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).

336, 5952, 27776, 60480, 97536, 196560, 455168, 8062976, 15713280, 97493760, 104282640, 402604032, 1597639680, 1878818816, 2959632000, 6499584000, 15923980800, 18979440480, 33281933312, 54027792000, 102953410560, 103078428672, 103448378880

Offset: 1

Author

Bernard Schott, Oct 09 2020

Keywords

• nonn

Comments

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).

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.

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)].

Examples

			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.
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.
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.
For a(6): 315 * sigma(315) = 351 * sigma(351) = 196560 is the smallest example with k and q both odd.
For a(18): 76230 * sigma(76230) = 80028 * sigma(80028) = 84942 * sigma(84942) = A337873(3266) = 18979440480.
  -> 1) for k=76230 and q=84942; with d=11^2, k/11^2=630 and q/11^2=702.
  630 * sigma(630) = 702 * sigma(702) = 1179360, hence (76230, 84942) is not a primitive solution;
  -> 2) for k=80028 and q=84942; with d=13, k/13=6156 and q/13=6534.
  6156 * sigma(6156) = 6534 * sigma(6534) = 104282640, hence (80028, 84942) is not a primitive solution; but
  -> 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).
		

References

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

Links

• Mac Tutor History of Mathematics, Leo Moser.

Crossrefs

Cf. A000203, A000668, A064987.

Subset of A337873.

Cf. A337874, A337876 (primitive solutions).

Programs

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;}
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;}
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);}
upto(10^12) \\ Michel Marcus, Oct 17 2020

Extensions

More terms from Jinyuan Wang, Oct 10 2020