cp's OEIS Frontend

A371418 The largest aliquot divisor of the sum of divisors of n; a(1) = 1.

%I A371418 #15 May 21 2025 18:58:34
%S A371418 1,1,2,1,3,6,4,5,1,9,6,14,7,12,12,1,9,13,10,21,16,18,12,30,1,21,20,28,
%T A371418 15,36,16,21,24,27,24,13,19,30,28,45,21,48,22,42,39,36,24,62,19,31,36,
%U A371418 49,27,60,36,60,40,45,30,84,31,48,52,1,42,72,34,63,48,72
%N A371418 The largest aliquot divisor of the sum of divisors of n; a(1) = 1.
%C A371418 Carmichael (1921) defined this arithmetic function for the purpose of studying periodic chains that are formed by repeatedly applying the mapping x -> a(x) starting at a given positive integer. This results in a sequence that is analogous to an aliquot sequence.
%C A371418 Periodic chains of cycle 1 are the fixed points of this sequence. 1 and the even perfect numbers (the even terms of A000396) are fixed points. Are there any other numbers k such that a(k) = k?
%C A371418 If a(k) = k and k is even, then a(k) is even and so is sigma(k), and therefore sigma(k) = 2*k and k is an even perfect number. If k is odd, then it is an odd multiperfect number, and no odd multiperfect number above 1 is known.
%C A371418 More specifically, if a(k) = k and k is odd, then k must be a square, and an m-multiperfect number (number k such that sigma(k) = m * k), with m being an odd prime number that is the least prime factor of sigma(k). For example, if there is an odd triperfect number (A005820) then it is a fixed point of this sequence.
%C A371418 Periodic chains of cycle 2 are amicable pairs (A371419 and A371420). Are there any longer cycles?
%H A371418 Amiram Eldar, <a href="/A371418/b371418.txt">Table of n, a(n) for n = 1..10000</a>
%H A371418 Robert D. Carmichael, <a href="https://doi.org/10.5951/MT.14.6.0305">Empirical Results in the Theory of Numbers</a>, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; <a href="https://www.jstor.org/stable/27950349">alternative link</a>. See p. 309.
%H A371418 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AliquotSequence.html">Aliquot Sequence</a>.
%H A371418 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiperfectNumber.html">Multiperfect Number</a>.
%H A371418 Wikipedia, <a href="http://en.wikipedia.org/wiki/Aliquot_sequence">Aliquot sequence</a>.
%F A371418 a(n) = A032742(A000203(n)).
%F A371418 a(n) = A000203(n)/A071189(n).
%F A371418 a(n) = A000203(n)/2 if n is in A028983 (i.e., n is not in A028982).
%F A371418 a(k) = 1 if and only if k = 1 or k is in A023194.
%e A371418 The sum of the divisors of 3 is 1 + 3 = 4. The divisors of 4 are 1, 2, 4. 2 is the largest aliquot divisor of 4. Therefore a(3) = 2.
%t A371418 r[n_] := n/FactorInteger[n][[1, 1]]; a[n_] := r[DivisorSigma[1, n]]; Array[a, 100]
%o A371418 (PARI) a(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}
%Y A371418 Cf. A000203, A000396, A005820, A023194, A028982, A028983, A032742, A071189, A371419, A371420, A371421, A371422, A371423.
%K A371418 nonn,easy
%O A371418 1,3
%A A371418 _Amiram Eldar_, Mar 23 2024