1,1

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.

If sigma(N) is the sum of positive divisors of N, then:

a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),

a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),

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.

This is used in the example below.

A284326 also generates this sequence under iteration. - Timothy L. Tiffin, Feb 22 2022

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

a(n+4) = a(n).

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

a(1) = 34;

a(2) = sigma(34) = 54;

a(3) = sigma(54) - 18 - 6 = 42;

a(4) = sigma(42) - 42 - 6 = 48;

a(5) = sigma(48) - 48 - 24 - 12 - 6 = 34 = a(1);

:

:

PadRight[{}, 100, {34, 54, 42, 48}] (* Paolo Xausa, Jan 29 2026 *)

(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

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274338, A274339, A274340, A274549, A275082, A284326.

Sequence in context: A092888 A217768 A217704 * A305061 A051969 A241520

Adjacent sequences: A274377 A274378 A274379 * A274381 A274382 A274383

nonn,easy

Timothy L. Tiffin, Jun 22 2016

approved