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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).
5
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
1,1
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.
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 101-102.
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Bernard Schott, Oct 20 2020
EXTENSIONS
More terms from Michel Marcus, Oct 20 2020
STATUS
approved