1 form a cycle, and if all members a_1, a_2, ... , a_k, a_1 of a cycle are even and not divisible by 3 and 4 then 12*a_1, 12*a_2, ... , 12*a_k, 12*a_1 form a cycle.
%C A336219 If all members a_1, a_2, ... , a_k, a_1 of a cycle are odd, divisible by 3, but not divisible by 5 and 9 then 15*a_1, 15*a_2, ... , 15*a_k, 15*a_1 form a cycle. No such cycles exist in the current data up to 27287260.
%F A336219 a(n) = A336216( 1 + Sum_{i=1..n-1} A336218(i) ).
%e A336219 Start numbers of cycles related by a factor of 10 or 12, respectively:
%e A336219 10:  (6, 60), (114, 1140), (2418, 24180), (18018, 180180), (67158, 671580), (1177722, 1777220), ...
%e A336219 12:  (142310, 1707720), (1077890, 12934680), (1156870, 13882440), (1475810, 17709720), ...
%t A336219 (* a336216 and support functions in A336216 *)
%t A336219 Map[First, a336216[100000]] (* a(1..16) *)
%Y A336219 Cf. A034448, A063919, A327157, A336216, A336218.
%K A336219 nonn
%O A336219 1,1
%A A336219 _Hartmut F. W. Hoft_, Jul 12 2020