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A274380 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 34.

%I A274380 #54 Mar 04 2022 14:20:03
%S A274380 34,54,42,48,34,54,42,48,34,54,42,48,34,54,42,48,34,54,42,48,34,54,42,
%T A274380 48,34,54,42,48,34,54,42,48,34,54,42,48,34,54,42,48,34,54,42,48,34,54,
%U A274380 42,48,34,54,42,48,34,54,42,48,34,54,42,48
%N A274380 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 34.
%C A274380 This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065). It appears to be one of two sequences of period (order, length) 4 that A187793 generates under iteration. The other one is A274340.
%C A274380 If sigma(N) is the sum of positive divisors of N, then:
%C A274380   a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
%C A274380   a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
%C A274380   a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
%C A274380 This is used in the example below.
%C A274380 A284326 also generates this sequence under iteration. - _Timothy L. Tiffin_, Feb 22 2022
%H A274380 Colin Barker, <a href="/A274380/b274380.txt">Table of n, a(n) for n = 1..1000</a>
%H A274380 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).
%F A274380 a(n+4) = a(n).
%F A274380 G.f.: 2*x*(17 + 27*x + 21*x^2 + 24*x^3) / ((1 - x)*(1 + x)*(1 + x^2)). - _Colin Barker_, Jan 30 2020
%e A274380 a(1) = 34;
%e A274380 a(2) = sigma(34) = 54;
%e A274380 a(3) = sigma(54) - 18 - 6 = 42;
%e A274380 a(4) = sigma(42) - 42 - 6 = 48;
%e A274380 a(5) = sigma(48) - 48 - 24 - 12 - 6 = 34 = a(1);
%e A274380   :
%e A274380   :
%o A274380 (PARI) Vec(2*x*(17 + 27*x + 21*x^2 + 24*x^3) / ((1 - x)*(1 + x)*(1 + x^2)) + O(x^80)) \\ _Colin Barker_, Jan 30 2020
%Y A274380 Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274338, A274339, A274340, A274549, A275082, A284326.
%K A274380 nonn,easy
%O A274380 1,1
%A A274380 _Timothy L. Tiffin_, Jun 22 2016