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

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

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 the only one of period (order, length) 10 that A187793 generates under iteration.

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.

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. A001065, A005100, A005101, A071395, A125310, A187793, A274339, A274340, A274380, A274549, A275082.

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