[go: up one dir, main page]
Search a number
apocalyptic numbers
A number of the form  $2^n$  is called apocalyptic if its digits contain "666" as a substring.

The smallest apocalyptic number is  $2^{157}$, which is equal to

\[ 182687704\underline{666}362864775460604089535377456991567872\,, \]
while  $2^{220}$ is the smallest apocalyptic number which contains two 666 groups, being equal to
1684996666696914987166688442938726917102321526408785780068975640576.
The smallest power of 2 with 3 groups is  $2^{931}$.

A number  $n$  such that  $2^n$  is apocalyptic is called apocalyptic power or apocalyptic exponent.

Between 1 and  $3\cdot10^6$  there are 3715 numbers which are non-apocalyptic exponents, the largest being 29784. In other words, it is highly probable that  $2^n$  for  $n\ge29785$  is an apocalyptic number.

Probably there are only 8 numbers, namely 2666, 3666, 5666, 6660, 6665, 6669, 11666, 26667 which contains 666 among their digits but are not apocalyptic exponents.

The first apocalyptic exponents are 157, 192, 218, 220, 222, 224, 226, 243, 245, 247, 251, 278, 285, 286, 287, 312, 355, 361, 366, 382, 384, 390, 394, 411, 434, 443, 478, 497, 499, 506 more terms

The smallest 3 × 3 magic square made of consecutive apocalyptic numbers is

149014781487
148214851488
148314921480

Below, the spiral pattern of apocalyptic numbers up to  $120^2$. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of apocalyptic numbers

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of apocalyptic exponents up to 30000
Useful links Mathworld, Apocalyptic Number
OEIS, Sequence A007356
Apocalyptic exponents up to 30000 can also be... (you may click on names or numbers and on + to get more values)

a-pointer 1933 2039 2053 + 29641 ABA 192 384 578 + 29768 aban 157 192 218 + 994 abundant 192 220 222 + 30000 Achilles 648 800 864 + 29768 admirable 222 224 366 + 29994 alt.fact. 4421 alternating 218 278 361 + 29898 amenable 157 192 220 + 30000 amicable 220 2620 2924 + 18416 anti-perfect 285 arithmetic 157 220 222 + 29999 astonishing 3078 3388 7119 + 23490 automorphic 9376 balanced p. 157 977 1103 + 29873 Bell 4140 21147 bemirp 10061 10091 16001 19001 betrothed 1925 2024 5775 + 20735 binomial 220 286 528 + 29890 brilliant 247 361 529 + 29999 c.decagonal 361 2101 2311 + 29261 c.heptagonal 841 1772 1933 + 29947 c.nonagonal 820 1081 1891 + 29890 c.octagonal 361 529 841 + 29929 c.pentagonal 226 1501 1626 + 29976 c.square 841 925 1105 + 29525 c.triangular 361 1135 1219 + 29611 cake 1160 1351 1562 + 29317 Canada 8549 16999 Carmichael 1105 2821 6601 + 29341 Carol 3967 16127 Catalan 4862 16796 Chen 157 251 443 + 29927 compositorial 192 17280 congruent 157 220 222 + 29999 constructible 192 384 1088 + 26214 cube 3375 5832 6859 + 29791 Cullen 2049 4609 10241 22529 Cunningham 224 226 528 + 29930 Curzon 245 278 285 + 29990 cyclic 157 247 251 + 29999 D-number 411 669 693 + 29919 d-powerful 224 226 994 + 29967 de Polignac 251 977 1019 + 29983 decagonal 540 1105 1701 + 29326 deceptive 2821 2981 3367 + 29341 deficient 157 218 226 + 29999 dig.balanced 226 278 312 + 29965 double fact. 384 3840 10395 droll 800 2240 5184 + 17280 Duffinian 243 245 247 + 29999 eban 2000 2002 2004 + 30000 economical 157 192 224 + 30000 emirp 157 937 983 + 19973 emirpimes 226 355 394 + 28999 enlightened 2304 2500 23328 + 25600 equidigital 157 192 224 + 30000 eRAP 2079 4233 4345 + 28518 esthetic 434 787 898 + 23456 Eulerian 247 1191 4083 + 16369 evil 192 222 226 + 29999 factorial 720 fibodiv 366 497 646 + 29998 Fibonacci 610 4181 6765 + 28657 Friedman 1285 1435 1503 + 29929 frugal 243 1280 1701 + 29929 gapful 192 220 286 + 30000 Gilda 220 660 5346 13064 Giuga 1722 good prime 251 541 929 + 29983 happy 192 226 478 + 29994 harmonic 1638 6200 8128 + 27846 Harshad 192 220 222 + 30000 heptagonal 286 540 970 + 29539 hex 1141 1261 2269 + 29701 hexagonal 861 1128 1653 + 29646 highly composite 720 840 7560 + 27720 hoax 355 361 382 + 29938 Hogben 157 871 931 + 29757 Honaker 1301 1933 2273 + 29423 house 434 1285 2234 + 28882 hyperperfect 1333 2133 3901 + 26977 iban 220 222 224 + 27777 iccanobiF 836 12815 idoneal 312 840 impolite 8192 16384 inconsummate 411 443 497 + 29995 insolite 11112 interprime 192 312 506 + 30000 Jacobsthal 10923 21845 Jordan-Polya 192 384 720 + 28800 junction 218 610 612 + 29936 Kaprekar 2223 2728 5050 + 22222 katadrome 540 541 610 + 9876 Kynea 287 1087 4223 16639 Lehmer 247 1105 1141 + 29341 Leyland 2530 4240 5392 + 23401 lonely 1340 1341 1343 + 24281 Lucas 9349 15127 24476 lucky 285 361 529 + 29995 Lynch-Bell 312 384 612 + 29736 m-pointer 1321 2131 11261 + 27211 magic 671 1105 1695 + 29679 magnanimous 245 394 443 + 28429 metadrome 157 245 247 + 26789 modest 218 222 411 + 29999 Moran 222 247 285 + 29984 Motzkin 5798 15511 narcissistic 1634 8208 9474 nialpdrome 220 222 411 + 30000 nonagonal 1639 2125 2301 + 29394 nude 222 224 312 + 29916 O'Halloran 660 924 oban 312 355 366 + 985 octagonal 1160 1281 1541 + 29800 odious 157 218 220 + 30000 Ormiston 18379 18397 19013 + 25031 palindromic 222 434 646 + 29992 palprime 787 929 10301 + 19991 pancake 497 529 667 + 29891 panconsummate 361 pandigital 714 894 898 + 29965 partition 1002 3010 3718 + 26015 pentagonal 247 287 925 + 29751 perfect 8128 pernicious 157 192 218 + 30000 Perrin 1497 2627 3480 + 24914 Pierpont 2593 3889 10369 + 18433 plaindrome 157 222 224 + 29999 Poulet 1105 2047 2701 + 29341 power 243 361 529 + 29929 powerful 243 361 529 + 29929 practical 192 220 224 + 30000 prim.abundant 222 366 582 + 29994 prime 157 251 443 + 29989 primeval 10079 10123 10136 + 13679 primorial 2310 pronic 506 650 702 + 29756 Proth 865 929 1217 + 29953 pseudoperfect 192 220 222 + 30000 repdigit 222 666 2222 + 22222 repfigit 3684 4788 7385 + 7909 repunit 157 820 871 + 29757 Rhonda 1568 2835 4752 + 25662 Ruth-Aaron 714 1682 1862 + 28810 Saint-Exupery 3840 4200 7500 + 21060 Sastry 528 6099 13224 self 222 312 411 + 29995 semiprime 218 226 247 + 29999 sliding 650 700 925 + 29000 Smith 355 382 648 + 29960 Sophie Germain 251 443 1019 + 29873 sphenic 222 285 286 + 29998 square 361 529 841 + 29929 star 541 937 1261 + 29821 straight-line 222 666 840 + 23456 strobogrammatic 1881 6009 6699 + 19861 strong prime 251 499 541 + 29983 subfactorial 14833 super Niven 220 540 660 + 30000 super-d 247 669 749 + 29981 superabundant 720 840 10080 + 27720 tau 384 564 612 + 30000 taxicab 13832 20683 tetrahedral 220 286 1140 + 29260 tetranacci 1490 2872 10671 20569 triangular 528 666 820 + 29890 tribonacci 927 5768 10609 19513 trimorphic 251 499 624 + 18751 truncatable prime 443 647 823 + 29399 twin 823 857 859 + 29881 Ulam 243 382 390 + 29987 undulating 434 646 686 + 29292 unprimeable 620 624 840 + 30000 untouchable 540 612 624 + 30000 upside-down 1199 1289 1739 + 29518 vampire 1435 1827 2187 wasteful 218 220 222 + 29999 weak prime 443 647 823 + 29989 weird 836 5830 7912 + 29470 Wieferich 3279 3511 7651 + 22953 Woodall 2047 4607 5119 + 24575 Zeisel 1885 4505 15387 + 27559 Zuckerman 224 312 384 + 28416 Zumkeller 192 220 222 + 30000 zygodrome 222 666 1133 + 22999