Every Number Has a Story to Tell:
Today’s story is about the number 12,345,654,321.
12,345,654,321 has 243 divisors, whose sum
is σ = 25,375,524,993. It is divisible
by: 1, 3, 7, 9, 11, 13, 21, 33, 37, 39, 49, 63, 77, 91, 99, 111, 117, 121, 143,
147, 169, 231, 259, 273, 333, 363, 407, 429, 441, 481, 507, 539, 637, 693, 777,
819, 847, 1001, 1089, 1183, 1221, 1287, 1369, 1443, 1521, 1573, 1617, 1813,
1859, 1911, 2331, 2541, 2849, 3003, 3367, 3549, 3663, 4107, 4329, 4477, 4719,
4851, 5291, 5439, 5577, 5733, 5929, 6253, 7007, 7623, 8281, 8547, 9009, 9583, 10101,
10647, 11011, 12321, 13013, 13431, 14157, 15059, 15873, 16317, 16731, 17787,
17797, 18759, 19943, 20449, 21021, 23569, 24843, 25641, 28749, 30303, 31339,
33033, 37037, 39039, 40293, 43771, 45177, 47619, 53361, 53391, 56277, 58201,
59829, 61347, 63063, 67081, 68783, 70707, 74529, 77077, 86247, 91091, 94017,
99099, 105413, 111111, 117117, 124579, 131313, 135531, 143143, 160173, 165649,
174603, 179487, 184041, 195767, 201243, 206349, 212121, 219373, 231231, 231361,
259259, 273273, 282051, 306397, 316239, 333333, 373737, 393939, 407407, 429429,
481481, 496947, 523809, 587301, 603729, 619047, 658119, 693693, 694083, 737891,
756613, 777777, 819819, 872053, 919191, 948717, 1002001, 1121211, 1159543,
1222221, 1288287, 1370369, 1444443, 1490841, 1619527, 1761903, 1974357,
2082249, 2153437, 2213673, 2269839, 2333331, 2544971, 2616159, 2757573,
2851849, 3006003, 3370367, 3478629, 3666663, 4111107, 4333329, 4858581,
5296291, 6460311, 6641019, 6809517, 7634913, 7848477, 8116801, 8555547,
9018009, 9592583, 10111101, 10435887, 11336689, 12333321, 14575743, 15074059,
15888873, 17814797, 19380933, 22904739, 24350403, 25666641, 27994681, 28777749,
30333303, 34010067, 37074037, 45222177, 47666619, 53444391, 73051209, 83984043,
86333247, 102030201, 105518413, 111222111, 124703579, 135666531, 160333173,
195962767, 251952129, 316555239, 333666333, 374110737, 587888301, 949665717,
1122332211, 1371739369, 1763664903, 4115218107, and 12345654321.
Its totient is φ = 5,759,994,240. The sum of its prime factors is 142 (or 71
counting only the distinct ones).
The product of its digits is 86,400, while
the sum of its digits is 36 (which is a perfect square).
The string “12345654321” is not found in
the first 200 million digits of pi.
It can be written as a sum of positive
squares in 4 ways, for example, as 51,279,921 + 12,294,374,400 = 7,1612
+ 110,8802 .
12,345,654,321 is a “hidden beast” number,
since 1 + 2 + 34 + 565 + 43 + 21 = 666.
12,345,654,321 is a Giza number because its
digits increase by one up to the middle digit, and then decrease by one to the
end – resembling the pyramids in Giza.
12,345,654,321 is an esthetic number in
base 10, because in such base it adjacent digits differ by 1.
It is a super-2 number, since 2 × 12,345,654,3212
(a number of 21 digits) contains 22 as substring.
It is an alternating number because its
digits alternate between odd and even.
It is a Wonderful Demlo number, and a Curzon
number.
It is not an unprimeable number, because it
can be changed into a prime (12,345,654,311) by changing a digit.
12,345,654,321 is a polite number, since it
can be written in 242 ways as a sum of consecutive naturals, for example, 333,666,315
+ ... + 333,666,351.
12,345,654,321 is an evil number, because
the sum of its binary digits is even.
Almost surely, 212345654321 is
an apocalyptic number.
12,345,654,321 is the 111,111st square
number. It is also a palindromic number
whose square root is also a palindromic number.
12,345,654,321 can also be expressed as the product five perfect
squares: 12,345,654,321 = 111,1112 = 32 * 72 *
112 * 132 * 372
12,345,654,321 is the 55,556-th centered
octagonal number.
12,345,654,321 is an amenable number.
12,345,654,321 is an abundant number, since
it is smaller than the sum of its proper divisors (13,029,870,672).
12,345,654,321 is a wasteful number, since
it uses less digits than its factorization.
12,345,654,321 * (1 + 2 + 3 + 4 + 5 + 6 + 5
+ 4 + 3 +2 + 1) = 444,443,555,556 = 666,6662.
But, until recently nobody knew about this
one:
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The reciprocal of 12,345,654,321 knows
the 81 times table
1/12345654321 =
0.
000000 000081 000162 000243 000324 000405 000486 000567 000648 000729 000810 000891 000972 001053 001134 001215 001296 001377 001458 001539 001620 001701 001782 001863 001944 002025 002106 002187 002268 002349 002430 002511 002592 002673 002754 002835 002916 002997 003078 003159 003240 003321 003402 003483 003564 003645 003726 003807 003888 003969 004050 004131 004212 004293 004374 004455 004536 004617 004698 004779 004860 004941 005022 005103 005184 005265 005346 005427 005508 005589 005670 005751 005832 005913 005994 006075 006156 006237 006318 006399 006480 006561 006642 006723 006804 006885 006966 007047 007128 007209 007290 007371 007452 007533 007614 007695 007776 007857 007938 008019 008100 008181 008262 008343 008424 008505 008586 008667 008748 008829 008910 008991 009072 009153 009234 009315 009396 009477 009558 009639 009720 009801 009882 009963 010044 010125 010206 010287 010368 010449 010530 010611 010692 010773 010854 010935 011016 011097 011178 011259 011340 011421 011502 011583 011664 011745 011826 011907 011988 012069 012150 012231 012312 012393 012474 012555 012636 012717 012798 012879 012960 013041 013122 013203 013284 013365 013446 013527 013608 013689 013770 013851 013932 014013 014094 014175 014256 014337 014418 014499 014580 014661 014742 014823 014904 014985 015066 015147 015228 015309 015390 015471 015552 015633 015714 015795 015876 015957 016038 016119 016200 016281 016362 016443 016524 016605 016686 016767 016848 016929 017010 017091 017172 017253 017334 017415 017496 017577 017658 017739 017820 017901 017982 018063 018144 018225 018306 018387 018468 018549 018630 018711 018792 018873 018954 019035 019116 019197 019278 019359 019440 019521 019602 019683 019764 019845 019926 020007 020088 020169 020250 020331 020412 020493 020574 020655 020736 020817 020898 020979 021060 021141 021222 021303 021384 021465 021546 021627 021708 021789 021870 021951 022032 022113 022194 022275 022356 022437 022518 022599 022680 022761 022842 022923 023004 023085 023166 023247 023328 023409 023490 023571 023652 023733 023814 023895 023976 024057 024138 024219 024300 024381 024462 024543 024624 024705 024786 024867 024948 025029 025110 025191 025272 025353 025434 025515 025596 025677 025758 025839 025920 026001 026082 026163 026244 026325 026406 026487 026568 026649 026730 026811 026892 026973 027054 027135 027216 027297 027378 027459 027540 027621 027702 027783 027864 027945 028026 028107 028188 028269 028350 028431 028512 028593 028674 028755 028836 028917 028998 029079 029160 029241 029322 029403 029484 029565 029646 029727 029808 029889 029970 030051 030132 030213 030294 030375 030456 030537 030618 030699 030780 030861 030942 031023 031104 031185 031266 031347 031428 031509 031590 031671 031752 031833 031914 031995 032076 032157 032238 032319 032400 032481 032562 032643 032724 032805 032886 032967 033048 033129 033210 033291 033372 033453 033534 033615 033696 033777 033858 033939 034020 034101 034182 034263 034344 034425 034506 034587 034668 034749 034830 034911 034992 035073 035154 035235 035316 035397 035478 035559 035640 035721 035802 035883 035964 036045 036126 036207 036288 036369 036450 036531 036612 036693 036774 036855 036936 037017 037098 037179 037260 037341 037422 037503 037584 037665 037746 037827 037908 037989 038070 038151 038232 038313 038394 038475 038556 038637 038718 038799 038880 038961 039042 039123 039204 039285 039366 039447 039528 039609 039690 039771 039852 039933 040014 040095 040176 040257 040338 040419 040500 040581 040662 040743 040824 040905 040986 041067 041148 041229 041310 041391 041472 041553 041634 041715 041796 041877 041958 042039 042120 042201 042282 042363 042444 042525 042606 042687 042768 042849 042930 043011 043092 043173 043254 043335 043416 043497 043578 043659 043740 043821 043902 043983 044064 044145 044226 044307 044388 044469 044550 044631 044712 044793 044874 044955 045036 045117 045198 045279 045360 045441 045522 045603 045684 045765 045846 045927 046008 046089 046170 046251 046332 046413 046494 046575 046656 046737 046818 046899 046980 047061 047142 047223 047304 047385 047466 047547 047628 047709 047790 047871 047952 048033 048114 048195 048276 048357 048438 048519 048600 048681 048762 048843 048924 049005 049086 049167 049248 049329 049410 049491 049572 049653 049734 049815 049896 049977 050058 050139 050220 050301 050382 050463 050544 050625 050706 050787 050868 050949 051030 051111 051192 051273 051354 051435 051516 051597 051678 051759 051840 051921 052002 052083 052164 052245 052326 052407 052488 052569 052650 ...
Terms are written in six digit strings.
Terms are accurate to the 650th non-zero
term.
This is the limit of my calculation
abilities – about 3900 digits.
However, I have reason to believe that this sequence continues up to
999,864 before making an error.
This is a
repeating fraction with a period of 222,222 (which is twice the square root
of 12,345,654,321).
This sequence is not listed in the OEIS
collection.
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Now you know things that Newton, Gauss,
Ramanujan, Hardy, Euler, and Einstein did not know. It’s kind of cool isn’t it?
David
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