Search a number
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  $10^{12}$.

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of Rhonda numbers

A graph displaying how many Rhonda numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of Rhonda numbers
Useful links Mathworld, Rhonda Number
Kevin Brown, Infinitely Many Rhondas
OEIS, Sequence A099542
OEIS, Sequence A217986
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 + 47241522 47545278 49752836 Achilles 1568 21353351168 admirable 15811444 alternating 5832 12985 + 561698541 852929216 927492925 amenable 1568 4752 + 976122541 977851641 983314513 anti-perfect 133857 apocalyptic 1568 2835 + 19435 25284 25662 arithmetic 4752 5439 + 9322158 9327135 9843255 binomial 244856171115 brilliant 54479 256973 + 761583287 819497951 858441359 c.decagonal 15354942361 c.octagonal 15625 342225 115154361 5547121441 c.square 836197345261 canyon 5439 8526 521356 congruent 5265 5439 + 9523575 9541116 9843255 cube 5832 15625 Cunningham 74511423 151225321128 Curzon 5265 8526 + 169225629 174735134 181585326 cyclic 35581 47265 + 8511529 8519579 9214557 d-powerful 5832 196355 + 5288236 8926562 9892552 de Polignac 259333 1583197 + 93727151 95542117 98655551 decagonal 2835 deficient 5265 5439 + 9523575 9843255 9892552 dig.balanced 2835 15625 + 196253928 197227345 197521734 Duffinian 12985 15625 + 8161573 8511529 8519579 economical 1568 2835 + 17312534 17544114 18552839 emirpimes 54479 7388951 + 56944751 78644543 98655551 equidigital 1568 2835 + 15548279 17312534 18552839 evil 2835 4752 + 987481215 994131595 999445823 Friedman 5832 15625 + 95232 334359 342225 frugal 15625 1456611 + 431365129 563262861 922188518 gapful 242352 681265 + 98511411271 98717511354 99218417257 happy 4752 5824 + 9158232 9213385 9322158 Harshad 4752 5439 + 9715116312 9715418961 9821117252 heptagonal 12251169 hoax 47265 56356 + 81568225 86623756 91485531 inconsummate 4752 5265 + 531729 625275 733125 interprime 2835 5439 + 75145212 76566796 77245119 junction 56718 95232 + 81335232 82528344 94265232 Lehmer 534391 683116554121 lucky 2835 19435 + 7355649 9214557 9843255 magnanimous 53176 metadrome 1568 modest 154462 259333 23757125 1272819512 1624142857 mountain 4752 5832 + 256973 456841 6754321 nialpdrome 885521 98655551 nonagonal 2455212471 172122239521 odious 1568 5265 + 976122541 983314513 984929951 palindromic 512219912215 pancake 148411964521 451113685297 pandigital 13521354 pentagonal 1451892 pernicious 1568 5265 + 9158232 9213385 9541116 plaindrome 1568 245579 2258999 power 5832 15625 342225 115154361 5547121441 powerful 1568 5832 + 115154361 5547121441 21353351168 practical 1568 4752 + 8519924 9158232 9541116 prim.abundant 154462 254624 342225 12422655 15811444 pseudoperfect 1568 2835 + 513744 521356 625275 Ruth-Aaron 224256 12367251 5224314432 self 5664 15698 + 973511325 976122541 977221215 semiprime 54479 245579 + 64551779 78644543 98655551 Smith 47265 53742 + 65238174 74152481 86148315 sphenic 15698 47594 + 58463354 63513353 97537655 square 15625 342225 115154361 5547121441 super-d 8526 12985 + 8161573 8248158 8511529 tau 5664 5824 + 745925364 821528388 925469316 triangular 244856171115 Ulam 178754 194526 + 5531169 6412952 9541116 unprimeable 15698 25284 + 9541116 9843255 9892552 untouchable 5664 5832 + 254334 512278 513744 vampire 13985257 wasteful 4752 5265 + 9541116 9843255 9892552 Zumkeller 2835 4752 + 55272 56718 95232 zygodrome 333355511122