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

1,1

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.

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.

Links

Table of n, a(n) for n=1..55.

Example

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.

Prog

(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

Crossrefs

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

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

Sequence in context: A167336 A328687 A003634 * A338385 A220100 A080910

Adjacent sequences: A338380 A338381 A338382 * A338384 A338385 A338386

Keyword

nonn,tabf

Author

Bernard Schott, Oct 26 2020

Status

approved