ABA numbers

A number that can be expressed as $A\cdot B^A$ , for $A,B>1$

For example, 648 is the first ABA number with two representations, $2\cdot18^2$ and $3\cdot6^3$ .

According to Robert Munafo, the smallest number with 3 such representations is 344373768, which is equal to $8\cdot9^8$ , $3\cdot486^3$ and $2\cdot13122^2$ .

It is not difficult to build larger examples of numbers with 3 representations, but they actually tend to be quite larger. For example,

$30233088000000=2\cdot3888000^2=3\cdot21600^3=5\cdot360^5\,,$

$198077054103584768=2\cdot314703872^2=7\cdot224^7=8\cdot112^8\,,$

or, for an odd example,

$3\cdot170445385984785854816436767578125^3=$

$=5\cdot19705225067138671875^5=81\cdot15^{81}.$

In general, it is possible to construct a number $n$ with an arbitrary number of representations, of the form

$n = p_1 \cdot {a_1}^{p_1} = p_2\cdot {a_2}^{p_2}=\cdots =p_k\cdot {a_k}^{p_k}\,,$

where the

are the prime numbers 2, 3, 5,.... and clearly $a_k = (n/p_k)^{1/p_k}$ . Expressing $n$

as $p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ , it is possible to find the exponents $e_i$

's, and thus $n$

, by solving the system of $2\cdot k$ modular equations

$\left\{\begin{array}{l} e_i \equiv 1 \pmod {p_i}\,,\\ e_i \equiv 0 \pmod {P_k/p_i}\,, \end{array}\right.$

where

is the product of the primes from $p_1$

and

ranges from $i$

For example, a number of 255 digits with 4 such representations is $2^{105}\cdot3^{70}\cdot5^{126}\cdot7^{120}$ and one of 2549 digits with 5 representations is $2^{1155}\cdot3^{1540}\cdot5^{1386}\cdot7^{330}\cdot11^{210}$ .

The smallest Pythagorean triples made of ABA numbers are (98304, 131072, 163840) and (229376, 786432, 819200) which correspond to $(3\cdot 32^3,\, 2\cdot 256^2,\, 5\cdot 8^5)$ and $(14\cdot 2^{14},\, 3\cdot64^3,\, 2\cdot 640^2)$ .

The first ABA numbers are 8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200 more terms

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

A graph displaying how many ABA numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p. Among the first 2252741 terms, 2245081 are even.

Useful links R. Munafo, Generalized Cullen and Woodall Numbers
OEIS, Sequence A171607
OEIS, Sequence A171606 (ABA in two or more ways)

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

aban 18 24 32 + 999879000192 abundant 18 24 72 + 50000000 Achilles 72 200 288 + 9999982312712 admirable 24 alternating 18 32 50 + 896761250 amenable 24 32 64 + 999939200 apocalyptic 192 384 578 + 29768 arithmetic 375 1029 1215 + 9923847 c.heptagonal 512072 33035808968 c.octagonal 81 15625 59049 + 8741371209561 c.triangular 64 cake 64 2048 canyon 324 512 648 + 9876541234568 compositorial 24 192 congruent 24 375 384 + 9961472 constructible 24 32 64 + 8796093022208 cube 64 512 5832 + 9552418372232 Cunningham 24 50 242 + 2584123765442 Curzon 18 50 81 + 199880018 d-powerful 24 375 2048 + 9954722 de Polignac 1856465 deficient 32 50 64 + 9981512 dig.balanced 50 384 450 + 199520288 double fact. 384 droll 72 800 5184 + 9748267139072 Duffinian 32 50 64 + 9999392 eban 32 50 64 + 6000000000000 economical 32 64 81 + 19983842 enlightened 2048 2500 23328 + 361881028863 equidigital 32 64 81 + 19983842 eRAP 24 98 171014018 + 28080708128 esthetic 32 98 evil 18 24 72 + 999670898 factorial 24 Friedman 128 1024 2048 + 968832 frugal 128 512 1024 + 999670898 gapful 160 192 200 + 100000000000 happy 32 192 338 + 9910152 Harshad 18 24 50 + 9996980000 heptagonal 18 81 1877922 194706720450 highly composite 24 hoax 160 2500 3042 + 98140050 house 32 iban 24 72 200 + 744200 idoneal 18 24 72 impolite 32 64 128 + 8796093022208 inconsummate 882 1536 2592 + 999698 interprime 18 50 64 + 99658962 Jordan-Polya 24 32 64 + 9906778275840 junction 1024 1215 1922 + 99828450 katadrome 32 50 64 + 98 Leyland 32 512 93312 + 20000000000 Lucas 18 lucky 1029 2187 10125 + 7381125 Lynch-Bell 24 128 162 + 9176328 magnanimous 32 50 98 + 512 metadrome 18 24 128 + 1568 modest 164738 624962 10224242 + 1734028611 Moran 18 mountain 160 162 192 + 13478432 nialpdrome 32 50 64 + 8820000000000 nonagonal 24 34171875 nude 24 128 162 + 496881288 oban 18 50 98 + 968 octagonal 78408 752875208 7229107670408 odious 32 50 64 + 999939200 palindromic 242 3993 20402 + 8163248423618 panconsummate 18 24 72 81 pandigital 722 1922 6002500 + 9743521608 pernicious 18 24 50 + 9999392 persistent 3047618592 4163098752 7025391648 + 96024519378 plaindrome 18 24 128 + 5555557777778 power 32 64 81 + 9986219369604 powerful 32 64 72 + 9999982312712 practical 18 24 32 + 9999392 prim.abundant 18 pronic 72 2450 83232 + 3765344262050 Proth 81 pseudoperfect 18 24 72 + 994050 Rhonda 1568 5832 15625 21353351168 Ruth-Aaron 24 50 1682 + 981778806450 Saint-Exupery 202500 1620000 9447840 + 8138191680000 self 64 288 512 + 999849762 sliding 200 2500 20000 + 2000000000000 Smith 648 2888 9522 + 99630728 square 64 81 324 + 9986219369604 super Niven 24 50 200 + 45000000000 super-d 81 338 1058 + 9981512 superabundant 24 tau 18 24 72 + 999313218 taxicab 760032072 48642052608 399929172552 + 6398866760832 tribonacci 24 81 trimorphic 24 375 uban 18 32 50 + 8056098000000 Ulam 18 72 324 + 9990450 undulating 242 unprimeable 200 324 512 + 9981512 untouchable 162 288 324 + 985608 upside-down 64 2738 8192 151959 vampire 2187 378450 10396800 + 4968449928 wasteful 18 24 50 + 9999392 Zuckerman 24 128 384 + 9916826112 Zumkeller 24 160 192 + 98304 zygodrome 447722888 11000004488 22244466888 + 8877886668800