A338384 -id:A338384 - cp's OEIS frontend

A338385 Table read by rows, in which the n-th row lists the primitive solutions (k, q), kA338384(n).

18, 27, 24, 32, 56, 64, 192, 224, 400, 500, 360, 432, 540, 648, 972, 2187, 1875, 3125, 1458, 1701, 1296, 1350, 2160, 2400, 5120, 5632, 2880, 3024, 3840, 4608, 4032, 4704, 3780, 5184, 10240, 16384, 8448, 9216, 20000, 25000, 15680, 16464, 15876, 25515, 20412, 23814

Offset: 1

Bernard Schott, Nov 09 2020

As the multiplicativity of tau(k) ensures an infinity of solutions to the general equation k*tau(k) = q*tau(q) (see A338382), Richard K. Guy asked if there is an infinity of primitive solutions. A solution (k, q) with m = k*tau(k) = q*tau(q) is primitive in the sense that (k', q') is not a solution for any k' = k/d, q' = q/d, m' = m/(d*tau(d)), d>1 with m' = k' * tau(k') = q' * tau(q').

Warning, Richard K. Guy asked if "there is an infinity of primitive solutions (for k*tau(k) = q*tau(q)), in the sense that (k', q') is not a solution for any k' = k/d, q' = q/d, d>1". It appears that this definition is not enough well defined, because some solutions as (4032, 4704), (20000, 25000), (20412, 23814),... that are primitive are not obtained in this case (see detailed example (20000, 25000) below). The mathematical explanation is that tau satisfies the relation tau(r*s) = tau(r) * tau(s) * (t/tau(t)) where t = gcd(r,s).

			The table begins:
    18,  27;
    24,  32;
    56,  64;
   192, 224;
   400, 500;
   360, 432;
   ...
1st row is (18, 27) because 18 * tau(18) = 27 * tau(27) = 108 = A338384(1).
4th row is (192, 224) because 192 * tau(192) = 224 * tau(24) = 2688 = A338384(4); Note that 168 * tau(168) = 192 * tau(192) = 224 * tau(24) = 2688 = A338382(8) but (168, 192) and (168, 224) are not primitive solutions (see detailed example in A338384).
5th row is (400, 500) because 400 * tau(400) = 500 * tau(500) = 6000.
20th row is (20000, 25000) although (20000/50, 25000/50) = (400, 500) and that (400, 500) is the 5th row. Explanation: A338384(20) = 600000 = 20000*tau(20000) = 25000*tau(25000) and this pair is primitive, because for d = 50, we get 600000/(50*tau(50)) = 2000 <>  (20000/50)*tau(20000/50) = (25000/50)*tau(25000/50) = 6000. To be exhaustive, the two other pairs linked with 600000: (15000, 20000) and (15000, 25000) are not primitive.

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.

Cf. A000005, A038040, A338381, A338382, A338383, A338384.

Cf. A337876 (similar for k*sigma(k)).

A338382 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.

Original entry on oeis.org

108, 192, 448, 1080, 1512, 1920, 2376, 2688, 2808, 3672, 4104, 4224, 4480, 4968, 4992, 6000, 6264, 6528, 6696, 7296, 7992, 8100, 8640, 8832, 8856, 9288, 9856, 10152, 11136, 11448, 11648, 11904, 12096, 12744, 12960, 13176, 14208, 14400, 14472, 15120, 15232, 15336

Offset: 1

Bernard Schott, Oct 23 2020

nonn

The map k -> k*tau(k) = m is not injective (A038040), this sequence lists in increasing order the integers m that have several preimages.

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, ...

			a(1) = 108 because 18 * tau(18) = 27 * tau(27) = 108.
a(2) = 192 because 24 * tau(24) = 32 * tau(32) = 192.
a(3) = 448 because 56 * tau(56) = 64 * tau(64) = 448.
a(8) = 2688 is the smallest term with 3 preimages because 168 * tau(168) = 192 * tau(192) = 224 * tau(224) = 2688.

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.

Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.

Cf. A000005, A038040, A327166, A338381, A338383, A338384, A338385.
Cf. A337873 (similar for k*sigma(k)).
Subsequence of A036438.

solNum[n_] := DivisorSum[n, 1 &, # * DivisorSigma[0, #] == n &]; Select[Range[16000], solNum[#] > 1 &] (* Amiram Eldar, Oct 23 2020 *)

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

More terms from Amiram Eldar, Oct 23 2020

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).

Original entry on oeis.org

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

Bernard Schott, Oct 26 2020

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.

			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.

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.

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

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

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

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A338385 Table read by rows, in which the n-th row lists the primitive solutions (k, q), kA338384(n).

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A338382 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.

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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).

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Author

Keywords

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Examples

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Programs

PARI