|
%I #32 Mar 31 2023 09:18:51
%S 12,14,48,62,60,70,112,124,132,154,160,189,156,182,192,254,204,238,
%T 228,266,240,310,276,322,315,351,300,350,348,406,336,372,434,448,508,
%U 444,518,492,574,516,602,564,658,528,682,560,620,636,742
%N 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).
%C 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.
%C 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].
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 101-102.
%e The table begins:
%e 12, 14;
%e 48, 62;
%e 60, 70;
%e 112, 124;
%e 132, 154;
%e 160, 189;
%e ...
%e 1st row is (12, 14) because 12 * sigma(12) = 14 * sigma(14) = 336 = A337873(1) with p = 2 and r = 3.
%e 2nd row is (48, 62) because 48 * sigma(48) = 62 * sigma(62) = 5952 = A337873(2) with p = 2 and r = 5.
%e 16th row is (336, 372, 434) because 336 * sigma(336) = 372 * sigma(372) = 434 * sigma(434) = 333312 = A337873(16).
%t 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 *)
%o (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
%Y Cf. A000203, A064987, A327153.
%Y Cf. A337873, A337875, A337876.
%K nonn,tabf
%O 1,1
%A _Bernard Schott_, Oct 06 2020
|
