A337874 Table read by rows, in which the n-th row lists all the preimages k, in increasing order, such that k*sigma(k) = A337873(n).

12, 14, 48, 62, 60, 70, 112, 124, 132, 154, 160, 189, 156, 182, 192, 254, 204, 238, 228, 266, 240, 310, 276, 322, 315, 351, 300, 350, 348, 406, 336, 372, 434, 448, 508, 444, 518, 492, 574, 516, 602, 564, 658, 528, 682, 560, 620, 636, 742

Offset: 1

Author

Bernard Schott, Oct 06 2020

Keywords

• nonn
• tabf

Comments

The map k -> k*sigma(k) = m is not injective (A064987) and this sequence lists, in increasing order of m, the preimages of the integers m that have more than one preimage.

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 first 2 examples].

Examples

			The table begins:
   12,  14;
   48,  62;
   60,  70;
  112, 124;
  132, 154;
  160, 189;
  ...
1st row is (12, 14) because 12 * sigma(12) = 14 * sigma(14) = 336 = A337873(1) with p = 2 and r = 3.
2nd row is (48, 62) because 48 * sigma(48) = 62 * sigma(62) = 5952 = A337873(2) with p = 2 and r = 5.
16th row is (336, 372, 434) because 336 * sigma(336) = 372 * sigma(372) = 434 * sigma(434) = 333312 = A337873(16).
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 101-102.

Crossrefs

Cf. A000203, A064987, A327153.

Cf. A337873, A337875, A337876.

Programs

Mathematica

m = 10^6; v = Table[{}, {m}]; Do[i = n*DivisorSigma[1, n]; If[i <= m, AppendTo[v[[i]], n]], {n, 1, Floor@Sqrt[m]}]; Select[v, Length[#] > 1 &] // Flatten (* Amiram Eldar, Oct 06 2020 *)

PARI

upto(n) = {m = Map(); res = List(); n = sqrtint(n); w = []; for(i = 1, n, c = i*sigma(i); if(mapisdefined(m, c), listput(res, c); l = mapget(m, c); listput(l, i); mapput(m, c, l) , mapput(m, c, List(i)); ) ); listsort(res, 1); v = select(x -> x <= (n+1)^2, res); for(i = 1, #v, w = concat(w, Vec(mapget(m, v[i]))) ); w } \\ David A. Corneth, Oct 07 2020