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A323753 Lesser member of primitive exponential amicable pairs.

%I A323753 #11 Feb 16 2019 21:11:12
%S A323753 90972,937692,4548600,44030448,46884600,453842928,712931184,906494400,
%T A323753 20907057600,34793179200,47646797328,53469838800,240707724300
%N A323753 Lesser member of primitive exponential amicable pairs.
%C A323753 Exponential amicable pair (m,n) is primitive if there is no prime number that is a unitary divisor of both m and n. All the other amicable pairs can be generated from primitive pairs by multiplying them with a squarefree integer coprime to each of the members of the pair. Hagis found the first 6 terms in 1988. Pedersen found the next 7 terms in 1999.
%C A323753 a(14) <= 588330137304.
%C A323753 The larger counterparts are in A323754.
%H A323753 Peter Hagis, Jr., <a href="http://dx.doi.org/10.1155/S0161171288000407">Some Results Concerning Exponential Divisors</a>, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-350.
%H A323753 Jan Munch Pedersen, <a href="http://62.198.248.44/aliquot/knwne2.htm">Known Exponential Amicable Pairs</a>.
%e A323753 (90972 = 2^2*3^2*7*19^2, 100548 = 2^2*3^3*7^2*19) are a primitive pair since they are an exponential amicable pair (A126165, A126166) and they do not have a common prime divisor with multiplicity 1 in both.
%e A323753 (454860, 502740) = 5 * (90972, 100548) are not a primitive pair since 5 divides both of them only once.
%t A323753 rad[n_] := Times @@ First /@ FactorInteger[n]; pf[n_] := Denominator[n/rad[n]^2]; esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; es[n_] := esigma[n] - n; s = {}; Do[m = es[n]; If[m > n && es[m] == n && CoprimeQ[pf[n], pf[m]], AppendTo[s, n]], {n, 1, 10^7}]; s (* after _Jean-François Alcover_ at A055231 and A051377 *)
%Y A323753 Cf. A051377, A054979, A049419, A054980, A126164, A126165, A126166, A126167, A323754.
%K A323753 nonn,more
%O A323753 1,1
%A A323753 _Amiram Eldar_, Jan 26 2019