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).

18, 27, 24, 32, 56, 64, 90, 135, 126, 189, 120, 160, 198, 297, 168, 192, 224, 234, 351, 306, 459, 342, 513, 264, 352, 280, 320, 414, 621, 312, 416, 400, 500, 522, 783, 408, 544, 558, 837, 456, 608, 666, 999, 450, 675, 360, 432, 552, 736, 738, 1107, 774, 1161, 616, 704

Offset: 1

Author

Bernard Schott, Oct 26 2020

Keywords

• nonn
• tabf

Comments

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.

Examples

			The table begins:
   18,  27;
   24,  32;
   56,  64;
   90, 135;
  126, 189;
  120, 160;
  198, 297;
  168, 192, 224;
  ...
1st row is (18, 27) because 18 * tau(18) = 27 * tau(27) = 108 = A338382(1).
2nd row is (24, 32) because 24 * tau(24) = 32 * tau(32) = 192 = A338382(2).
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.
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B12, p. 102-103.
• D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 168, page 127.

Crossrefs

Cf. A000005, A038040, A327166, A338381, A338382, A338384, A338385.

Cf. A337874 (similar for k*sigma(k)).

Programs

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