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

Author

Timothy L. Tiffin, Jun 22 2016

Keywords

• nonn
• easy

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.

Examples

			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).
		

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).

Crossrefs

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

Programs

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

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