Thursday, October 22, 2015

Every Number Has a Story to Tell - 12,345,654,321

10/22/2015

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.