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A274338 The 10-cycle of the iterated sum of deficient divisors function. 6
52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This cycle is generated in a similar way to the aliquot sequences (or sociable chains) that are generated by the sum of proper divisors function.  This cycle of length (or order) 10 appears to be the longest cycle generated by the sum of deficient divisors function.

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, a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number, a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either abundant or perfect, and so forth.

LINKS

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

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

FORMULA

a(n+10) = a(n).

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

EXAMPLE

a(1) = 52, a(2) = sigma(52) = 98, a(3) = sigma(98) = 171, a(4) = sigma(171) = 260, a(5) = sigma(260)-260-20 = 308, a(6) = sigma(308)-308-28 = 336, a(7) = 1+2+3+4+7+8+14+16+21 = 76 [since 336 has more abundant divisors than deficient ones], a(8) = sigma(76) = 140, a(9) = sigma(140)-140-70-28-20 = 78, a(10) = sigma(78)-78-6 = 84, a(11) = sigma(84)-84-42-28-12-6 = 52 = a(1), ...

PROG

(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

(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

CROSSREFS

Cf. A125310, A187793, A274339, A274340, A274380, A274549.

Sequence in context: A234099 A026067 A039475 * A094552 A236461 A044141

Adjacent sequences:  A274335 A274336 A274337 * A274339 A274340 A274341

KEYWORD

nonn,easy

AUTHOR

Timothy L. Tiffin, Jun 22 2016

STATUS

approved

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Last modified January 25 16:01 EST 2022. Contains 350572 sequences. (Running on oeis4.)