## Sunday, June 21, 2015

### Math Trivia

6/21/2015

Next month will be the 124th Carnival of Mathematics, so let's prepare a little for the occasion.

124 = 2 * 2 * 31 (this is its prime factorization).
It has 6 divisors (1, 2, 4, 31, 62, 124), whose sum is σ = 224. Its totient is φ = 60.  The sum of its prime factors is 35 (or 33 counting only the distinct ones).  The product of its digits is 8, while the sum is 7.
124 is an amenable, composite, congruent, deficient, even, iccanobiF, nude, pernicious, polite, odious, untouchable and wasteful number.
124 is nontrivially palindromic and repdigit when written in base 5: 4445.
It is one of the 548 Lynch-Bell numbers.
It is a plaindrome in base 5, base 7, base 9, base 10, base 14 and base 16.  It is a nialpdrome in base 2, base 5, base 12, base 13 and base 15.  It is a zygodrome in base 2 and base 5.
24 is the hypotenuse of at least one Pythagorean triangle.
124 is a 3-almost prime.
124 is a Loeschian number.
124 is the nearest integer to imaginary part of 41st zero of Riemann zeta function.
124 can be expressed as the sum of 4 (but no fewer) non-zero squares.
There 2,841,940,500 partitions of 124.
124 straight slices can cut a cake into 317,875 pieces (the 124th Cake number).
124 straight cuts can divide a pizza or pancake into 7751 pieces (the 124th Lazy Caterer number).
Mordell's equation (y^2 = x^3 + n) has no integral solutions when n = 124.
124 is a Stella Octangula number.
124 is divisible by every digit in 124.
There are 124 benzenoids with 23 hexagons, C_(2v) symmetry and containing 69 carbon atoms.
124 cannot be expressed as the hypotenuse of a Pythagorean triple.
There are 124 squares and rectangles after 20 stages in the toothpick structure.  This structure contains 207 toothpicks.  At stage 124, the structure will contain 8971 toothpicks.
124 is a Belgian-2 number, a Belgian-4 number, a Belgian-5 number, and a Belgian-9 number.
124 is the sum of eight consecutive primes: 124 = 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29.
124 = 27 - 22.
124 = 52 + 52 + 52 + 72.
The string 124 occurs at position 1080, counting from the first digit after the decimal point (the “3.” is not counted).  This string occurs 199,797 times in the first 200 million digits of Pi.
124 is a palindromic and repdigit number when written in base 30: 4430.
124 is the smallest number whose sum of digits as well as the number itself are one less than the cube of a prime. [Gupta]
The sum of the 3rd to 10th prime numbers (5 + 7 + ... + 29 = 124). [Thoms]
124 = sigma(1! * 2! * 4!). Note that 124 is the smallest multi-digit number with this property. [Firoozbakht]
124 can be expressed as the difference of two squares: 124 = 322 - 302.
124 can be expressed as the sum of two prime numbers in 5 different ways: 124 = 113 + 11 = 107 + 17 = 101 + 23 = 83 + 41 = 71 + 53.
124 written as a Greek numeral is ρκδʹ.
124 written as a Chinese numeral is 一百二十四.
124 written as a Hebrew numeral is קכד.
124 written as an Arabic numeral is ١٢٤.
124 written as a Roman numeral is CXXIV.
124 written as a Maya numeral is:

▃▃▃▃▃
●●●●
124 is a 22-gonal number.
124 is the short leg of at least one Pythagorean triple:  1242 + 19202 = 19242.
The 124th pair of Amicable numbers are 13,813,150, and 14,310,050.
The 124th set of Sociable numbers are 100,805,144,361,379,855,289,068, 103,831,608,305,414,552,892,692, 106,948,916,760,522,865,797,868, and 103,831,603,731,649,764,832,532.

 The decimal expansion of the inverse of 806,451,612,903,224,193,548,387,096,775 produces a digital sequence that shows the multiples of 124, written in 15 digit strings (each term is separated by spaces to make reading easier). 1/806451612903224193548387096775 = 0. 000000000000000  000000000000124  000000000000248  000000000000372  000000000000496  000000000000620  000000000000744  000000000000868  000000000000992  000000000001116  000000000001240  000000000001364  000000000001488  000000000001612  000000000001736  000000000001860  000000000001984  000000000002108  000000000002232  000000000002356  000000000002480  000000000002604  000000000002728  000000000002852  000000000002976  000000000003100  000000000003224  000000000003348  000000000003472  000000000003596  000000000003720  000000000003844  000000000003968  000000000004092  000000000004216  000000000004340  000000000004464  000000000004588  000000000004712  000000000004836  000000000004960  000000000005084  000000000005208  000000000005332  000000000005456  000000000005580  000000000005704  000000000005828  000000000005952  000000000006076  000000000006200  000000000006324  000000000006448  000000000006572  000000000006696  000000000006820  000000000006944  000000000007068  000000000007192  000000000007316  000000000007440  000000000007564  000000000007688  000000000007812  000000000007936  000000000008060  000000000008184  000000000008308  000000000008432  000000000008556  000000000008680  000000000008804  000000000008928  000000000009052  000000000009176  000000000009300  000000000009424  000000000009548  000000000009672  000000000009796  000000000009920  000000000010044  000000000010168  000000000010292  000000000010416  000000000010540  000000000010664  000000000010788  000000000010912  000000000011036  000000000011160  000000000011284  000000000011408  000000000011532  000000000011656  000000000011780  000000000011904  000000000012028  000000000012152  000000000012276  000000000012400  000000000012524  000000000012648  000000000012772  000000000012896  000000000013020  000000000013144  000000000013268  000000000013392  000000000013516  000000000013640  000000000013764  000000000013888  000000000014012  000000000014136  000000000014260  000000000014384  000000000014508  000000000014632  000000000014756  000000000014880  000000000015004  000000000015128  000000000015252  000000000015376  000000000015500  000000000015624  000000000015748  000000000015872  000000000015996  000000000016120  000000000016244  000000000016368  000000000016492  000000000016616  000000000016740  000000000016864  000000000016988  000000000017112  000000000017236  000000000017360  000000000017484  000000000017608  000000000017732  000000000017856  000000000017980  000000000018104  000000000018228  000000000018352  000000000018476  000000000018600  000000000018724  000000000018848  000000000018972  000000000019096  000000000019220  000000000019344  000000000019468  000000000019592  000000000019716  000000000019840  000000000019964  000000000020088  000000000020212  000000000020336  000000000020460  000000000020584  000000000020708  000000000020832  000000000020956  000000000021080  000000000021204  000000000021328  000000000021452  000000000021576  000000000021700  000000000021824  000000000021948  000000000022072  000000000022196  000000000022320  000000000022444  000000000022568  000000000022692  000000000022816  000000000022940  000000000023064  000000000023188  000000000023312  000000000023436  000000000023560  000000000023684  000000000023808  000000000023932  000000000024056  000000000024180  000000000024304  000000000024428  000000000024552  000000000024676  000000000024800  000000000024924  000000000025048  000000000025172  000000000025296  000000000025420  000000000025544  000000000025668  000000000025792  000000000025916  000000000026040  000000000026164  000000000026288  000000000026412  000000000026536  000000000026660  000000000026784  000000000026908  000000000027032  000000000027156  000000000027280  000000000027404  000000000027528  000000000027652  000000000027776  000000000027900  000000000028024  000000000028148  000000000028272  000000000028396  000000000028520  000000000028644  000000000028768  000000000028892  000000000029016  000000000029140  000000000029264  000000000029388  000000000029512  000000000029636  000000000029760  000000000029884  000000000030008  000000000030132  000000000030256  000000000030380  000000000030504  000000000030628  000000000030752  000000000030876  000000000031000  000000000031124  000000000031248  000000000031372  000000000031496  000000000031620  000000000031744  000000000031868  000000000031992  000000000032116  000000000032240  …

 The digital expansion of the inverse of 999,999,999,999,999,999,999,876 produces a sequence of digits showing the powers of 124, beginning with 1240, written in 24 digit strings.  The first 25 terms are accurate. 1/999999999999999999999876 = 0. 000000000000000000000001  000000000000000000000124  000000000000000000015376  000000000000000001906624  000000000000000236421376  000000000000029316250624  000000000003635215077376  000000000450766669594624  000000055895067029733376  000006930988311686938624  000859442550649180389376  106570876280498368282637  214788658781797667047014  …

 The digital expansion of the inverse of 999,999,998,999,999,997,999,999,996 produces a digit sequence that shows the terms of a Fibonacci like sequence (Tribonacci 1,2,4): a(0) = a(1) = 0, a(2) = 1, and when n>2 then a(n) = 1*a(n-1) + 2*a(n-2) + 4*a(n-3).  The terms are separated by spaces to make reading them easier. 1/999999998999999997999999996 = 0. 000000000  000000000  000000001  000000001  000000003  000000009  000000019  000000049  000000123  000000297  000000739  000001825  000004491  000011097  000027379  000067537  000166683  000411273  001014787  002504065  006178731  015246009  037619731  092826673  229050171  …