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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

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