[go: up one dir, main page]
Search a number
Achilles numbers
A number which is powerful but not a perfect power, like 72 = 23 ⋅ 32.

There are infinite pairs of consecutive Achilles numbers, the smallest being 5425069447 = 73 ⋅ 412 ⋅ 972, 5425069448 = 23 ⋅ 260412.

Actually, Richard B. Stanley has proved that each integer can be expressed in infinite ways as the difference of two coprime Achilles numbers.

Every number greater that 2370 can be expressed as the sum of Achilles numbers.

The smallest 3 × 3 magic square of Achilles numbers is

9126001825200109512 23⋅33⋅52⋅132 24⋅33⋅52⋅132 23⋅34⋅132
1460169491041752192 =  25⋅33⋅132 24⋅33⋅133 27⋅34⋅132
178869673008985608 23⋅33⋅72⋅132 24⋅33⋅132 23⋅36⋅132

The first Achilles numbers are 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267 more terms

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

A graph displaying how many Achilles numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of Achilles numbers
Useful links Mathworld, Achilles Number
Wikipedia, Achilles number
OEIS, Sequence A052486
Achilles numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 72 200 288 + 9999982312712 aban 72 108 200 + 999879000192 abundant 72 108 200 + 50000000 admirable 1372 476656 570375 alternating 72 432 3456 + 981418707 amenable 72 108 200 + 999939200 apocalyptic 648 800 864 + 29768 arithmetic 432 500 864 + 9948825 binomial 29403 74024028 27665282700 + 1665678432600 c.heptagonal 512072 33035808968 2414198307851 c.square 9870125 c.triangular 78528691 compositorial 5267275776000 congruent 500 864 1125 + 9984600 Cunningham 288 675 9800 + 15061377048200 Curzon 1125 6125 36125 + 197506125 d-powerful 1323 5324 8712 + 9279432 de Polignac 704099 757687 1119371 + 97749971 decagonal 4000 469567 1026675 38637897507 deficient 675 1125 1323 + 9981512 dig.balanced 108 500 675 + 199895904 droll 72 800 6272 + 49127043891200 Duffinian 392 800 968 + 9999392 eban 2000 4000 32000 + 50000000000000 economical 1125 1152 1323 + 20000000 enlightened 23328 25000 219488 + 361881028863 equidigital 1125 1152 1323 + 19961856 eRAP 660534263 1605631112 2005604116 + 28080708128 esthetic 432 3456 evil 72 108 288 + 999581472 Friedman 2592 10368 15125 + 995328 frugal 10125 10368 10976 + 999581472 gapful 108 200 500 + 99999796428 happy 392 1125 1152 + 9940500 Harshad 72 108 200 + 9999898832 heptagonal 28252567 hex 12172437058327 hexagonal 74024028 1024060098375 hoax 16000 25000 39200 + 95520600 Hogben 7906143973 iban 72 200 1323 + 771147 idoneal 72 inconsummate 432 675 972 + 996872 interprime 72 108 288 + 99904500 Jordan-Polya 72 288 432 + 49941577728000 junction 5324 14112 16928 + 99715239 Kaprekar 5292 katadrome 72 432 864 972 Leyland 93312 20000000000 17832200896512 lucky 1323 7803 8575 + 9948123 Lynch-Bell 432 648 864 + 9176328 metadrome 1568 3456 12348 modest 238003927 272004488 918015147 + 1976030303 nialpdrome 72 200 432 + 50000000000000 nonagonal 28125 34171875 674528548509 nude 288 432 648 + 498389976 oban 500 675 800 968 octagonal 78408 128066400 752875208 + 7229107670408 odious 200 392 648 + 999939200 palindromic 36963 2138312 2177712 + 40516277261504 panconsummate 72 pandigital 108 128547 254043 + 9876523104 pentagonal 16008300 34346991901 270990840125 pernicious 72 200 288 + 9999392 persistent 1324890675 1854207963 3042571896 + 98175243600 plaindrome 288 1125 1568 + 44677778888 powerful 72 108 200 + 50000000000000 practical 72 108 200 + 9999392 prim.abundant 570375 3172468 4636684 + 99198099 pronic 72 83232 456300 + 3331356865200 pseudoperfect 72 108 200 + 998784 repunit 7906143973 Rhonda 1568 21353351168 Ruth-Aaron 9800 1193983 4929800 + 725957408000 Saint-Exupery 1620000 5467500 25312500 + 49914606307500 self 108 288 648 + 999760328 sliding 200 2000 20000 + 20000000000000 Smith 648 2888 17496 + 99630728 straight-line 432 864 3456 super Niven 200 500 800 + 45000000000 super-d 3087 10584 17496 + 9984600 tau 72 108 288 + 999045000 taxicab 28149336 225194688 760032072 + 49991146569000 tetranacci 108 triangular 29403 74024028 27665282700 + 1665678432600 trimorphic 2109375 uban 72 2000000 5000000 + 50000000000000 Ulam 72 800 864 + 9980928 unprimeable 200 1352 6728 + 9984600 untouchable 288 1944 3872 + 990584 upside-down 22128988 22817137939288 vampire 10396800 10716300 11926728 + 5894137611 wasteful 72 108 200 + 9999392 Zuckerman 432 6912 14112 + 9916826112 Zumkeller 108 432 500 + 98784 zygodrome 665500 5556600 6655000 + 44885588800000