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A338383 Table read by rows, in which the n-th row lists all the preimages k, in increasing order, such that k*tau(k) = A338382(n).

%I A338383 #22 Mar 31 2023 08:14:39
%S A338383 18,27,24,32,56,64,90,135,126,189,120,160,198,297,168,192,224,234,351,
%T A338383 306,459,342,513,264,352,280,320,414,621,312,416,400,500,522,783,408,
%U A338383 544,558,837,456,608,666,999,450,675,360,432,552,736,738,1107,774,1161,616,704
%N A338383 Table read by rows, in which the n-th row lists all the preimages k, in increasing order, such that k*tau(k) = A338382(n).
%C A338383 The map k -> k*tau(k) = m is not injective (A038040) and this sequence lists, in increasing order of m, the preimages of the integers m that have more than one preimage.
%D A338383 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B12, p. 102-103.
%D A338383 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 168, page 127.
%e A338383 The table begins:
%e A338383    18,  27;
%e A338383    24,  32;
%e A338383    56,  64;
%e A338383    90, 135;
%e A338383   126, 189;
%e A338383   120, 160;
%e A338383   198, 297;
%e A338383   168, 192, 224;
%e A338383   ...
%e A338383 1st row is (18, 27) because 18 * tau(18) = 27 * tau(27) = 108 = A338382(1).
%e A338383 2nd row is (24, 32) because 24 * tau(24) = 32 * tau(32) = 192 = A338382(2).
%e A338383 8th row is (168, 192, 224), because 168 * tau(168) = 192 * tau(192) = 224 * tau(224) = 2688 = A338382(8); it is the first row with 3 preimages.
%o A338383 (PARI) upto(n) = {m = Map(); res = List(); n = n\2; w = []; for(i = 1, n, c = i*numdiv(i); if(mapisdefined(m, c), listput(res, c); l = mapget(m, c); listput(l, i); mapput(m, c, l) , mapput(m, c, List(i)); ) ); listsort(res, 1); v = select(x -> x <= 2*(n+1), res); for(i = 1, #v, w = concat(w, Vec(mapget(m, v[i]))) ); w; } \\ _Michel Marcus_, Oct 27 2020
%Y A338383 Cf. A000005, A038040, A327166, A338381, A338382, A338384, A338385.
%Y A338383 Cf. A337874 (similar for k*sigma(k)).
%K A338383 nonn,tabf
%O A338383 1,1
%A A338383 _Bernard Schott_, Oct 26 2020