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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The application k -> k*tau(k) = m is not injective (A038040), this sequence proposes 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.

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

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Last modified October 26 18:30 EDT 2021. Contains 348268 sequences. (Running on oeis4.)