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A274380
The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 34.
6
34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48
OFFSET
1,1
COMMENTS
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
FORMULA
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
EXAMPLE
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);
:
:
PROG
(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
KEYWORD
nonn,easy
AUTHOR
Timothy L. Tiffin, Jun 22 2016
STATUS
approved