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.

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

1,1

Comments

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, ...

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..42.

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A000005, A038040, A327166, A338381, A338383, A338384, A338385.

Cf. A337873 (similar for k*sigma(k)).

Subsequence of A036438.

Sequence in context: A245032 A208088 A323548 * A338384 A344702 A044340

Adjacent sequences: A338379 A338380 A338381 * A338383 A338384 A338385

Keyword

nonn

Author

Bernard Schott, Oct 23 2020

Extensions

More terms from Amiram Eldar, Oct 23 2020

Status

approved