OFFSET
1,1
COMMENTS
The map k -> k*sigma(k) = m is not injective (A064987), this sequence lists in increasing order the integers m that have several preimages.
These terms m satisfy A327153(m) > 1.
If 2^p-1 and 2^r-1 are distinct Mersenne primes (A000668), then k = (2^p-1)* 2^(r-1) and q = (2^r-1) * 2^(p-1) satisfy k*sigma(k) = q*sigma(q) = m = (2^p-1) * (2^r-1) * 2^(p+r-1) [see examples a(1) and a(2)].
The multiplicativity of sigma(k) ensures an infinity of solutions and thus of terms m [see example a(3)].
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 101-102.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
EXAMPLE
For a(1): 12 * sigma(12) = 14 * sigma(14) = 336 with p=2 and r=3.
For a(2): 48 * sigma(48) = 62 * sigma(62) = 5952 with p=2 and r=5.
For a(3): 60 * sigma(60) = 70 * sigma(70) = 10080 with 60/12 = 70/14 = 5.
a(16) = 333312 is the smallest term with 3 preimages because 336 * sigma(336) = 372 * sigma(372) = 434 * sigma(434) = 333312.
MATHEMATICA
m = 2*10^6; v = Table[0, {m}]; Do[i = n*DivisorSigma[1, n]; If[i <= m, v[[i]]++], {n, 1, Floor@Sqrt[m]}]; Position[v, _?(# > 1 &)] // Flatten (* Amiram Eldar, Sep 28 2020 *)
PROG
(PARI) upto(n) = {m = Map(); res = List(); n = sqrtint(n); for(i = 1, n, c = i*sigma(i); if(mapisdefined(m, c), listput(res, c); mapput(m, c, mapget(m, c) + 1) , mapput(m, c, 1); ) ); listsort(res, 1); select(x -> x <= (n+1)^2, res) } \\ David A. Corneth, Sep 27 2020
(PARI) isok(m) = {my(nb=0); fordiv(m, d, if (d*sigma(d) == m, nb++; if (nb>1, return(1))); ); return (0); } \\ Michel Marcus, Sep 29 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bernard Schott, Sep 27 2020
EXTENSIONS
More terms from David A. Corneth, Sep 27 2020
STATUS
approved