

A338381


Smallest number m such that tau(k) * k = m has exactly n solutions when tau(k) is the number of divisors of k.


4




OFFSET

1,2


COMMENTS

The application k > k*tau(k) = m is not injective (A038040), this sequence proposes the smallest integers m that have exactly n preimages.
This sequence is not increasing with a(5) < a(4).
a(6) <= 4124832465600000, a(7) <= 33195080318400000.  David A. Corneth, Oct 28 2020
a(6) <= 1508867287200000, a(8) <= 2544150895374925200000, a(9) <= 55487699012097891000000.  Daniel Suteu, Oct 28 2020


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..5.


EXAMPLE

a(1) = 1 because 1 * tau(1) = 1.
a(2) = 108 because 18 * tau(18) = 27 * tau(27) = 108 and 108 is the smallest number with 2 preimages.
a(3) = 2688 because 168 * tau(168) = 192 * tau(192) = 224 * tau(224) = 2688 and 2688 is the smallest number with 3 preimages.
a(4) = 21000000 and the corresponding 4 values of k are: 210000, 350000, 375000, 500000.
a(5) = 8400000 and the corresponding 5 values of k are: 105000, 120000, 140000, 175000, 200000. Thanks to Amiram Eldar for these values of k for a(4) and a(5).


PROG

(PARI) isok(k, n) = sumdiv(k, d, d*numdiv(d) == k) == n;
a(n) = my(k=1); while (! isok(k, n), k++); k; \\ Michel Marcus, Oct 28 2020


CROSSREFS

Cf. A000005, A038040, A327166.
Cf. A212490 (similar for k*sigma(k)).
Sequence in context: A183357 A264681 A221018 * A269182 A244877 A223153
Adjacent sequences: A338378 A338379 A338380 * A338382 A338383 A338384


KEYWORD

nonn,more


AUTHOR

Bernard Schott, Oct 23 2020


EXTENSIONS

a(4)a(5) from Amiram Eldar, Oct 23 2020


STATUS

approved



