Monday, January 16, 2017

Number of the day: 656292204869201

Properties of the number 656292204869201:

656292204869201 is prime.
656292204869201 has 5 antidivisors and 656292204869200 totatives.
656292204869201 has an emirpimes digit sum 62 in base 10.
656292204869201 = 3281461024346012 - 3281461024346002 is the difference of 2 nonnegative squares in 1 way.
656292204869201 is the difference of 2 positive pentagonal numbers in 1 way.
656292204869201 = 125104552 + 223557762 is the sum of 2 positive squares in 1 way.
656292204869201 = 175402 + 273992 + 256181802 is the sum of 3 positive squares.
6562922048692012 = 3432692362551512 + 5593618592761602 is the sum of 2 positive squares in 1 way.
6562922048692012 is the sum of 3 positive squares.
656292204869201 is a divisor of 16211640730512173 - 1.
656292204869201 = '65629' + '2204869201' is the concatenation of 2 prime numbers.
656292204869201 = '65' + '6292204869201' is the concatenation of 2 semiprime numbers.
656292204869201 is an emirp in (at least) the following bases: 2, 13, 61, 75, 79, and 99.

Sunday, January 15, 2017

Number of the day: 12962

Sofia Kovalevskaya was born on this day 167 years ago.

Properties of the number 12962:

12962 = 2 × 6481 is semiprime and squarefree.
12962 has 2 distinct prime factors, 4 divisors, 13 antidivisors and 6480 totatives.
12962 has an oblong digit sum 20 in base 10.
Reversing the decimal digits of 12962 results in a prime.
12962 = 712 + 892 is the sum of 2 positive squares in 1 way.
12962 = 72 + 122 + 1132 is the sum of 3 positive squares.
129622 = 28802 + 126382 is the sum of 2 positive squares in 1 way.
129622 is the sum of 3 positive squares.
12962 is a divisor of 60110 - 1.
12962 = '129' + '62' is the concatenation of 2 emirpimes.
12962 is an emirpimes in (at least) the following bases: 2, 6, 8, 13, 18, 23, 27, 29, 30, 31, 36, 41, 42, 43, 47, 50, 51, 55, 58, 60, 61, 62, 66, 71, 77, 78, 82, 85, 86, 87, 88, 95, 96, and 98.
12962 is palindromic in (at least) the following bases: 72, 80, -62, -81, -90, and -96.

The number 12962 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A005901: Number of points on surface of cuboctahedron (or icosahedron): a(0) = 1; for n > 0, a(n) = 10n^2 + 2. Also coordination sequence for f.c.c. or A_3 or D_3 lattice.
A010022: a(0) = 1, a(n) = 40*n^2 + 2 for n>0.
A099792: Positions of records for terms in the continued fraction of the Glaisher-Kinkelin constant A.
A107317: Semiprimes of the form 2*(m^2 + m + 1) (implying that m^2 + m + 1 is a prime).
A139485: a(1)=1. For m>=0 and 1<=k<=2^m, a(2^m +k) = a(k) + sum{j=1 to 2^m) a(j).
A212074: Beach-Williams Pell numbers of type 2p (p prime).
A213359: Sum of all parts that are not the smallest part (counted with multiplicity) of all partitions of n.
A230831: T(n,k)=Number of nXk 0..3 arrays x(i,j) with each element horizontally, vertically or antidiagonally next to at least one element with value (x(i,j)+1) mod 4 and at least one element with value (x(i,j)-1) mod 4, no adjacent elements equal, and upper left element zero
A238728: Number of standard Young tableaux with n cells where the largest value n is contained in the last row.
A239729: Number of partitions p of n such that if h = min(p), then h is an (h,2)-separator of p; see Comments.

Saturday, January 14, 2017

Number of the day: 184127285037141

Alfred Tarski was born on this day 116 years ago.

Properties of the number 184127285037141:

184127285037141 = 32 × 137 × 149332753477 is composite and not squarefree.
184127285037141 has 3 distinct prime factors, 12 divisors, 35 antidivisors and 121855526836416 totatives.
184127285037141 = 920636425185712 - 920636425185702 = 306878808395252 - 306878808395222 = 102292936131792 - 102292936131702 = 6719973907152 - 6719973905782 = 2239991304212 - 2239991300102 = 746663773552 - 746663761222 is the difference of 2 nonnegative squares in 6 ways.
184127285037141 is the difference of 2 positive pentagonal numbers in 2 ways.
184127285037141 = 51809462 + 125413352 = 40849502 + 129398792 is the sum of 2 positive squares in 2 ways.
184127285037141 = 74902 + 155212 + 135693402 is the sum of 3 positive squares.
1841272850371412 = 1299519588058202 + 1304428821273092 = 1182715407537842 + 1411194520357652 = 1057175174421002 + 1507536520321412 = 158100879372842 + 1834472627611652 is the sum of 2 positive squares in 4 ways.
1841272850371412 is the sum of 3 positive squares.
184127285037141 is a divisor of 5472164242804 - 1.
184127285037141 = '1841' + '27285037141' is the concatenation of 2 semiprime numbers.

Friday, January 13, 2017

Number of the day: 23434639

Properties of the number 23434639:

23434639 = 29 × 53 × 79 × 193 is the 21960788th composite number and is squarefree.
23434639 has 4 distinct prime factors, 16 divisors, 31 antidivisors and 21805056 totatives.
23434639 has a semiprime digit sum 34 in base 10.
23434639 has a Fibonacci digit sum 34 in base 10.
23434639 = 117173202 - 117173192 = 4040602 - 4040312 = 2211082 - 2210552 = 1483602 - 1482812 = 608082 - 606152 = 83922 - 68552 = 62602 - 39692 = 48922 - 7052 is the difference of 2 nonnegative squares in 8 ways.
23434639 is the difference of 2 positive pentagonal numbers in 6 ways.
23434639 is not the sum of 3 positive squares.
234346392 = 115351852 + 203990642 = 112259002 + 205708892 = 9828392 + 234140202 = 154358892 + 176328002 = 68164362 + 224213852 = 60577202 + 226381612 = 162771602 + 168593112 = 57152552 + 227270362 = 71541612 + 223159202 = 161618202 + 169699112 = 58700952 + 226875362 = 47570642 + 229467352 = 123805642 + 198973352 is the sum of 2 positive squares in 13 ways.
234346392 is the sum of 3 positive squares.
23434639 is a divisor of 853156 - 1.
23434639 = '2343463' + '9' is the concatenation of 2 semiprime numbers.

Thursday, January 12, 2017

Number of the day: 975761

Properties of the number 975761:

975761 = 101 × 9661 is semiprime and squarefree.
975761 has 2 distinct prime factors, 4 divisors, 21 antidivisors and 966000 totatives.
975761 has a semiprime digit sum 35 in base 10.
Reversing the decimal digits of 975761 results in an emirpimes.
975761 = 4878812 - 4878802 = 48812 - 47802 is the difference of 2 nonnegative squares in 2 ways.
975761 is the difference of 2 positive pentagonal numbers in 2 ways.
975761 = 6312 + 7602 = 6202 + 7692 is the sum of 2 positive squares in 2 ways.
975761 = 512 + 942 + 9822 is the sum of 3 positive squares.
9757612 = 1932202 + 9564392 = 1794392 + 9591202 = 2069612 + 9535602 = 140392 + 9756602 is the sum of 2 positive squares in 4 ways.
9757612 is the sum of 3 positive squares.
975761 is a divisor of 139100 - 1.
975761 is an emirpimes in (at least) the following bases: 2, 4, 7, 8, 9, 10, 17, 22, 23, 27, 38, 39, 41, 52, 55, 56, 59, 65, 71, 73, 78, 82, 83, 84, 85, 90, 92, 93, 95, 96, and 98.

Wednesday, January 11, 2017

Number of the day: 3742

Properties of the number 3742:

3742 = 2 × 1871 is semiprime and squarefree.
3742 has 2 distinct prime factors, 4 divisors, 9 antidivisors and 1870 totatives.
Reversing the decimal digits of 3742 results in a prime.
3742 is the sum of 2 positive triangular numbers.
3742 is the difference of 2 positive pentagonal numbers in 2 ways.
3742 = 142 + 392 + 452 is the sum of 3 positive squares.
37422 is the sum of 3 positive squares.
3742 is a divisor of 1915 - 1.
3742 is an emirpimes in (at least) the following bases: 2, 4, 5, 7, 8, 12, 13, 17, 18, 20, 21, 32, 34, 37, 41, 43, 44, 46, 47, 49, 50, 54, 60, 65, 71, 72, 73, 74, 76, 80, 82, 83, 85, 86, 89, 91, 94, 95, 99, and 100.
3742 is palindromic in (at least) the following bases: 16, -44, -55, and -87.
3742 in base 16 = e9e and consists of only the digits '9' and 'e'.
3742 in base 30 = 44m and consists of only the digits '4' and 'm'.
3742 in base 43 = 211 and consists of only the digits '1' and '2'.

The number 3742 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A002703: Sets with a congruence property.
A079222: Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the six-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).
A080855: (9*n^2-3*n+2)/2.
A084849: a(n) = 1 + n + 2*n^2.
A125264: Numbers n such that n^10 + 9 is prime.
A185264: Number of disconnected 6-regular simple graphs on n vertices with girth at least 4.
A189912: Extended Motzkin numbers, Sum{k>=0} C(n,k)C(k), C(k) the extended Catalan number A057977(k).
A214203: Number of rooted planar binary unlabeled trees with n leaves and caterpillar index <= 5.
A243366: Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, k<=0<=max(0,floor(n/2)-1), read by rows.
A249508: Lengths of complete iterations (direct and reverse branches) of the Oldenburger-Kolakoski sequence A000002.

Tuesday, January 10, 2017

Number of the day: 53952

Properties of the number 53952:

53952 = 26 × 3 × 281 is the 48454th composite number and is not squarefree.
53952 has 3 distinct prime factors, 28 divisors, 9 antidivisors and 17920 totatives.
53952 is the difference of 2 nonnegative squares in 10 ways.
53952 = 562 + 1042 + 2002 is the sum of 3 positive squares.
539522 = 307202 + 443522 is the sum of 2 positive squares in 1 way.
539522 is the sum of 3 positive squares.
53952 is a divisor of 898 - 1.
53952 is palindromic in (at least) base -43.
53952 in base 3 = 2202000020 and consists of only the digits '0' and '2'.
53952 in base 42 = UOO and consists of only the digits 'O' and 'U'.

The number 53952 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A080877: a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=2.
A080878: a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=3.
A098648: Expansion of (1-3*x)/(1 - 6*x + 4*x^2).
A149377: Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (-1, 0, 1), (1, -1, 0), (1, 1, -1), (1, 1, 0)}
A205071: Number of (n+1)X8 0..1 arrays with the number of rightwards and downwards edge increases in each 2X2 subblock differing from the number in all its horizontal and vertical neighbors
A234645: Sum of the divisors of n^3+1.