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A337875
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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).
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4
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336, 5952, 27776, 60480, 97536, 196560, 455168, 8062976, 15713280, 97493760, 104282640, 402604032, 1597639680, 1878818816, 2959632000, 6499584000, 15923980800, 18979440480, 33281933312, 54027792000, 102953410560, 103078428672, 103448378880
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OFFSET
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1,1
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COMMENTS
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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)].
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 101-102.
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LINKS
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EXAMPLE
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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).
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PROG
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(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); }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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