Wednesday, September 20, 2017

Number of the day: 129

Properties of the number 129:

129 = 3 × 43 is semiprime and squarefree.
129 has 2 distinct prime factors, 4 divisors, 5 antidivisors and 84 totatives.
129 has an oblong digit sum 12 in base 10.
Reversing the decimal digits of 129 results in an emirpimes.
129 = 13 + 43 + 43 is the sum of 3 positive cubes in 1 way.
129 = 652 - 642 = 232 - 202 is the difference of 2 nonnegative squares in 2 ways.
129 is the difference of 2 positive pentagonal numbers in 1 way.
129 = 22 + 52 + 102 is the sum of 3 positive squares.
1292 is the sum of 3 positive squares.
129 is a proper divisor of 1732 - 1.
129 is an emirpimes in (at least) the following bases: 3, 5, 7, 9, 10, 12, 14, 15, 17, 19, 21, 23, 24, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 45, 51, 55, 60, 69, 71, 72, 78, 80, 84, 89, 91, 93, 94, 98, and 100.
129 is palindromic in (at least) the following bases: 2, 6, 42, -7, -16, -32, and -64.
129 in base 6 = 333 and consists of only the digit '3'.

The number 129 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000051: a(n) = 2^n + 1.
A000930: Narayana's cows sequence: a(0) = a(1) = a(2) = 1; thereafter a(n) = a(n-1) + a(n-3).
A000959: Lucky numbers.
A001358: Semiprimes (or biprimes): products of two primes.
A001969: Evil numbers: numbers with an even number of 1's in their binary expansion.
A005408: The odd numbers: a(n) = 2*n + 1.
A006881: Squarefree semiprimes: Numbers that are the product of two distinct primes.
A007504: Sum of first n primes.
A008585: a(n) = 3*n.
A016813: a(n) = 4n + 1.

Tuesday, September 19, 2017

Number of the day: 962716

Ky Fan was born on this day 103 years ago.

Properties of the number 962716:

962716 = 22 × 229 × 1051 is the 886891th composite number and is not squarefree.
962716 has 3 distinct prime factors, 12 divisors, 13 antidivisors and 478800 totatives.
962716 has an emirp digit sum 31 in base 10.
962716 has sum of divisors equal to 1693720 which is a triangular number.
Reversing the decimal digits of 962716 results in a prime.
962716 = 2406802 - 2406782 = 12802 - 8222 is the difference of 2 nonnegative squares in 2 ways.
962716 is the difference of 2 positive pentagonal numbers in 2 ways.
962716 is not the sum of 3 positive squares.
9627162 = 2522402 + 9290842 is the sum of 2 positive squares in 1 way.
9627162 is the sum of 3 positive squares.
962716 is a proper divisor of 137370 - 1.

Monday, September 18, 2017

Number of the day: 19499565

Adrien-Marie Legendre was born on this day 265 years ago.

Properties of the number 19499565:

19499565 = 3 × 5 × 61 × 101 × 211 is the 18258760th composite number and is squarefree.
19499565 has 5 distinct prime factors, 32 divisors, 35 antidivisors and 10080000 totatives.
19499565 is the difference of 2 nonnegative squares in 16 ways.
19499565 is the difference of 2 positive pentagonal numbers in 7 ways.
19499565 = 1002 + 2772 + 44062 is the sum of 3 positive squares.
194995652 = 83790212 + 176075282 = 50665322 + 188298512 = 114167882 + 158079092 = 38613002 + 191134352 = 3513152 + 194964002 = 72446852 + 181038002 = 129739682 + 145571012 = 101362292 + 166580282 = 119788922 + 153863312 = 116997392 + 155996522 = 86948882 + 174537092 = 132341312 + 143209922 = 35163152 + 191799002 is the sum of 2 positive squares in 13 ways.
194995652 is the sum of 3 positive squares.
19499565 is a proper divisor of 10760 - 1.
19499565 = '194' + '99565' is the concatenation of 2 semiprime numbers.

Sunday, September 17, 2017

Number of the day: 85501

Bernhard Riemann was born on this day 191 years ago.

Properties of the number 85501:

85501 is a cyclic number.
85501 = 13 × 6577 is semiprime and squarefree.
85501 has 2 distinct prime factors, 4 divisors, 19 antidivisors and 78912 totatives.
85501 has a prime digit sum 19 in base 10.
Reversing the decimal digits of 85501 results in an emirpimes.
85501 = 427512 - 427502 = 32952 - 32822 is the difference of 2 nonnegative squares in 2 ways.
85501 is the sum of 2 positive triangular numbers.
85501 is the difference of 2 positive pentagonal numbers in 2 ways.
85501 = 1742 + 2352 = 1502 + 2512 is the sum of 2 positive squares in 2 ways.
85501 = 212 + 462 + 2882 is the sum of 3 positive squares.
855012 = 249492 + 817802 = 84242 + 850852 = 405012 + 753002 = 328852 + 789242 is the sum of 2 positive squares in 4 ways.
855012 is the sum of 3 positive squares.
85501 is a proper divisor of 35312 - 1.
85501 = '85' + '501' is the concatenation of 2 semiprime numbers.
85501 is an emirpimes in (at least) the following bases: 2, 8, 9, 10, 13, 21, 27, 28, 31, 35, 36, 37, 38, 39, 40, 42, 44, 45, 49, 53, 54, 58, 66, 71, 78, 80, 81, 83, 85, 86, 87, 88, 89, and 97.
85501 is palindromic in (at least) the following bases: -14, and -71.
85501 in base 14 = 23233 and consists of only the digits '2' and '3'.

Saturday, September 16, 2017

Number of the day: 11292

Properties of the number 11292:

11292 = 22 × 3 × 941 is the 9926th composite number and is not squarefree.
11292 has 3 distinct prime factors, 12 divisors, 7 antidivisors and 3760 totatives.
11292 has an emirpimes digit sum 15 in base 10.
11292 has a triangular digit sum 15 in base 10.
11292 has a triangular digit product 36 in base 10.
11292 = 342 + … + 412 is the sum of at least 2 consecutive positive squares in 1 way.
11292 = 28242 - 28222 = 9442 - 9382 is the difference of 2 nonnegative squares in 2 ways.
11292 is the sum of 2 positive triangular numbers.
11292 is the difference of 2 positive pentagonal numbers in 1 way.
11292 is not the sum of 3 positive squares.
112922 = 69602 + 88922 is the sum of 2 positive squares in 1 way.
112922 is the sum of 3 positive squares.
11292 is a proper divisor of 974 - 1.
11292 = '1129' + '2' is the concatenation of 2 prime numbers.
11292 is palindromic in (at least) the following bases: 30, -23, and -71.
11292 in base 30 = cgc and consists of only the digits 'c' and 'g'.
11292 in base 47 = 55C and consists of only the digits '5' and 'C'.

The number 11292 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A006550: n+8*C(n,2)+30*C(n,3)+62*C(n,4)+75*C(n,5)+30*C(n,6).
A034997: Number of Generalized Retarded Functions in Quantum Field Theory.
A101713: Indices of primes in sequence defined by A(0) = 43, A(n) = 10*A(n-1) - 27 for n > 0.
A137200: Number of ways to tile an n X 1 strip with 1 X 1 squares and 2 X 1 dominoes with the restriction that no three consecutive tiles are of the same type.
A212584: Nonnegative walks of length n on the x-axis starting at the origin using steps {1,-1} and visiting no point more than twice.
A213754: Principal diagonal of the convolution array A213753.
A227333: T(n,k)=Number of nXk 0,1 arrays indicating 2X2 subblocks of some larger (n+1)X(k+1) binary array having two adjacent 1s and two adjacent 0s, with rows and columns of the latter in lexicographically nondecreasing order
A271580: Magic sums of 4 X 4 magic squares composed of squares.
A289191: Number of polygonal tiles with n sides with two exits per side and n edges connecting pairs of exits, with no edges between exits on the same side and non-isomorphic under rotational symmetry.
A289343: a(n) = A289191(prime(n)).

Friday, September 15, 2017

Number of the day: 894518

Properties of the number 894518:

894518 = 2 × 447259 is semiprime and squarefree.
894518 has 2 distinct prime factors, 4 divisors, 15 antidivisors and 447258 totatives.
894518 has a semiprime digit sum 35 in base 10.
Reversing the decimal digits of 894518 results in a sphenic number.
894518 = 72 + 382 + 9452 is the sum of 3 positive squares.
8945182 is the sum of 3 positive squares.
894518 is a proper divisor of 853926 - 1.
894518 = '894' + '518' is the concatenation of 2 sphenic numbers.
894518 is an emirpimes in (at least) the following bases: 2, 7, 8, 16, 26, 27, 30, 32, 36, 38, 40, 41, 43, 52, 62, 67, 70, 75, 76, 82, 84, 87, 88, 90, 91, 98, 99, and 100.

Thursday, September 14, 2017

Number of the day: 8347

Properties of the number 8347:

8347 is a cyclic number.
8347 = 17 × 491 is semiprime and squarefree.
8347 has 2 distinct prime factors, 4 divisors, 25 antidivisors and 7840 totatives.
8347 has a semiprime digit sum 22 in base 10.
Reversing the decimal digits of 8347 results in an emirpimes.
8347 = 41742 - 41732 = 2542 - 2372 is the difference of 2 nonnegative squares in 2 ways.
8347 is the sum of 2 positive triangular numbers.
8347 is the difference of 2 positive pentagonal numbers in 2 ways.
8347 = 72 + 272 + 872 is the sum of 3 positive squares.
83472 = 39282 + 73652 is the sum of 2 positive squares in 1 way.
83472 is the sum of 3 positive squares.
8347 is a proper divisor of 10110 - 1.
8347 = '83' + '47' is the concatenation of 2 prime numbers.
8347 is an emirpimes in (at least) the following bases: 5, 6, 8, 9, 10, 21, 26, 28, 33, 34, 35, 37, 38, 43, 45, 50, 54, 55, 56, 57, 59, 60, 62, 66, 68, 70, 73, 79, 82, 84, 88, 90, 98, and 99.
8347 is palindromic in (at least) the following bases: 31, 78, -43, and -56.
8347 in base 7 = 33223 and consists of only the digits '2' and '3'.
8347 in base 31 = 8l8 and consists of only the digits '8' and 'l'.
8347 in base 34 = 77h and consists of only the digits '7' and 'h'.

The number 8347 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A005676: Sum C(n-k,4*k), k = 0..n.
A007042: Diagonal of partition triangle A047812.
A009962: Coordination sequence for sigma-CrFe, Position Xa.
A031899: Lucky numbers with size of gaps equal to 16 (upper terms).
A035107: First differences give (essentially) A028242.
A128034: a(0)=a(1)=1. a(n) = the multiple of n which is > a(n-1)+a(n-2) and is <= a(n-1)+a(n-2)+n.
A240012: Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 3.
A268325: Number of length-(n+1) 0..6 arrays with new values introduced in sequential order, and with new repeated values introduced in sequential order, both starting with zero.
A272223: Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 441", based on the 5-celled von Neumann neighborhood.
A279754: Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 197", based on the 5-celled von Neumann neighborhood.