A377417 Numbers t such that t + s(t) = s(s(t)), where s(t) is the sum of aliquot divisors of t (A001065(t)).

1272, 4632, 31632, 266712, 805152, 4897392, 94177392, 2928675264

Offset: 1

Author

Lechoslaw Ratajczak, Oct 27 2024

Keywords

• nonn
• more

Comments

Equivalently, numbers t which satisfy the equation t = sigma(sigma(t) - t) - 2*(sigma(t) - t), where sigma(t) is the sum of divisors of t (A000203(t)).

The sum of aliquot divisors is also called the sum of aliquot parts, the aliquot sum or the sum of proper divisors (A001065).

a(9) > 10^10 (if it exists).

Let s^[m](t) denote m-fold iteration of s(t) (i.e., s^[0](t) = t and s^[m](t) = s(s^[m-1](t))).

The following table shows consecutive solutions t (t <= z) of the equation s^[m](t) = Sum_{i=0..m-1} s^[i](t) for m = 3,4,5,...,30 (solutions for m = 1 are perfect numbers, solutions for m = 2 are in the data section):

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m | z | t | (m+1)-tuple (s^[0](t), s^[1](t),...,s^[m](t))

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3 | 10^9 | 8880 | 8880, 19392, 32424, 60696 = 8880+19392+32424

| | 1468584 | 1468584, 2933016, 4399584, 8801184 = 1468584+2933016+4399584

4 | 10^9 | 285816 | 285816, 428784, 679032, 1160208, 2553840 = 285816+428784+

| | | +679032+1160208

5 | 10^9 | 3280592 | 3280592, 4415344, 4196952, 7343928, 12546072, 31782888 =

| | | = 3280592+4415344+4196952+7343928+12546072

6,...,30 | 10^8 | - |

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Examples

			a(1) = 1272 because s(1272) = 1968, s(1968) = 3240 = 1272 + 1968.
a(2) = 4632 because s(4632) = 7008, s(7008) = 11640 = 4632 + 7008.
a(3) = 31632 because s(31632) = 50208, s(50208) = 81840 = 31632 + 50208.
		

Crossrefs

Cf. A000203, A000396, A001065, A003416, A051027, A063990.

Programs

Maxima

(n:1, for t:2 thru 100000000 do
(x:divsum(t)-t, y:divsum(x)-x,
if t+x=y then
(print(n, "" , t ), n:n+1)));

PARI

isok(t) = t == sigma(sigma(t) - t) - 2*(sigma(t) - t); \\ Michel Marcus, Oct 29 2024