%I #6 Mar 23 2024 08:23:33
%S 12,48,112,160,192,448,1984,12288,28672,126976,196608,458752,520192,
%T 786432,1835008,2031616,8126464,8323072,33292288,536805376,2147221504,
%U 3221225472,7516192768,33285996544,34359476224,136365211648
%N Lesser member of Carmichael's variant of amicable pair: numbers k < m such that s(k) = m and s(m) = k, where s(k) = A371418(k).
%C Analogous to amicable numbers (A002025 and A002046) with the largest aliquot divisor of the sum of divisors (A371418) instead of the sum of aliquot divisors (A001065).
%C Carmichael (1921) proposed this function (A371418) for the purpose of studying periodic chains that are formed by repeatedly applying the mapping x -> A371418(x). The chains of cycle 2 are analogous to amicable numbers.
%C Carmichael noted that if q < p are two different Mersenne exponents (A000043), then 2^(p-1)*(2^q-1) and 2^(q-1)*(2^p-1) are an amicable pair. With the 51 Mersenne exponents that are currently known it is possible to calculate 51 * 50 / 2 = 1275 amicable pairs. (160, 189) is a pair that is not of this "Mersenne form". Are there any other pairs like it? There are no other such pairs with lesser member below a(26).
%C a(27) <= 8795019280384.
%C The greater counterparts are in A371420.
%H 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.
%e 12 is a term since A371418(12) = 14 > 12, and A371418(14) = 12.
%t r[n_] := n/FactorInteger[n][[1, 1]]; s[n_] := r[DivisorSigma[1, n]]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 1, 10^6}]; seq
%o (PARI) f(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}
%o lista(nmax) = {my(m); for(n = 1, nmax, m = f(n); if(m > n && f(m) == n, print1(n, ", ")));}
%Y Cf. A000043, A001065, A002046, A337874, A337876, A371418, A371420.
%Y Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.
%K nonn,more
%O 1,1
%A _Amiram Eldar_, Mar 23 2024