cp's OEIS Frontend

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

%I A377417 #10 May 30 2025 23:16:10
%S A377417 1272,4632,31632,266712,805152,4897392,94177392,2928675264
%N A377417 Numbers t such that t + s(t) = s(s(t)), where s(t) is the sum of aliquot divisors of t (A001065(t)).
%C A377417 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)).
%C A377417 The sum of aliquot divisors is also called the sum of aliquot parts, the aliquot sum or the sum of proper divisors (A001065).
%C A377417 a(9) > 10^10 (if it exists).
%C A377417 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))).
%C A377417 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):
%C A377417 ---------------------------------------------------------------------------------------------
%C A377417      m   |   z  |    t    |     (m+1)-tuple (s^[0](t), s^[1](t),...,s^[m](t))
%C A377417 ---------------------------------------------------------------------------------------------
%C A377417 3        | 10^9 | 8880    | 8880, 19392, 32424, 60696 = 8880+19392+32424
%C A377417          |      | 1468584 | 1468584, 2933016, 4399584, 8801184 = 1468584+2933016+4399584
%C A377417 4        | 10^9 | 285816  | 285816, 428784, 679032, 1160208, 2553840 = 285816+428784+
%C A377417          |      |         | +679032+1160208
%C A377417 5        | 10^9 | 3280592 | 3280592, 4415344, 4196952, 7343928, 12546072, 31782888 =
%C A377417          |      |         | = 3280592+4415344+4196952+7343928+12546072
%C A377417 6,...,30 | 10^8 |    -    |
%C A377417 ---------------------------------------------------------------------------------------------
%e A377417 a(1) = 1272 because s(1272) = 1968, s(1968) = 3240 = 1272 + 1968.
%e A377417 a(2) = 4632 because s(4632) = 7008, s(7008) = 11640 = 4632 + 7008.
%e A377417 a(3) = 31632 because s(31632) = 50208, s(50208) = 81840 = 31632 + 50208.
%o A377417 (Maxima)
%o A377417 (n:1, for t:2 thru 100000000 do
%o A377417 (x:divsum(t)-t, y:divsum(x)-x,
%o A377417 if t+x=y then
%o A377417 (print(n, "" , t ), n:n+1)));
%o A377417 (PARI) isok(t) = t == sigma(sigma(t) - t) - 2*(sigma(t) - t); \\ _Michel Marcus_, Oct 29 2024
%Y A377417 Cf. A000203, A000396, A001065, A003416, A051027, A063990.
%K A377417 nonn,more
%O A377417 1,1
%A A377417 _Lechoslaw Ratajczak_, Oct 27 2024