The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A337873 Numbers m such that the equation m = k*sigma(k) has more than one solution. 6
 336, 5952, 10080, 27776, 44352, 60480, 61152, 97536, 102816, 127680, 178560, 185472, 196560, 260400, 292320, 333312, 455168, 472416, 578592, 635712, 758016, 785664, 833280, 961632, 1083264, 1179360, 1189440, 1270752, 1330560, 1530816, 1717632, 1815072, 1821312, 1834560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The application k -> k*sigma(k) = m is not injective (A064987), this sequence proposes 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 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 Cf. A000203, A000668, A064987. Cf. A327153. Subsequence of A327165. Cf. A212490, A337874 (preimages), A337875 (primitive terms). Sequence in context: A251066 A269050 A184557 * A337875 A223446 A229697 Adjacent sequences:  A337870 A337871 A337872 * A337874 A337875 A337876 KEYWORD nonn,easy AUTHOR Bernard Schott, Sep 27 2020 EXTENSIONS More terms from David A. Corneth, Sep 27 2020 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 08:30 EST 2021. Contains 349543 sequences. (Running on oeis4.)