cp's OEIS Frontend

If {x,y} are used as the generators of primitive Pythagorean triples (a,b,c) where a=y^2-x^2, b=2x*y and c=x^2+y^2, then the above sequence enumerates all PPT's as a 1-to-1 mapping into the integers.

Rationals a/b (lowest form) can be mapped uniquely into a triangular array A226314(n)/A054531(n).

By Schinzel's Hypothesis H the functions f_1=ax-1 and f_2=bx-1 have infinite values of x such that f_1 and f_2 are simultaneously prime. Hence a/b can be expressed using two primes p and q as a/b=(p+1)/(q+1). This sequence determines the least x for generating p=f_1 and q=f_2 with the sequence index n selecting a/b from the triangular array A226314(n)/A054531(n).

The triangle array A226314(n)/A054531(n) that enumerates all positive rationals x/y can be generalized to enumerate all ordered pairs {x, y} where x and y are natural numbers. For example, A243808 uses a subset of this triangular array to enumerate all primitive Pythagorean triples (PPT).

A006991(n) is the sequence of primitive congruent numbers and by definition there exists a PPT whose area is equal to k^2*A006991(n) for some integer k. a(n) is an enumeration of these PPT's and is a measure of the number of Pythagorean triangles that have to be searched to find a PPT with the least hypotenuse that has an area equal to k^2*A006991(n). If {x, y} are the generators of a PPT (a, b, c) where a = y^2-x^2, b = 2x*y, c=y^2+x^2 then its area = x*y(y^2-x^2). The Mathematica program limits searches to the first 12.5 million generated PPT's. All other results have been obtained from tables cataloged by Hisanori Mishima (see Links).

Rationals a/b (lowest form) can be mapped 1-to-1 to a positive integer n where a/b is the n-th term of the triangular array A226314(n)/A054531(n). Consider two function of x, f_1 = ax-1 and f_2 = bx-1. Then by Schinzel's Hypothesis H there are infinite values of x such that f_1 and f_2 are simultaneously prime allowing a/b to be expressed using two primes p and q as a/b=(p+1)/(q+1).

By choosing the least x for generating p=f_1 and q=f_2 (see A278635) it is possible to find a unique prime rational p/q that maps to rational a/b. If n is the sequence index that selects the rational a/b from the triangular array A226314(n)/A054531(n), then a(n) selects the prime rationals p/q from the same array.

Also sum of the orders of the elements in a cyclic group with n elements, i.e., row sums of A054531. - Avi Peretz (njk(AT)netvision.net.il), Mar 31 2001

Also inverse Moebius transform of EulerPhi(n^2), A002618.

Sequence is multiplicative with a(p^e) = (p^(2*e+1)+1)/(p+1). Example: a(10) = a(2)*a(5) = 3*21 = 63.

a(n) is the number of pairs (a, b) such that the equation ax = b is solvable in the ring (Zn, +, x). See the Mathematical Reflections link. - Michel Marcus, Jan 07 2017

From Jake Duzyk, Jun 06 2023: (Start)

These are the "contraharmonic means" of the improper divisors of square integers (inclusive of 1 and the square integer itself).

Permitting "Contraharmonic Divisor Numbers" to be defined analogously to Øystein Ore's Harmonic Divisor Numbers, the only numbers for which there exists an integer contraharmonic mean of the divisors are the square numbers, and a(n) is the n-th integer contraharmonic mean, expressible also as the sum of squares of divisors of n^2 divided by the sum of divisors of n^2. That is, a(n) = sigma_2(n^2)/sigma(n^2).

(a(n) = A001157(k)/A000203(k) where k is the n-th number such that A001157(k)/A000203(k) is an integer, i.e., k = n^2.)

This sequence is an analog of A001600 (Harmonic means of divisors of harmonic numbers) and A102187 (Arithmetic means of divisors of arithmetic numbers). (End)

Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, ...). - Gary W. Adamson, May 20 2007

Characteristic function of A169581: a(A169581(n)) = 1; a(A169582(n)) = 0. - Reinhard Zumkeller, Dec 02 2009

The function T(n,k) = T(k,n) is defined for k > n but only the values for 1 <= k <= n as a triangular array are listed here.

T(n,k) = |K(n-k|k)| where K(i|j) is the Kronecker symbol. - Peter Luschny, Aug 05 2012

Twice the sum over the antidiagonals, starting with entry T(n-1,1), for n >= 3, is the same as the row n sum (i.e., phi(n): 2*Sum_{k=1..floor(n/2)} T(n-k,k) = phi(n), n >= 3). - Wolfdieter Lang, Apr 26 2013

The number of zeros in the n-th row of the triangle is cototient(n) = A051953(n). - Omar E. Pol, Apr 21 2017

This triangle is the j = 1 sub-triangle of A349221(n,k) = Sum_{j>=1} [k|binomial(n-1,k-1) AND gcd(n,k) = j], n >= 1, 1 <= k <= n, where [] is the Iverson bracket. - Richard L. Ollerton, Dec 14 2021

Continued fraction for sqrt(3).

Also coefficient of the highest power of q in the expansion of the polynomial nu(n) defined by: nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(1,1), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

nu(0)=1 nu(1)=1; nu(2)=2; nu(3)=3+q; nu(4)=5+3q+2q^2; nu(5)=8+7q+6q^2+4q^3+q^4; nu(6)=13+15q+16q^2+14q^3+11q^4+5q^5+2q^6.

From Jaroslav Krizek, May 28 2010: (Start)

a(n) = denominators of arithmetic means of the first n positive integers for n >= 1.

See A026741(n+1) or A145051(n) - denominators of arithmetic means of the first n positive integers. (End)

From R. J. Mathar, Feb 16 2011: (Start)

This is a prototype of multiplicative sequences defined by a(p^e)=1 for odd primes p, and a(2^e)=c with some constant c, here c=2. They have Dirichlet generating functions (1+(c-1)/2^s)*zeta(s).

Examples are A153284, A176040 (c=3), A040005 (c=4), A021070, A176260 (c=5), A040011, A176355 (c=6), A176415 (c=7), A040019, A021059 (c=8), A040029 (c=10), A040041 (c=12). (End)

a(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A000325(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012

For n > 0: denominators of row sums of the triangular enumeration of rational numbers A226314(n,k) / A054531(n,k), 1 <= k <= n; see A226555 for numerators. - Reinhard Zumkeller, Jun 10 2013

From Jianing Song, Nov 01 2022: (Start)

For n > 0, a(n) is the minimal gap of distinct numbers coprime to n. Proof: denote the minimal gap by b(n). For odd n we have A058026(n) > 0, hence b(n) = 1. For even n, since 1 and -1 are both coprime to n we have b(n) <= 2, and that b(n) >= 2 is obvious.

The maximal gap is given by A048669. (End)

The function T(n,k) = T(k,n) is defined for all integer k,n but only the values for 1 <= k <= n as a triangular array are listed here.

For each divisor d of n, the number of d's in row n is phi(n/d). Furthermore, if {a_1, a_2, ..., a_phi(n/d)} is the set of positive integers <= n/d that are relatively prime to n/d then T(n,a_i * d) = d. - Geoffrey Critzer, Feb 22 2015

Starting with any row n and working downwards, consider the infinite rectangular array with k = 1..n. A repeating pattern occurs every A003418(n) rows. For example, n=3: A003418(3) = 6. The 6-row pattern starting with row 3 is {1,1,3}, {1,2,1}, {1,1,1}, {1,2,3}, {1,1,1}, {1,2,1}, and this pattern repeats every 6 rows, i.e., starting with rows {9,15,21,27,...}. - Bob Selcoe and Jamie Morken, Aug 02 2017

From Robert G. Wilson v, May 10 2002: (Start)

The first term of the k-th row is k. The first leading diagonal contains all 1's. The second leading diagonal contains twice the triangular numbers = n*(n-1).

For p prime, the sum of the p-th row is (p^3 - p^2 + 2)/2.

Proof: The p-th row is p, 2*p, 3*p, ..., (p-2)*p, (p-1)*p, 1. The sum of the row = p*(1 + 2 + 3 + ... + (p-2) + (p-1)) + 1 = p*(p-1)*p/2 + 1 = (p^3 - p^2 + 2)/2. (End) [Edited by Petros Hadjicostas, May 27 2020]

In the square array where T(i,j) = T(j,i), the natural extension of the triangle, each set of rows and columns with common indices [d1, d2, ..., ds] define a group multiplication table on their grid, if the d1, d2, ..., ds are the set of divisors of a squarefree number [A. Jorza]. - R. J. Mathar, May 03 2007

T(n,k) is the minimum number of squares necessary to fill a rectangle with sides of length n and k. - Stefano Spezia, Oct 06 2018