A337873
Numbers m such that the equation m = k*sigma(k) has more than one solution.
Original entry on oeis.org
336, 5952, 10080, 27776, 44352, 60480, 61152, 97536, 102816, 127680, 178560, 185472, 196560, 260400, 292320, 333312, 455168, 472416, 578592, 635712, 758016, 785664, 833280, 961632, 1083264, 1179360, 1189440, 1270752, 1330560, 1530816, 1717632, 1815072, 1821312, 1834560
Offset: 1
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.
For a(3): 60 * sigma(60) = 70 * sigma(70) = 10080 with 60/12 = 70/14 = 5.
a(16) = 333312 is the smallest term with 3 preimages because 336 * sigma(336) = 372 * sigma(372) = 434 * sigma(434) = 333312.
- Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 101-102.
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m = 2*10^6; v = Table[0, {m}]; Do[i = n*DivisorSigma[1, n]; If[i <= m, v[[i]]++], {n, 1, Floor@Sqrt[m]}]; Position[v, ?(# > 1 &)] // Flatten (* _Amiram Eldar, Sep 28 2020 *)
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upto(n) = {m = Map(); res = List(); n = sqrtint(n); for(i = 1, n, c = i*sigma(i); if(mapisdefined(m, c), listput(res, c); mapput(m, c, mapget(m, c) + 1) , mapput(m, c, 1); ) ); listsort(res, 1); select(x -> x <= (n+1)^2, res) } \\ David A. Corneth, Sep 27 2020
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isok(m) = {my(nb=0); fordiv(m, d, if (d*sigma(d) == m, nb++; if (nb>1, return(1)));); return (0);} \\ Michel Marcus, Sep 29 2020
A212327
Numbers k such that A001065(x)*x = k has at least two solutions.
Original entry on oeis.org
36, 62480, 141440, 1245335, 1432640, 2286080, 6680960, 7660880, 27931280, 39685376, 116636864, 179299575, 318047135, 318523136, 358491735, 533718135, 709131500, 1119849500, 1122571695, 1814416175, 2081125376, 3565970135, 3991520000, 4141021500, 4483640576
Offset: 1
For k = 36, A001065(6)*6 = 36, A001065(9)*9 = 36, therefore 36 is a term.
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q[k_] := DivisorSum[k, 1 &, # * (DivisorSigma[1, #] - #) == k &] > 1; Select[Range[23*10^5], q] (* Amiram Eldar, Jul 01 2025 *)
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isok(k) = {my(d = divisors(k, 1), c = 0); for(i = 1, #d, if(d[i][1] * (sigma(d[i][2]) - d[i][1]) == k, c++; if(c == 2, break))); c == 2;} \\ Amiram Eldar, Jul 01 2025
A338381
Smallest number m such that tau(k) * k = m has exactly n solutions when tau(k) is the number of divisors of k.
Original entry on oeis.org
1, 108, 2688, 21000000, 8400000
Offset: 1
a(1) = 1 because 1 * tau(1) = 1.
a(2) = 108 because 18 * tau(18) = 27 * tau(27) = 108 and 108 is the smallest number with 2 preimages.
a(3) = 2688 because 168 * tau(168) = 192 * tau(192) = 224 * tau(224) = 2688 and 2688 is the smallest number with 3 preimages.
a(4) = 21000000 and the corresponding 4 values of k are: 210000, 350000, 375000, 500000.
a(5) = 8400000 and the corresponding 5 values of k are: 105000, 120000, 140000, 175000, 200000. Thanks to _Amiram Eldar_ for these values of k for a(4) and a(5).
- Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B12, p. 102-103.
- D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 168, page 127.
Cf.
A212490 (similar for k*sigma(k)).
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isok(k, n) = sumdiv(k, d, d*numdiv(d) == k) == n;
a(n) = my(k=1); while (! isok(k,n), k++); k; \\ Michel Marcus, Oct 28 2020
A212373
Numbers n such that A001065(x)*x = n has at least two solutions, where x is an abundant number.
Original entry on oeis.org
141440, 2286080, 6680960, 1119849500, 2081125376, 3991520000, 4141021500, 9644638208, 23664804800, 32662630400, 37516855536, 67994319888, 577461690368, 618169892864, 627198842816, 4132702579824, 4949713492400, 5025386326400, 5579119296000, 9013476012156
Offset: 1
Example: For n=141440, A001065(340)*340 = 141440, A001065(320)*320 = 141440, A001065(340) > 340, A001065(320) > 320, therefore 141440 is included in this sequence.
A212489
Numbers n such that A001065(x)*x = n has at least two solutions, where x is a deficient number.
Original entry on oeis.org
1245335, 318047135, 358491735, 533718135, 709131500, 1122571695, 1814416175, 3565970135, 4486354631, 14336906175, 14827262351, 22805269551, 36360557831, 43971297884, 72370166375, 99254203895, 102204949847, 145262865020, 156161459559, 173741271935, 231267964895
Offset: 1
Example: For n=1245335, A001065(1955)*1955 = 1245335, A001065(2093)*2093 = 1245335, A001065(1955) < 1955, A001065(2093) < 2093, therefore 1245335 is included in this sequence.
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