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A338385 Table read by rows, in which the n-th row lists the primitive solutions (k, q), k such that k*tau(k) = q*tau(q) = A338384(n).

%I A338385 #45 Dec 06 2020 08:36:32
%S A338385 18,27,24,32,56,64,192,224,400,500,360,432,540,648,972,2187,1875,3125,
%T A338385 1458,1701,1296,1350,2160,2400,5120,5632,2880,3024,3840,4608,4032,
%U A338385 4704,3780,5184,10240,16384,8448,9216,20000,25000,15680,16464,15876,25515,20412,23814
%N A338385 Table read by rows, in which the n-th row lists the primitive solutions (k, q), k<q, such that k*tau(k) = q*tau(q) = A338384(n).
%C A338385 As the multiplicativity of tau(k) ensures an infinity of solutions to the general equation k*tau(k) = q*tau(q) (see A338382), Richard K. Guy asked if there is an infinity of primitive solutions. A solution (k, q) with m = k*tau(k) = q*tau(q) is  primitive in the sense that (k', q') is not a solution for any k' = k/d, q' = q/d, m' = m/(d*tau(d)), d>1 with m' = k' * tau(k') = q' * tau(q').
%C A338385 Warning, Richard K. Guy asked if "there is an infinity of primitive solutions (for k*tau(k) = q*tau(q)), in the sense that (k', q') is not a solution for any k' = k/d, q' = q/d, d>1". It appears that this definition is not enough well defined, because some solutions as (4032, 4704), (20000, 25000), (20412, 23814),... that are primitive are not obtained in this case (see detailed example (20000, 25000) below). The mathematical explanation is that tau satisfies the relation tau(r*s) = tau(r) * tau(s) * (t/tau(t)) where t = gcd(r,s).
%D A338385 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B12, p. 102-103.
%D A338385 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 168, page 127.
%e A338385 The table begins:
%e A338385     18,  27;
%e A338385     24,  32;
%e A338385     56,  64;
%e A338385    192, 224;
%e A338385    400, 500;
%e A338385    360, 432;
%e A338385    ...
%e A338385 1st row is (18, 27) because 18 * tau(18) = 27 * tau(27) = 108 = A338384(1).
%e A338385 4th row is (192, 224) because 192 * tau(192) = 224 * tau(24) = 2688 = A338384(4); Note that 168 * tau(168) = 192 * tau(192) = 224 * tau(24) = 2688 = A338382(8) but (168, 192) and (168, 224) are not primitive solutions (see detailed example in A338384).
%e A338385 5th row is (400, 500) because 400 * tau(400) = 500 * tau(500) = 6000.
%e A338385 20th row is (20000, 25000) although (20000/50, 25000/50) = (400, 500) and that (400, 500) is the 5th row. Explanation: A338384(20) = 600000 = 20000*tau(20000) = 25000*tau(25000) and this pair is primitive, because for d = 50, we get 600000/(50*tau(50)) = 2000 <>  (20000/50)*tau(20000/50) = (25000/50)*tau(25000/50) = 6000. To be exhaustive, the two other pairs linked with 600000: (15000, 20000) and (15000, 25000) are not primitive.
%Y A338385 Cf. A000005, A038040, A338381, A338382, A338383, A338384.
%Y A338385 Cf. A337876 (similar for k*sigma(k)).
%K A338385 nonn,tabf
%O A338385 1,1
%A A338385 _Bernard Schott_, Nov 09 2020