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tau or refactorable numbers
R.E.Kennedy and C.N.Cooper called  $n$  a tau number if  $n$  is divisible by the number  $\tau(n)$  of its divisors, and they proved that the natural density of tau numbers is zero.

Later, S.Colton, by means of his automatic concept formation program HR, re-discovered them and called these numbers refactorable.

S.Colton proved that there are infinite tau numbers, since for every prime  $p$ the number  $p^{p-1}$  is refactorable and that all the odd tau numbers are squares and that tau numbers are congruent to 0, 1, 2 or 4  $\pmod 8$.

Moreover, Colton proved that, if  $m$  is the product of  $k$  distinct primes, then there are  $k!$  tau numbers with  $m$  divisors. For example, the tau numbers with  $30=2\cdot3\cdot5$  divisors are 720, 1200, 1620, 4050, 7500 and 11250.

It is easy to see that, if  $(m,n)=1$  and  $m$  and  $n$  are tau numbers, then  $m\cdot n$  is also a tau number.

J.Zelinsky proved in 2002 that there are no 3 consecutive tau numbers. However, it is conjectured that there are infinite pairs of consecutive tau numbers, which, as Colton proved, always contain a square. The first ones are (1, 2), (8, 9), (1520, 1521) and (50624, 50625).

Below, the spiral pattern of tau numbers up to  $250^2$. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of tau numbers

The smallest Pythagorean triple of tau numbers is (40, 96, 104). Is there a primitive such triple?

The first tau numbers are 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225 more terms

Tau numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 18 24 72 + 999313218 aban 12 18 24 + 1000000000 abundant 12 18 24 + 49999968 Achilles 72 108 288 + 999045000 admirable 12 24 40 + 45532800 alternating 12 18 36 + 949810761 amenable 12 24 36 + 1000000000 amicable 9363584 9437056 25596544 + 967887488 apocalyptic 384 564 612 + 30000 arithmetic 56 60 96 + 9999992 astonishing 204 2132532 automorphic 376 625 Bell 4140 betrothed 8892 9504 199760 + 313472880 binomial 36 56 84 + 999246160 c.heptagonal 3928 8576 13672 + 993315968 c.nonagonal 136 12880 32896 + 999246160 c.octagonal 225 441 625 + 998117649 c.pentagonal 276 856 2176 + 999750016 c.triangular 136 4456 8104 + 999814960 cake 232 2952 4992 + 944720768 Catalan 132 compositorial 24 17280 696729600 congruent 24 56 60 + 9999992 constructible 12 24 40 + 858980352 cube 21952 32768 64000 + 1000000000 Cunningham 24 80 288 + 999128880 Curzon 18 441 1089 + 195692121 d-powerful 24 132 372 + 9995424 de Polignac 40401 62001 660969 + 87665769 decagonal 232 2232 14220 + 997501680 deficient 128 136 152 + 9999992 dig.balanced 12 56 108 + 199999784 double fact. 384 645120 10321920 droll 72 240 672 + 995328000 Duffinian 36 128 225 + 9903609 eban 36 40 56 + 66064044 economical 128 384 448 + 19999872 enlightened 2176 2560 2370816 + 250000000 equidigital 384 448 640 + 19999360 eRAP 24 4716 10620 + 998593920 esthetic 12 56 232 + 898989876 Eulerian 524268 2203488 8388584 evil 12 18 24 + 999999920 factorial 24 720 5040 + 479001600 fibodiv 488 7288 10408 + 984380344 Fibonacci 46368 Friedman 128 625 1260 + 995364 frugal 128 625 1250 + 999981056 gapful 108 132 180 + 1000000000 Gilda 152 880 4484776 happy 376 536 632 + 10000000 harmonic 672 30240 23569920 + 714954240 Harshad 12 18 24 + 999999648 heptagonal 18 3744 5688 + 996772608 hexagonal 276 2016 2556 + 996342480 highly composite 12 24 36 + 735134400 hoax 84 136 424 + 99995120 house 25884 56560 146312 + 958696256 iban 12 24 40 + 777704 idoneal 12 18 24 + 240 impolite 128 32768 inconsummate 84 276 372 + 999896 interprime 12 18 56 + 99999600 Jordan-Polya 12 24 36 + 995328000 junction 204 612 808 + 99999760 Kaprekar 5292 7272 82656 + 11111112 katadrome 40 60 72 + 987654320 Leyland 4240 20412 268436240 lonely 1344 31432 47326800 Lucas 18 lucky 4761 12321 15129 + 9941409 Lynch-Bell 12 24 36 + 9781632 magic 2056 6924 2048080 + 702464560 magnanimous 12 56 136 + 995955112 metadrome 12 18 24 + 1234568 modest 444 824 872 + 998844444 Moran 18 84 152 + 99960012 Motzkin 310572 18199284 narcissistic 93084 nialpdrome 40 60 72 + 1000000000 nonagonal 24 204 396 + 991564704 nude 12 24 36 + 499999968 O'Halloran 12 36 60 + 204 oban 12 18 36 + 996 octagonal 40 96 225 + 998713056 odious 56 84 88 + 1000000000 palindromic 88 232 252 + 889949988 pancake 56 232 904 + 999469696 panconsummate 12 18 24 + 72 pandigital 108 156 180 + 381367044 partition 56 792 5604 + 541946240 pentagonal 12 376 852 + 995353520 pernicious 12 18 24 + 9999992 Perrin 12 plaindrome 12 18 24 + 777788888 power 36 128 225 + 1000000000 powerful 36 72 108 + 1000000000 practical 12 18 24 + 10000000 prim.abundant 12 18 56 + 99956096 pronic 12 56 72 + 999539840 Proth 225 1089 16641 + 67125249 pseudoperfect 12 18 24 + 999920 repdigit 88 444 444444444 repfigit 2208 3684 4788 298320 repunit 40 156 3280 + 943531280 Rhonda 5664 5824 342225 + 925469316 Ruth-Aaron 24 104 492 + 997962144 Saint-Exupery 60 480 1620 + 994882500 Sastry 328 110213248 333052608 self 108 132 288 + 999999544 self-describing 10313316 14212332 14313312 + 33322424 sliding 2000 2504 25040 + 801250000 Smith 636 852 1284 + 99999920 square 36 225 441 + 998117649 straight-line 444 468 852 + 444444444 strobogrammatic 88 96 808 + 969808696 super Niven 12 24 36 + 1000000000 super-d 348 1056 1068 + 9999072 superabundant 12 24 36 + 735134400 taxicab 110656 314496 513856 + 988048000 tetrahedral 56 84 560 + 986652720 tetranacci 56 108 2872 39648 triangular 36 136 276 + 999246160 tribonacci 24 504 66012 410744 trimorphic 24 376 625 + 2890624 uban 12 18 36 + 1000000000 Ulam 18 36 72 + 10000236 undulating 232 252 424 + 696969696 unprimeable 204 328 516 + 10000000 untouchable 88 96 248 + 999928 upside-down 852 2648 4376 + 899357112 vampire 1260 120600 156240 + 96399072 wasteful 12 18 24 + 9999992 Woodall 80 1948616 Zuckerman 12 24 36 + 996292224 Zumkeller 12 24 40 + 99968 zygodrome 88 444 22000 + 999992244