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%I #37 Nov 17 2023 11:57:21
%S 108,192,448,1080,1512,1920,2376,2688,2808,3672,4104,4224,4480,4968,
%T 4992,6000,6264,6528,6696,7296,7992,8100,8640,8832,8856,9288,9856,
%U 10152,11136,11448,11648,11904,12096,12744,12960,13176,14208,14400,14472,15120,15232,15336
%N Numbers m such that the equation m = k*tau(k) has more than one solution, where tau(k) is the number of divisors of k.
%C The map k -> k*tau(k) = m is not injective (A038040), this sequence lists in increasing order the integers m that have several preimages.
%C There are primitive terms that generate an infinity of terms because of the multiplicativity of tau(k); for example, a(1) = 108 and with t such that gcd(t,6) = 1, every m = 108*(t*tau(t)) is another term; in particular, with p prime > 3, every m = 216*p is another term: 1080, 1512, 2376, ...
%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.
%H Passawan Noppakaew and Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Pongsriiam/pong43.html">Product of Some Polynomials and Arithmetic Functions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
%e a(1) = 108 because 18 * tau(18) = 27 * tau(27) = 108.
%e a(2) = 192 because 24 * tau(24) = 32 * tau(32) = 192.
%e a(3) = 448 because 56 * tau(56) = 64 * tau(64) = 448.
%e a(8) = 2688 is the smallest term with 3 preimages because 168 * tau(168) = 192 * tau(192) = 224 * tau(224) = 2688.
%t solNum[n_] := DivisorSum[n, 1 &, # * DivisorSigma[0, #] == n &]; Select[Range[16000], solNum[#] > 1 &] (* _Amiram Eldar_, Oct 23 2020 *)
%o (PARI) isok(m) = {my(nb=0); fordiv(m, d, if (d*numdiv(d) == m, nb++; if (nb>1, return(1))); ); return (0); } \\ _Michel Marcus_, Oct 24 2020
%Y Cf. A000005, A038040, A327166, A338381, A338383, A338384, A338385.
%Y Cf. A337873 (similar for k*sigma(k)).
%Y Subsequence of A036438.
%K nonn
%O 1,1
%A _Bernard Schott_, Oct 23 2020
%E More terms from _Amiram Eldar_, Oct 23 2020