Tuesday, January 1, 2019

Number of the day: 2019

Happy New Year!

Properties of the number 2019:

2019 = 3 × 673 is semiprime and squarefree.
2019 has 2 distinct prime factors, 4 divisors, 7 antidivisors and 1344 totatives.
2019 has an oblong digit sum 12 in base 10.
2019 = 10102 - 10092 = 3382 - 3352 is the difference of 2 nonnegative squares in 2 ways.
2019 is the sum of 2 positive triangular numbers.
2019 is the difference of 2 positive pentagonal numbers in 1 way.
2019 = 132 + 252 + 352 is the sum of 3 positive squares.
20192 = 11552 + 16562 is the sum of 2 positive squares in 1 way.
20192 is the sum of 3 positive squares.
2019 is a proper divisor of 9296 - 1.
2019 = '201' + '9' is the concatenation of 2 semiprime numbers.
2019 is an emirpimes in (at least) the following bases: 4, 6, 8, 11, 13, 23, 25, 29, 36, 40, 42, 44, 45, 46, 47, 51, 53, 55, 57, 63, 64, 65, 67, 69, 71, 75, 87, 91, 93, and 96.
2019 is palindromic in (at least) the following bases: 19, 24, -15, and -28.
2019 in base 19 = 5b5 and consists of only the digits '5' and 'b'.
2019 in base 23 = 3ii and consists of only the digits '3' and 'i'.
2019 in base 24 = 3c3 and consists of only the digits '3' and 'c'.
2019 in base 44 = 11d and consists of only the digits '1' and 'd'.

The number 2019 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A028878: a(n) = (n+3)^2 - 6.
A037015: Numbers n with property that, reading binary expansion of n from right to left, run lengths strictly increase.
A046254: a(1) = 4; a(n) is smallest number >= a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.
A075003: Floor[ concatenation of n+1 and n divided by 2 ].
A100037: Positions of occurrences of the natural numbers as second subsequence in A100035.
A100287: First occurrence of n in A100002; the least k such that A100002(k) = n.
A112558: Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 4 multiples of n-1, n-2, ..., 1, for n>=1.
A113747: Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 10 multiples of n-1, n-2, ..., 1, for n>=1.
A127423: a(1) = 1; for n>1, a(n) = n concatenated with n-1.
A201847: T(n,k)=Number of zero-sum nXk -1..1 arrays with every element equal to at least one horizontal or vertical neighbor

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