

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
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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. 102103.
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



