A337876 Table read by rows, in which the n-th row lists all the primitive solutions k, in increasing order, such that k*sigma(k) = A337875(n).

12, 14, 48, 62, 112, 124, 160, 189, 192, 254, 315, 351, 448, 508, 1984, 2032, 2560, 2728, 5580, 5616, 6156, 6534, 12288, 16382, 22464, 22860, 28672, 32764, 28800, 34000, 42000, 51200, 46500, 51200, 51200, 54250, 72800, 95697, 76230, 80028, 126976, 131056, 119700, 189875

Offset: 1

Author

Bernard Schott, Oct 20 2020

Keywords

• nonn
• tabf

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). This sequence lists in increasing order of m the primitive solutions (k, q).

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 first 2 examples]. Hence, there exists an infinity of primitive solutions if the sequence A000043 of Mersenne exponents is infinite.

Examples

			The table begins:
   12,   14;
   48,   62;
  112,  124;
  160,  189;
  192,  254;
  315,  351;
  ...
1st row is (12, 14) because 12 * sigma(12) = 14 * sigma(14) = 336 = A337875(1) with p = 2 and r = 3.
2nd row is (48, 62) because 48 * sigma(48) = 62 * sigma(62) = 5952 = A337875(2) with p = 2 and r = 5.
16th row is (42000, 51200), (46500, 51200), (51200, 54250) because 42000 * sigma(42000) = 51200 * sigma(51200), 46500 * sigma(46500) = 51200 * sigma(51200) and  51200 * sigma(51200) = sigma54250 * sigma(54250) = 649999584000 = A337875(16). These 3 primitive solutions corresponding to the smallest m = 649999584000 have been found by _Michel Marcus_. The three other possible solutions (42000, 46500), (42000, 54250), (46500, 54250) are not primitive.
18th row is (76230, 80028) because 76230 * sigma(76230) = 80028 * sigma(80028) = 18979440480 = A337875(18). Note that 76230 * sigma(76230) = 80028 * sigma(80028) = 84942 * sigma(84942) = 18979440480 = A337873(3266) but (76230, 84942) and (80028, 84942) are not primitive solutions (see detailed example in A337875). These case have been found by _Jinyuan Wang_.
		

References

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

Crossrefs

Cf. A000203, A000668, A064987.

Cf. A337873, A337874, A337875.

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)); Vec(findprim(res, mx));}
upto(10^11) \\ Michel Marcus, Oct 20 2020

Extensions

More terms from Michel Marcus, Oct 20 2020