%I #19 Nov 08 2020 23:06:21
%S 108,192,448,2688,6000,8640,12960,17496,18750,20412,32400,86400,
%T 112640,120960,138240,169344,181440,245760,304128,600000,658560,
%U 714420,857304,979776,1350000,1632960,1778112,2073600,2361960,3359232,3500000,4561920,7112448
%N Integers that can be written m = k*tau(k) = q*tau(q) where (k, q) is a primitive solution of this equation and tau(k) is the number of divisors of k.
%C As the multiplicativity of tau(k) ensures an infinity of solutions to the general equation m = k*tau(k) (see A338382), Richard K. Guy asked if, as for k*sigma(k) = q*sigma(q) (A337875, A337876), k*tau(k) = q*tau(q) has an infinity of primitive solutions, in the sense that (k', q') is not a solution for any k' = k/d, q' = q/d, d>1 (see reference Guy's book and 3rd example). The answer to this question seems not to be known today.
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B12, p. 102-103.
%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 168, page 127.
%e -> For a(1): 18 * tau(18) = 27 * tau(27) = 108.
%e -> For a(2): 24 * tau(24) = 32 * tau(32) = 192.
%e -> Why 1080 = A338382(4) is not a term? 90 * tau(90) = 135 * tau(135) = 1080 but as 90/5 = 18 and 135/5 = 27, this solution that is generated by the first example is not primitive.
%e -> For a(4) : 168 * tau(168) = 192 * tau(192) = 224 * tau(224) = A338382(8) = 2688.
%e 1) for k=168 and q=192; with d=3, k/3=56 and q/3=64, with 56 * tau(56) = 64 * tau(64) = 448 = a(3), hence (168, 192) is not a primitive solution;
%e 2) for k=168 and q=224; with d=7, k/7=24 and q/7=32, with 24 * tau(24) = 32 * tau(32) = 192 = a(2), hence (24, 32) is not a primitive solution; but
%e 3) for k=192 and q=224, there is no common divisor d such that 192/d and 224/d can satisfy (192/d)*tau(192/d) = (224/d)*tau(224/d), so (192, 224) is a primitive solution linked to m = 2688 that is the term a(4).
%o (PARI) is(n) = {my(l, d); l = List(); d = divisors(n); for(i = 1, #d, if(d[i]*numdiv(d[i]) == n, listput(l, d[i]); ) ); forvec(x = vector(2, i, [1, #l]), if(isprimitive(l[x[1]], l[x[2]], n), return(1) ) , 2 ); 0 }
%o isprimitive(m, n, t) = { my(g = gcd(m, n), d = divisors(g)); for(i = 2, #d, if(m/d[i]*numdiv(m/d[i]) == t/d[i]/numdiv(d[i]) && n/d[i]*numdiv(n/d[i]) == t/d[i]/numdiv(d[i]), return(0) ) ); 1 } \\ _David A. Corneth_, Nov 06 2020
%Y Subsequence of A338382.
%Y Cf. A000005, A038040, A338381, A338383, A338385, A338519.
%Y Cf. A337875 (similar for k*sigma(k))
%K nonn
%O 1,1
%A _Bernard Schott_, Nov 03 2020
%E More terms from _David A. Corneth_, Nov 04 2020