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A337875 Integers that can be written m = k*sigma(k) = q*sigma(q) where (k, q) is a primitive solution of this equation and sigma(m) is the sum of divisors of (m).

%I A337875 #54 Dec 08 2020 02:36:51
%S A337875 336,5952,27776,60480,97536,196560,455168,8062976,15713280,97493760,
%T A337875 104282640,402604032,1597639680,1878818816,2959632000,6499584000,
%U A337875 15923980800,18979440480,33281933312,54027792000,102953410560,103078428672,103448378880
%N A337875 Integers that can be written m = k*sigma(k) = q*sigma(q) where (k, q) is a primitive solution of this equation and sigma(m) is the sum of divisors of (m).
%C A337875 As the multiplicativity of sigma(k) ensures an infinity of solutions to the general equation m = k*sigma(k) (see A337873), Leo Moser asked if k*sigma(k) = q*sigma(q) has an infinity of primitive solutions, in the sense that (k', q') is not a solution for any k' = k/d, q' = q/d, d>1 (see References and 3rd example).
%C A337875 A subset of primitive solutions: 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)]. Hence, there exists an infinity of primitive solutions if the sequence A000043 of Mersenne exponents is infinite.
%C A337875 There exist terms m in A337873 that have three solutions like A337873(16) = 333312 = 336 * sigma(336) = 372 * sigma(372) = 434 * sigma(434) whose solutions (336,372), (336,434) and (372,434) are not primitive, but _Jinyuan Wang_ has found some terms m in A337873 with 3 preimages as A337873(3266) = 18979440480 from which one pair is primitive and the two others not primitive [see example a(18)].
%D A337875 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 101-102.
%H A337875 Mac Tutor History of Mathematics, <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Moser_Leo/">Leo Moser</a>.
%e A337875 For a(1): 12 * sigma(12) = 14 * sigma(14) = 336 with p=2 and r=3.
%e A337875 For a(2): 48 * sigma(48) = 62 * sigma(62) = 5952 with p=2 and r=5.
%e A337875 10080 is not a term: 60 * sigma(60) = 70 * sigma(70) = 10080 but as 60/5 = 12 and 70/5 = 14, hence, this solution that is generated by the first example is not primitive.
%e A337875 For a(4): 160 * sigma(160) = 189 * sigma(189) = 60480 is the smallest example with gcd(k,q) = 1 with k = 2^5*5 = 160 and q = 3^3*7 = 189.
%e A337875 For a(6): 315 * sigma(315) = 351 * sigma(351) = 196560 is the smallest example with k and q both odd.
%e A337875 For a(18): 76230 * sigma(76230) = 80028 * sigma(80028) = 84942 * sigma(84942) = A337873(3266) = 18979440480.
%e A337875   -> 1) for k=76230 and q=84942; with d=11^2, k/11^2=630 and q/11^2=702.
%e A337875   630 * sigma(630) = 702 * sigma(702) = 1179360, hence (76230, 84942) is not a primitive solution;
%e A337875   -> 2) for k=80028 and q=84942; with d=13, k/13=6156 and q/13=6534.
%e A337875   6156 * sigma(6156) = 6534 * sigma(6534) = 104282640, hence (80028, 84942) is not a primitive solution; but
%e A337875   -> 3) for k=76230 and q=80028, there is no common divisor d such that k/d and q/d can satisfy (k/d)*sigma(k/d) = (q/d)*sigma(q/d), so (76239, 80028) is a primitive solution linked to m = 18979440480 that is the term a(18).
%o A337875 (PARI) process(x, y, resp) = {my(vresp = Vec(resp)); for (i=1, #vresp, if (x/vresp[i][1] == y/vresp[i][2], return(resp));); listput(resp, [x, y]); resp;}
%o A337875 findprim(res, mx) = {my(mp = Map()); my(resp = List()); for (i=1, #res, my(vx = mapget(mx, res[i])); for (j=1, #vx-1, for (k=j+1, #vx, resp = process(vx[j], vx[k], resp);););); resp;}
%o A337875 upto(n) = {my(m = Map(), mx = Map(), res = List(), n = sqrtint(n), resp);for(i = 1, n, my(c = i*sigma(i)); if(mapisdefined(m, c), listput(res, c); mapput(m, c, mapget(m, c) + 1); mapput(mx, c, concat(mapget(mx, c), i)), mapput(m, c, 1); mapput(mx, c, [i]);)); listsort(res, 1); res = Vec(select(x -> x <= (n+1)^2, res)); resp = findprim(res, mx); vresp = Vec(resp); vecsort(vector(#vresp, k, vresp[k][1]*sigma(vresp[k][1])),,8);}
%o A337875 upto(10^12) \\ _Michel Marcus_, Oct 17 2020
%Y A337875 Cf. A000203, A000668, A064987.
%Y A337875 Subset of A337873.
%Y A337875 Cf. A337874, A337876 (primitive solutions).
%K A337875 nonn
%O A337875 1,1
%A A337875 _Bernard Schott_, Oct 09 2020
%E A337875 More terms from _Jinyuan Wang_, Oct 10 2020