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Rhonda numbers
A number  $n$  is a base-b Rhonda number if the product of its digits, when represented in base  $b$, is equal to  $b$  times the sum of its prime factors.

For example,  $1568 =2^5\cdot 7^2$  is a base-10 Rhonda number because

\[1\cdot5\cdot6\cdot8 = 10\cdot(2+2+2+2+2+7+7).\]

Rhonda numbers exist only in composite bases. Indeed, the product of the digits of a number in a prime base  $b$ cannot be divisible by  $b$, since every digit is smaller than  $b$.

Kevin Brown (see link below) has proved that there are infinite Rhonda numbers.

The first base-10 Rhonda numbers are 1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, 15625, 15698, 19435, 25284 more terms

1000 is the smallest Rhonda number in two bases, namely 16 and 36, since  $1000=2^3\cdot 5^3$,  $1000=(3,14,8)_{16}=(27,28)_{36}$  and we have  $3\cdot14\cdot8 = 16\cdot(2\cdot3+3\cdot5)$  for base 16 and  $27\cdot28=36\cdot(2\cdot3+3\cdot5)$  for base 36.

The first smallest Rhonda numbers with respect to 1, 2,...,10 bases are 560, 1000, 10200, 5670, 63945, 158400, 322920, 140800, 1200420, 889200.

You can download a text file containing the 64507 base-10 Rhonda numbers up to  $10^{12}$.

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

ABA 1568 5832 15625 21353351168 abundant 1568 2835 4752 5664 5824 + 49752836 Achilles 1568 21353351168 admirable 15811444 alternating 5832 12985 185614 4161454 4525236 + 927492925 amenable 1568 4752 5265 5664 5824 + 983314513 anti-perfect 133857 apocalyptic 1568 2835 4752 5439 5824 + 25662 arithmetic 4752 5439 5664 5824 8526 + 9843255 binomial 244856171115 brilliant 54479 256973 285359 7388951 14746583 + 858441359 c.decagonal 15354942361 c.octagonal 15625 342225 115154361 5547121441 c.square 836197345261 congruent 5265 5439 8526 12985 25662 + 9843255 cube 5832 15625 Cunningham 74511423 151225321128 Curzon 5265 8526 15698 47265 47594 + 181585326 cyclic 35581 47265 54479 118465 172577 + 9214557 d-powerful 5832 196355 226953 242352 316285 + 9892552 de Polignac 259333 1583197 2156517 3132853 8519579 + 98655551 decagonal 2835 deficient 5265 5439 12985 15625 15698 + 9892552 dig.balanced 2835 15625 25662 34935 52374 + 197521734 Duffinian 12985 15625 19435 54479 172577 + 8519579 economical 1568 2835 5832 12985 15625 + 18552839 emirpimes 54479 7388951 18552839 55997261 56944751 + 98655551 equidigital 1568 2835 5832 12985 224256 + 18552839 evil 2835 4752 5664 5832 8526 + 999445823 Friedman 5832 15625 15698 95232 334359 342225 frugal 15625 1456611 1584117 11225871 11528125 + 922188518 gapful 242352 681265 733125 954664 1292656 + 99218417257 happy 4752 5824 8526 15625 47265 + 9322158 Harshad 4752 5439 5832 8526 25284 + 9821117252 heptagonal 12251169 hoax 47265 56356 733125 749425 1556814 + 91485531 inconsummate 4752 5265 8526 25662 53742 + 733125 interprime 2835 5439 5664 5824 25662 + 77245119 junction 56718 95232 259333 342225 392533 + 94265232 Lehmer 534391 683116554121 lucky 2835 19435 33475 45951 47265 + 9843255 magnanimous 53176 metadrome 1568 modest 154462 259333 23757125 1272819512 1624142857 nialpdrome 885521 98655551 nonagonal 2455212471 172122239521 odious 1568 5265 5439 5824 15625 + 984929951 palindromic 512219912215 pancake 148411964521 451113685297 pandigital 13521354 pentagonal 1451892 pernicious 1568 5265 5824 15625 47265 + 9541116 plaindrome 1568 245579 2258999 power 5832 15625 342225 115154361 5547121441 powerful 1568 5832 15625 342225 115154361 + 21353351168 practical 1568 4752 5664 5824 5832 + 9541116 prim.abundant 154462 254624 342225 12422655 15811444 pseudoperfect 1568 2835 4752 5664 5824 + 625275 Ruth-Aaron 224256 12367251 5224314432 self 5664 15698 25284 52374 56356 + 977221215 semiprime 54479 245579 256973 285359 2258999 + 98655551 Smith 47265 53742 133857 226953 272952 + 86148315 sphenic 15698 47594 172577 178754 196355 + 97537655 square 15625 342225 115154361 5547121441 super-d 8526 12985 33475 35581 45951 + 8511529 tau 5664 5824 342225 4525236 5826176 + 925469316 triangular 244856171115 Ulam 178754 194526 521356 668185 1378658 + 9541116 unprimeable 15698 25284 53742 55272 154462 + 9892552 untouchable 5664 5832 52374 53742 55272 + 513744 vampire 13985257 wasteful 4752 5265 5439 5664 5824 + 9892552 Zumkeller 2835 4752 5664 5824 8526 + 95232 zygodrome 333355511122