OFFSET
1,2
COMMENTS
The map k -> k*tau(k) = m is not injective (A038040), this sequence lists 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. 102-103.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 168, page 127.
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
KEYWORD
nonn,more
AUTHOR
Bernard Schott, Oct 23 2020
EXTENSIONS
a(4)-a(5) from Amiram Eldar, Oct 23 2020
STATUS
approved