## 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

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.