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
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
KEYWORD
nonn
AUTHOR
Bernard Schott, Oct 23 2020
EXTENSIONS
More terms from Amiram Eldar, Oct 23 2020
STATUS
approved