cp's OEIS Frontend

A274338 The period 10 sequence of the iterated sum of deficient divisors function (A187793) starting at 52.

%I A274338 #50 Mar 04 2022 08:44:51
%S A274338 52,98,171,260,308,336,76,140,78,84,52,98,171,260,308,336,76,140,78,
%T A274338 84,52,98,171,260,308,336,76,140,78,84,52,98,171,260,308,336,76,140,
%U A274338 78,84,52,98,171,260,308,336,76,140,78,84,52,98,171,260,308,336,76,140,78
%N A274338 The period 10 sequence of the iterated sum of deficient divisors function (A187793) starting at 52.
%C A274338 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 the only one of period (order, length) 10 that A187793 generates under iteration.
%C A274338 If sigma(N) is the sum of positive divisors of N, then:
%C A274338   a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
%C A274338   a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
%C A274338   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 A274338 This is used in the example below.
%H A274338 Colin Barker, <a href="/A274338/b274338.txt">Table of n, a(n) for n = 1..1000</a>
%H A274338 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,1).
%F A274338 a(n+10) = a(n).
%F A274338 G.f.: x*(52 + 98*x + 171*x^2 + 260*x^3 + 308*x^4 + 336*x^5 + 76*x^6 + 140*x^7 + 78*x^8 + 84*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)). - _Colin Barker_, Jan 30 2020
%e A274338 a(1) = 52;
%e A274338 a(2) = sigma(52) = 98;
%e A274338 a(3) = sigma(98) = 171;
%e A274338 a(4) = sigma(171) = 260;
%e A274338 a(5) = sigma(260) - 260 - 20 = 308;
%e A274338 a(6) = sigma(308) - 308 - 28 = 336;
%e A274338 a(7) = 1 + 2 + 3 + 4 + 7 + 8 + 14 + 16 + 21 = 76 [since 336 has more abundant divisors than deficient ones];
%e A274338 a(8) = sigma(76) = 140;
%e A274338 a(9) = sigma(140) - 140 - 70 - 28 - 20 = 78;
%e A274338 a(10) = sigma(78) - 78 - 6 = 84;
%e A274338 a(11) = sigma(84) - 84 - 42 - 28 - 12 - 6 = 52 = a(1).
%o A274338 (PARI) a(n)=n=n%10; if(n>0, sumdiv(a(n-1),d,if(sigma(d,-1)<2,d,0)), 84) \\ _Charles R Greathouse IV_, Jun 23 2016
%o A274338 (PARI) Vec(x*(52 + 98*x + 171*x^2 + 260*x^3 + 308*x^4 + 336*x^5 + 76*x^6 + 140*x^7 + 78*x^8 + 84*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^50)) \\ _Colin Barker_, Jan 30 2020
%Y A274338 Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274339, A274340, A274380, A274549, A275082.
%K A274338 nonn,easy
%O A274338 1,1
%A A274338 _Timothy L. Tiffin_, Jun 22 2016