Integer n; n = k*d(n), d(n) is the number of divisors of n

 
 
n   is a positive integer and   d(n)   the number of divisors of   n

Let’s find the lowest   n   such that   n = k \, d(n),     k \leq 100

 
Here’s a partial list
 

k = 1   …   2 = 1 × d(2)
k = 2   …   8 = 2 × d(8)
k = 3   …   9 = 3 × d(9)
k = 4   …   36 = 4 × d(36)
k = 5   …   40 = 5 × d(40)
k = 6   …   72 = 6 × d(72)
k = 7   …   56 = 7 × d(56)
k = 8   …   80 = 8 × d(80)
k = 9   …   108 = 9 × d(108)

k = 10   …   180 = 10 × d(180)
k = 11   …   88 = 11 × d(88)
k = 12   …   240 = 12 × d(240)
k = 13   …   156 = 13 × d(156)
k = 14   …   252 = 14 × d(252)
k = 15   …   360 = 15 × d(360)
k = 16   …   128 = 16 × d(128)
k = 17   …   136 = 17 × d(136)

 
 
Complete the list.

 
 
Paul found:

There are no solutions for   k = 18, 27, 30, 45, 63, 64, 72, 99
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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2 Responses to Integer n; n = k*d(n), d(n) is the number of divisors of n

  1. paul says:

    There are a few gaps atm, but here are the rest, missing ones at

    {18, 27, 30, 45, 63, 64, 72, 99}

    136 k = 17 .. 136 = 17 x d(136)

    152 k = 19 .. 152 = 19 x d(152)
    480 k = 20 .. 480 = 20 x d(480)
    504 k = 21 .. 504 = 21 x d(504)
    396 k = 22 .. 396 = 22 x d(396)
    184 k = 23 .. 184 = 23 x d(184)
    384 k = 24 .. 384 = 24 x d(384)
    225 k = 25 .. 225 = 25 x d(225)
    468 k = 26 .. 468 = 26 x d(468)

    560 k = 28 .. 560 = 28 x d(560)
    232 k = 29 .. 232 = 29 x d(232)

    248 k = 31 .. 248 = 31 x d(248)
    448 k = 32 .. 448 = 32 x d(448)
    792 k = 33 .. 792 = 33 x d(792)
    612 k = 34 .. 612 = 34 x d(612)
    1260 k = 35 .. 1260 = 35 x d(1260)
    864 k = 36 .. 864 = 36 x d(864)
    296 k = 37 .. 296 = 37 x d(296)
    684 k = 38 .. 684 = 38 x d(684)
    936 k = 39 .. 936 = 39 x d(936)
    640 k = 40 .. 640 = 40 x d(640)
    328 k = 41 .. 328 = 41 x d(328)
    1680 k = 42 .. 1680 = 42 x d(1680)
    344 k = 43 .. 344 = 43 x d(344)
    880 k = 44 .. 880 = 44 x d(880)

    828 k = 46 .. 828 = 46 x d(828)
    376 k = 47 .. 376 = 47 x d(376)
    1152 k = 48 .. 1152 = 48 x d(1152)
    441 k = 49 .. 441 = 49 x d(441)
    1800 k = 50 .. 1800 = 50 x d(1800)
    1224 k = 51 .. 1224 = 51 x d(1224)
    1040 k = 52 .. 1040 = 52 x d(1040)
    424 k = 53 .. 424 = 53 x d(424)
    972 k = 54 .. 972 = 54 x d(972)
    1980 k = 55 .. 1980 = 55 x d(1980)
    896 k = 56 .. 896 = 56 x d(896)
    1368 k = 57 .. 1368 = 57 x d(1368)
    1044 k = 58 .. 1044 = 58 x d(1044)
    472 k = 59 .. 472 = 59 x d(472)
    1920 k = 60 .. 1920 = 60 x d(1920)
    488 k = 61 .. 488 = 61 x d(488)
    1116 k = 62 .. 1116 = 62 x d(1116)

    2340 k = 65 .. 2340 = 65 x d(2340)
    2640 k = 66 .. 2640 = 66 x d(2640)
    536 k = 67 .. 536 = 67 x d(536)
    1360 k = 68 .. 1360 = 68 x d(1360)
    1656 k = 69 .. 1656 = 69 x d(1656)
    3360 k = 70 .. 3360 = 70 x d(3360)
    568 k = 71 .. 568 = 71 x d(568)

    584 k = 73 .. 584 = 73 x d(584)
    1332 k = 74 .. 1332 = 74 x d(1332)
    2700 k = 75 .. 2700 = 75 x d(2700)
    1520 k = 76 .. 1520 = 76 x d(1520)
    2772 k = 77 .. 2772 = 77 x d(2772)
    3120 k = 78 .. 3120 = 78 x d(3120)
    632 k = 79 .. 632 = 79 x d(632)
    2240 k = 80 .. 2240 = 80 x d(2240)
    1944 k = 81 .. 1944 = 81 x d(1944)
    1476 k = 82 .. 1476 = 82 x d(1476)
    664 k = 83 .. 664 = 83 x d(664)
    2688 k = 84 .. 2688 = 84 x d(2688)
    3060 k = 85 .. 3060 = 85 x d(3060)
    1548 k = 86 .. 1548 = 86 x d(1548)
    2088 k = 87 .. 2088 = 87 x d(2088)
    1408 k = 88 .. 1408 = 88 x d(1408)
    712 k = 89 .. 712 = 89 x d(712)
    4320 k = 90 .. 4320 = 90 x d(4320)
    3276 k = 91 .. 3276 = 91 x d(3276)
    1840 k = 92 .. 1840 = 92 x d(1840)
    2232 k = 93 .. 2232 = 93 x d(2232)
    1692 k = 94 .. 1692 = 94 x d(1692)
    3420 k = 95 .. 3420 = 95 x d(3420)
    4032 k = 96 .. 4032 = 96 x d(4032)
    776 k = 97 .. 776 = 97 x d(776)
    3528 k = 98 .. 3528 = 98 x d(3528)

    2000 k = 100 .. 2000 = 100 x d(2000)

    Paul.

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