cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9, 2, 8, 2, 8
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

If the canonical factorization of n into prime powers is Product p^e(p) then d(n) = Product (e(p) + 1). More generally, for k > 0, sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1) is the sum of the k-th powers of the divisors of n.
Number of ways to write n as n = x*y, 1 <= x <= n, 1 <= y <= n. For number of unordered solutions to x*y=n, see A038548.
Note that d(n) is not the number of Pythagorean triangles with radius of the inscribed circle equal to n (that is A078644). For number of primitive Pythagorean triangles having inradius n, see A068068(n).
Number of factors in the factorization of the polynomial x^n-1 over the integers. - T. D. Noe, Apr 16 2003
Also equal to the number of partitions p of n such that all the parts have the same cardinality, i.e., max(p)=min(p). - Giovanni Resta, Feb 06 2006
Equals A127093 as an infinite lower triangular matrix * the harmonic series, [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, May 10 2007
For odd n, this is the number of partitions of n into consecutive integers. Proof: For n = 1, clearly true. For n = 2k + 1, k >= 1, map each (necessarily odd) divisor to such a partition as follows: For 1 and n, map k + (k+1) and n, respectively. For any remaining divisor d <= sqrt(n), map (n/d - (d-1)/2) + ... + (n/d - 1) + (n/d) + (n/d + 1) + ... + (n/d + (d-1)/2) {i.e., n/d plus (d-1)/2 pairs each summing to 2n/d}. For any remaining divisor d > sqrt(n), map ((d-1)/2 - (n/d - 1)) + ... + ((d-1)/2 - 1) + (d-1)/2 + (d+1)/2 + ((d+1)/2 + 1) + ... + ((d+1)/2 + (n/d - 1)) {i.e., n/d pairs each summing to d}. As all such partitions must be of one of the above forms, the 1-to-1 correspondence and proof is complete. - Rick L. Shepherd, Apr 20 2008
Number of subgroups of the cyclic group of order n. - Benoit Jubin, Apr 29 2008
Equals row sums of triangle A143319. - Gary W. Adamson, Aug 07 2008
Equals row sums of triangle A159934, equivalent to generating a(n) by convolving A000005 prefaced with a 1; (1, 1, 2, 2, 3, 2, ...) with the INVERTi transform of A000005, (A159933): (1, 1,-1, 0, -1, 2, ...). Example: a(6) = 4 = (1, 1, 2, 2, 3, 2) dot (2, -1, 0, -1, 1, 1) = (2, -1, 0, -2, 3, 2) = 4. - Gary W. Adamson, Apr 26 2009
Number of times n appears in an n X n multiplication table. - Dominick Cancilla, Aug 02 2010
Number of k >= 0 such that (k^2 + k*n + k)/(k + 1) is an integer. - Juri-Stepan Gerasimov, Oct 25 2015
The only numbers k such that tau(k) >= k/2 are 1,2,3,4,6,8,12. - Michael De Vlieger, Dec 14 2016
a(n) is also the number of partitions of 2*n into equal parts, minus the number of partitions of 2*n into consecutive parts. - Omar E. Pol, May 03 2017
From Tomohiro Yamada, Oct 27 2020: (Start)
Let k(n) = log d(n)*log log n/(log 2 * log n), then lim sup k(n) = 1 (Hardy and Wright, Chapter 18, Theorem 317) and k(n) <= k(6983776800) = 1.537939... (the constant A280235) for every n (Nicolas and Robin, 1983).
There exist infinitely many n such that d(n) = d(n+1) (Heath-Brown, 1984). The number of such integers n <= x is at least c*x/(log log x)^3 (Hildebrand, 1987) but at most O(x/sqrt(log log x)) (Erdős, Carl Pomerance and Sárközy, 1987). (End)
Number of 2D grids of n congruent rectangles with two different side lengths, in a rectangle, modulo rotation (cf. A038548 for squares instead of rectangles). Also number of ways to arrange n identical objects in a rectangle (NOT modulo rotation, cf. A038548 for modulo rotation); cf. A007425 and A140773 for the 3D case. - Manfred Boergens, Jun 08 2021
The constant quoted above from Nicolas and Robin, 6983776800 = 2^5 * 3^3 * 5^2 * 7 * 11 * 13 * 17 * 19, appears arbitrary, but interestingly equals 2 * A095849(36). That second factor is highly composite and deeply composite. - Hal M. Switkay, Aug 08 2025

Examples

			G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 2*x^7 + 4*x^8 + 3*x^9 + ...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
  • G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 345, Exercise XXI(16). MR0121327 (22 #12066)
  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 55.
  • G. H. Hardy and E. M. Wright, revised by D. R. Heath-Brown and J. H. Silverman, An Introduction to the Theory of Numbers, 6th ed., Oxford Univ. Press, 2008.
  • K. Knopp, Theory and Application of Infinite Series, Blackie, London, 1951, p. 451.
  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Chap. II. (For inequalities, etc.)
  • S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. Has many references to this sequence. - N. J. A. Sloane, Jun 02 2014
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • B. Spearman and K. S. Williams, Handbook of Estimates in the Theory of Numbers, Carleton Math. Lecture Note Series No. 14, 1975; see p. 2.1.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 285.
  • E. C. Titchmarsh, The Theory of Functions, Oxford, 1938, p. 160.
  • Terence Tao, Poincaré's Legacies, Part I, Amer. Math. Soc., 2009, see pp. 31ff for upper bounds on d(n).

Links

Crossrefs

See A002183, A002182 for records. See A000203 for the sum-of-divisors function sigma(n).
For partial sums see A006218.
Cf. A007427 (Dirichlet Inverse), A001227, A001826, A001842, A005237, A005238, A006558, A006601, A019273, A027750, A034296, A039665, A049051, A049820, A051731, A061017, A066446, A091202, A091220, A106737, A115361, A127093, A129372, A129510, A143319, A156552, A159933, A159934, A163280, A183063, A237665, A263730.
Factorizations into given number of factors: writing n = x*y (A038548, unordered, A000005, ordered), n = x*y*z (A034836, unordered, A007425, ordered), n = w*x*y*z (A007426, ordered).
Cf. A000010, A095849.
Cf. A098198 (Dgf at s=2), A183030 (Dgf at s=3), A183031 (Dgf at s=3).

Programs

  • GAP
    List([1..150],n->Tau(n)); # Muniru A Asiru, Mar 05 2019
  • Haskell
    divisors 1 = [1]
divisors n = (1:filter ((==0) . rem n)
               [2..n `div` 2]) ++ [n]
a = length . divisors
-- James Spahlinger, Oct 07 2012
  • Haskell
    a000005 = product . map (+ 1) . a124010_row  -- Reinhard Zumkeller, Jul 12 2013
  • Julia
    function tau(n)
    i = 2; num = 1
    while i * i <= n
        if rem(n, i) == 0
            e = 0
            while rem(n, i) == 0
                e += 1
                n = div(n, i)
            end
            num *= e + 1
        end
        i += 1
    end
    return n > 1 ? num + num : num
end
println([tau(n) for n in 1:104])  # Peter Luschny, Sep 03 2023
  • Magma
    [ NumberOfDivisors(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
  • Maple
    with(numtheory): A000005 := tau; [ seq(tau(n), n=1..100) ];
  • Mathematica
    Table[DivisorSigma[0, n], {n, 100}] (* Enrique Pérez Herrero, Aug 27 2009 *)
CoefficientList[Series[(Log[1 - q] + QPolyGamma[1, q])/(q Log[q]), {q, 0, 100}], q] (* Vladimir Reshetnikov, Apr 23 2013 *)
a[ n_] := SeriesCoefficient[ (QPolyGamma[ 1, q] + Log[1 - q]) / Log[q], {q, 0, Abs@n}]; (* Michael Somos, Apr 25 2013 *)
a[ n_] := SeriesCoefficient[ q/(1 - q)^2 QHypergeometricPFQ[ {q, q}, {q^2, q^2}, q, q^2], {q, 0, Abs@n}]; (* Michael Somos, Mar 05 2014 *)
a[n_] := SeriesCoefficient[q/(1 - q) QHypergeometricPFQ[{q, q}, {q^2}, q, q], {q, 0, Abs@n}] (* Mats Granvik, Apr 15 2015 *)
With[{M=500},CoefficientList[Series[(2x)/(1-x)-Sum[x^k (1-2x^k)/(1-x^k),{k,M}],{x,0,M}],x]] (* Mamuka Jibladze, Aug 31 2018 *)
  • MuPAD
    numlib::tau (n)$ n=1..90 // Zerinvary Lajos, May 13 2008
  • PARI
    {a(n) = if( n==0, 0, numdiv(n))}; /* Michael Somos, Apr 27 2003 */
  • PARI
    {a(n) = n=abs(n); if( n<1, 0, direuler( p=2, n, 1 / (1 - X)^2)[n])}; /* Michael Somos, Apr 27 2003 */
  • PARI
    {a(n)=polcoeff(sum(m=1, n+1, sumdiv(m, d, (-log(1-x^(m/d) +x*O(x^n) ))^d/d!)), n)} \\ Paul D. Hanna, Aug 21 2014
  • Python
    from sympy import divisor_count
for n in range(1, 20): print(divisor_count(n), end=', ') # Stefano Spezia, Nov 05 2018
  • Sage
    [sigma(n, 0) for n in range(1, 105)]  # Zerinvary Lajos, Jun 04 2009

Formula

If n is written as 2^z*3^y*5^x*7^w*11^v*... then a(n)=(z+1)*(y+1)*(x+1)*(w+1)*(v+1)*...
a(n) = 2 iff n is prime.
G.f.: Sum_{n >= 1} a(n) x^n = Sum_{k>0} x^k/(1-x^k). This is usually called THE Lambert series (see Knopp, Titchmarsh).
a(n) = A083888(n) + A083889(n) + A083890(n) + A083891(n) + A083892(n) + A083893(n) + A083894(n) + A083895(n) + A083896(n).
a(n) = A083910(n) + A083911(n) + A083912(n) + A083913(n) + A083914(n) + A083915(n) + A083916(n) + A083917(n) + A083918(n) + A083919(n).
Multiplicative with a(p^e) = e+1. - David W. Wilson, Aug 01 2001
a(n) <= 2 sqrt(n) [see Mitrinovich, p. 39, also A046522].
a(n) is odd iff n is a square. - Reinhard Zumkeller, Dec 29 2001
a(n) = Sum_{k=1..n} f(k, n) where f(k, n) = 1 if k divides n, 0 otherwise (Mobius transform of A000012). Equivalently, f(k, n) = (1/k)*Sum_{l=1..k} z(k, l)^n with z(k, l) the k-th roots of unity. - Ralf Stephan, Dec 25 2002
G.f.: Sum_{k>0} ((-1)^(k+1) * x^(k * (k + 1)/2) / ((1 - x^k) * Product_{i=1..k} (1 - x^i))). - Michael Somos, Apr 27 2003
a(n) = n - Sum_{k=1..n} (ceiling(n/k) - floor(n/k)). - Benoit Cloitre, May 11 2003
a(n) = A032741(n) + 1 = A062011(n)/2 = A054519(n) - A054519(n-1) = A006218(n) - A006218(n-1) = 1 + Sum_{k=1..n-1} A051950(k+1). - Ralf Stephan, Mar 26 2004
G.f.: Sum_{k>0} x^(k^2)*(1+x^k)/(1-x^k). Dirichlet g.f.: zeta(s)^2. - Michael Somos, Apr 05 2003
Sequence = M*V where M = A129372 as an infinite lower triangular matrix and V = ruler sequence A001511 as a vector: [1, 2, 1, 3, 1, 2, 1, 4, ...]. - Gary W. Adamson, Apr 15 2007
Sequence = M*V, where M = A115361 is an infinite lower triangular matrix and V = A001227, the number of odd divisors of n, is a vector: [1, 1, 2, 1, 2, 2, 2, ...]. - Gary W. Adamson, Apr 15 2007
Row sums of triangle A051731. - Gary W. Adamson, Nov 02 2007
Sum_{n>0} a(n)/(n^n) = Sum_{n>0, m>0} 1/(n*m). - Gerald McGarvey, Dec 15 2007
Logarithmic g.f.: Sum_{n>=1} a(n)/n * x^n = -log( Product_{n>=1} (1-x^n)^(1/n) ). - Joerg Arndt, May 03 2008
a(n) = Sum_{k=1..n} (floor(n/k) - floor((n-1)/k)). - Enrique Pérez Herrero, Aug 27 2009
a(s) = 2^omega(s), if s > 1 is a squarefree number (A005117) and omega(s) is: A001221. - Enrique Pérez Herrero, Sep 08 2009
a(n) = A048691(n) - A055205(n). - Reinhard Zumkeller, Dec 08 2009
For n > 1, a(n) = 2 + Sum_{k=2..n-1} floor((cos(Pi*n/k))^2). And floor((cos(Pi*n/k))^2) = floor(1/4 * e^(-(2*i*Pi*n)/k) + 1/4 * e^((2*i*Pi*n)/k) + 1/2). - Eric Desbiaux, Mar 09 2010, corrected Apr 16 2011
a(n) = 1 + Sum_{k=1..n} (floor(2^n/(2^k-1)) mod 2) for every n. - Fabio Civolani (civox(AT)tiscali.it), Mar 12 2010
From Vladimir Shevelev, May 22 2010: (Start)
(Sum_{d|n} a(d))^2 = Sum_{d|n} a(d)^3 (J. Liouville).
Sum_{d|n} A008836(d)*a(d)^2 = A008836(n)*Sum_{d|n} a(d). (End)
a(n) = sigma_0(n) = 1 + Sum_{m>=2} Sum_{r>=1} (1/m^(r+1))*Sum_{j=1..m-1} Sum_{k=0..m^(r+1)-1} e^(2*k*Pi*i*(n+(m-j)*m^r)/m^(r+1)). - A. Neves, Oct 04 2010
a(n) = 2*A038548(n) - A010052(n). - Reinhard Zumkeller, Mar 08 2013
Sum_{n>=1} a(n)*q^n = (log(1-q) + psi_q(1)) / log(q), where psi_q(z) is the q-digamma function. - Vladimir Reshetnikov, Apr 23 2013
a(n) = Product_{k = 1..A001221(n)} (A124010(n,k) + 1). - Reinhard Zumkeller, Jul 12 2013
a(n) = Sum_{k=1..n} A238133(k)*A000041(n-k). - Mircea Merca, Feb 18 2013
G.f.: Sum_{k>=1} Sum_{j>=1} x^(j*k). - Mats Granvik, Jun 15 2013
The formula above is obtained by expanding the Lambert series Sum_{k>=1} x^k/(1-x^k). - Joerg Arndt, Mar 12 2014
G.f.: Sum_{n>=1} Sum_{d|n} ( -log(1 - x^(n/d)) )^d / d!. - Paul D. Hanna, Aug 21 2014
2*Pi*a(n) = Sum_{m=1..n} Integral_{x=0..2*Pi} r^(m-n)( cos((m-n)*x)-r^m cos(n*x) )/( 1+r^(2*m)-2r^m cos(m*x) )dx, 0 < r < 1 a free parameter. This formula is obtained as the sum of the residues of the Lambert series Sum_{k>=1} x^k/(1-x^k). - Seiichi Kirikami, Oct 22 2015
a(n) = A091220(A091202(n)) = A106737(A156552(n)). - Antti Karttunen, circa 2004 & Mar 06 2017
a(n) = A034296(n) - A237665(n+1) [Wang, Fokkink, Fokkink]. - George Beck, May 06 2017
G.f.: 2*x/(1-x) - Sum_{k>0} x^k*(1-2*x^k)/(1-x^k). - Mamuka Jibladze, Aug 29 2018
a(n) = Sum_{k=1..n} 1/phi(n / gcd(n, k)). - Daniel Suteu, Nov 05 2018
a(k*n) = a(n)*(f(k,n)+2)/(f(k,n)+1), where f(k,n) is the exponent of the highest power of k dividing n and k is prime. - Gary Detlefs, Feb 08 2019
a(n) = 2*log(p(n))/log(n), n > 1, where p(n)= the product of the factors of n = A007955(n). - Gary Detlefs, Feb 15 2019
a(n) = (1/n) * Sum_{k=1..n} sigma(gcd(n,k)), where sigma(n) = sum of divisors of n. - Orges Leka, May 09 2019
a(n) = A001227(n)*(A007814(n) + 1) = A001227(n)*A001511(n). - Ivan N. Ianakiev, Nov 14 2019
From Richard L. Ollerton, May 11 2021: (Start)
a(n) = A038040(n) / n = (1/n)*Sum_{d|n} phi(d)*sigma(n/d), where phi = A000010 and sigma = A000203.
a(n) = (1/n)*Sum_{k=1..n} phi(gcd(n,k))*sigma(n/gcd(n,k))/phi(n/gcd(n,k)). (End)
From Ridouane Oudra, Nov 12 2021: (Start)
a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*cos(2*k*n*Pi/j);
a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*e^(2*k*n*Pi*i/j), where i^2=-1. (End)

Extensions

Incorrect formula deleted by Ridouane Oudra, Oct 28 2021

A120062 Number of triangles with integer sides a <= b <= c having integer inradius n.

Original entry on oeis.org

1, 5, 13, 18, 15, 45, 24, 45, 51, 52, 26, 139, 31, 80, 110, 89, 33, 184, 34, 145, 185, 103, 42, 312, 65, 96, 140, 225, 36, 379, 46, 169, 211, 116, 173, 498, 38, 123, 210, 328, 44, 560, 60, 280, 382, 134, 64, 592, 116, 228, 230, 271, 47, 452, 229, 510, 276, 134, 54
Offset: 1

Views

Author

Hugo Pfoertner, Jun 11 2006

Keywords

Comments

It is conjectured that the longest possible side c of a triangle with integer sides and inradius n is given by A057721(n) = n^4 + 3*n^2 + 1.
For n >= 1, a(n) >= 1 because triangle (a, b, c) = (n^2 + 2, n^4 + 2*n^2 + 1, n^4 + 3*n^2 + 1) has inradius n. - David W. Wilson, Jun 17 2006
Previous name was "Number of triangles with integer sides a<=bA362669); so, now effectively, a(10) = 52. - Bernard Schott, Apr 24 2023

Examples

			a(1)=1: {3,4,5} is the only triangle with integer sides and inradius 1.
a(2)=5: {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17} are the only triangles with integer sides and inradius 2.
a(4)=A120252(1)+A120252(2)+A120252(4)=1+4+13 because 1, 2 and 4 are the factors of 4. The 1 primitive triangle with inradius n=1 is (3,4,5). The 4 primitive triangles with n=2 are (5,12,13), (9,10,17), (7,15,20), (6,25,29). The 13 primitive triangles with n=4 are (13,14,15), (15,15,24), (11,25,30), (15,26,37), (10,35,39), (9,40,41), (33,34,65), (25,51,74), (9,75,78), (11,90,97), (21,85,104), (19,153,170), (18,289,305). (Primitive means GCD(a, b, c, n)=1.)

Links

Crossrefs

Cf. A078644 [Pythagorean triangles with inradius n], A057721 [n^4+3*n^2+1].
Let S(n) be the set of triangles with integer sides a<=b<=c and inradius n. Then:
A120062(n) gives number of triangles in S(n).
A120261(n) gives number of triangles in S(n) with gcd(a, b, c) = 1.
A120252(n) gives number of triangles in S(n) with gcd(a, b, c, n) = 1.
A005408(n) = 2n+1 gives shortest short side a of triangles in S(n).
A120064(n) gives shortest middle side b of triangles in S(n).
A120063(n) gives shortest long side c of triangles in S(n).
A120570(n) gives shortest perimeter of triangles in S(n).
A120572(n) gives smallest area of triangles in S(n).
A058331(n) = 2n^2+1 gives longest short side a of triangles in S(n).
A082044(n) = n^4+2n^2+1 gives longest middle side b of triangles in S(n).
A057721(n) = n^4+3n^2+1 gives longest long side c of triangles in S(n).
A120571(n) = 2n^4+6n^2+4 gives longest perimeter of triangles in S(n).
A120573(n) = gives largest area of triangles in S(n).
Cf. A120252 [primitive triangles with integer inradius], A120063 [minimum of longest sides], A057721 [maximum of longest sides], A120064 [minimum of middle sides], A082044 [maximum of middle sides], A005408 [minimum of shortest sides], A058331 [maximum of shortest sides], A007237 [number of triangles with integer sides and area = n times perimeter].
Cf. A362669, A362670.

Programs

  • Mathematica
    (* See link above. *)

Formula

The even-numbered terms are given by a(2*n)=A007237(n).
a(n) = Sum_{k|n} A120252(k).

Extensions

More terms from Graeme McRae and Hugo Pfoertner, Jun 12 2006
Name corrected by Bernard Schott, Apr 24 2023

A361689 The number of divisors of 2*n^2.

Original entry on oeis.org

2, 4, 6, 6, 6, 12, 6, 8, 10, 12, 6, 18, 6, 12, 18, 10, 6, 20, 6, 18, 18, 12, 6, 24, 10, 12, 14, 18, 6, 36, 6, 12, 18, 12, 18, 30, 6, 12, 18, 24, 6, 36, 6, 18, 30, 12, 6, 30, 10, 20, 18, 18, 6, 28, 18, 24, 18, 12, 6, 54, 6, 12, 30, 14, 18, 36, 6, 18, 18, 36, 6, 40, 6, 12, 30, 18, 18, 36, 6, 30
Offset: 1

Views

Author

R. J. Mathar, Mar 20 2023

Keywords

Links

Crossrefs

Cf. A000005, A078644, A048691, A007814, A275367.

Programs

  • Maple
    A361689 := proc(n)
        numtheory[tau](2*n^2) ;
end proc:
seq(A361689(n),n=1..80) ;
  • PARI
    a(n) = numdiv(2*n^2); \\ Michel Marcus, Mar 21 2023

Formula

a(n) = A000005(2*n^2) = 2*A078644(n).
a(n) = 2*A048691(n) if n odd, = 2*(1+A007814(n)) *A275367(n) if n even.

A156688 The total number of distinct Pythagorean triples with an area numerically equal to n times their perimeters.

Original entry on oeis.org

2, 3, 6, 4, 6, 9, 6, 5, 10, 9, 6, 12, 6, 9, 18, 6, 6, 15, 6, 12, 18, 9, 6, 15, 10, 9, 14, 12, 6, 27, 6, 7, 18, 9, 18, 20, 6, 9, 18, 15, 6, 27, 6, 12, 30, 9, 6, 18, 10, 15, 18, 12, 6, 21, 18, 15, 18, 9, 6, 36, 6, 9, 30, 8, 18, 27, 6, 12, 18, 27, 6, 25, 6, 9, 30, 12, 18, 27, 6, 18, 18, 9, 6, 36, 18, 9, 18, 15, 6, 45, 18, 12, 18, 9, 18
Offset: 1

Views

Author

Ant King, Feb 18 2009

Keywords

Comments

The members of this sequence are also 1/2 the number of divisors of 8n^2. The corresponding results for primitive triangles only are in A068068.
Also, the total number of distinct "areas with equal border", that is: Let x, y be positive integers so that the area xy equals the border around it with thickness n. As a formula it is: 2xy = (x+2n)(y+2n). To compare with the original, the areas at thickness 5 are 11x210, 12x110, 14x60, 15x50, 18x35, 20x30. - Juhani Heino, Jul 22 2012

Examples

			There are 6 Pythagorean triples whose area is 5 times their perimeters - (21,220,221), (22,120,122), (24,70,74), (25,60,65),(28,45,53) and (30,40,50) - hence a(5)=6.

References

  • Chi, Henjin and Killgrove, Raymond; Problem 1447, Crux Math 15(5), May 1989.
  • Chi, Henjin and Killgrove, Raymond; Solution to Problem 1447, Crux Math 16(7), September 1990.

Links

Crossrefs

Cf. A000005, A068068.

Programs

  • Mathematica
    1/2 DivisorSigma[0,8#^2] &/@Range[75]
  • PARI
    A156688(n) = (numdiv(8*n*n)/2); \\ Antti Karttunen, Sep 27 2018

Formula

a(n) = A000005(8n^2)/2 = A078644(2n).

Extensions

More terms from Antti Karttunen, Sep 27 2018

A386980 Number of acute Heronian triangles with integer inradius n.

Original entry on oeis.org

0, 0, 1, 1, 0, 4, 0, 2, 2, 2, 0, 6, 0, 1, 4, 3, 0, 8, 0, 6, 7, 2, 0, 17, 1, 0, 2, 8, 0, 14, 0, 3, 6, 1, 4, 17, 0, 0, 4, 12, 0, 27, 0, 4, 13, 1, 0, 27, 1, 4, 2, 4, 0, 13, 5, 14, 2, 0, 0, 32, 0, 0, 14, 4, 3, 18, 0, 5, 3, 15, 0, 41, 0, 0, 10, 4, 7, 16, 0, 18, 3, 0, 0, 60, 2, 0, 2, 18, 0, 39, 9
Offset: 1

Views

Author

Frank M Jackson, Aug 11 2025

Keywords

Comments

If a Heronian triangle has an inradius n, and sides (x, y, z), where x <= y <= z, then the triangle is acute iff n < (x+y-z)/2.
The only Heronian triangle with inradius 1 is the right triangle (3, 4, 5). Also, it has been proved that other than n = 3, all acute Heronian triangles have no prime inradii. For n = 3, the Heronian triangle has sides (10, 10, 12).
Empirically, it appears that the remaining occurrences of zero counts (other than 1 and the primes excluding 3) are inradii of the form 2p where p is in the set 13, 19, 29 and all other primes > 29.
The number of right integer triangles with inradius n is given by A078644, the number of obtuse Heronian triangles with inradius n is given by A386981 and the total number of Heronian triangles with inradius n is given by A120062.

Examples

			a(6) = 4, and the 4 acute Heronian triangles with inradius 6 have sides (15, 34, 35), (17, 25, 28), (17, 25, 26), (20, 20, 24).

Links

Crossrefs

Cf. A078644, A120062, A386981.

Programs

  • Mathematica
    (* See link above. *)

A386981 Number of obtuse Heronian triangles with integer inradius n.

Original entry on oeis.org

0, 3, 9, 14, 12, 35, 21, 39, 44, 44, 23, 124, 28, 73, 97, 81, 30, 166, 31, 130, 169, 95, 39, 283, 59, 90, 131, 208, 33, 347, 43, 160, 196, 109, 160, 466, 35, 117, 197, 304, 41, 515, 57, 267, 354, 127, 61, 550, 110, 214, 219, 258, 44, 425, 215, 484, 265, 128, 51, 977, 41, 138, 582, 269, 169, 603, 48, 325, 252, 564, 47, 1058, 65, 133, 445, 341
Offset: 1

Views

Author

Frank M Jackson, Aug 11 2025

Keywords

Comments

If a Heronian triangle has an inradius n, and sides (x, y, z), where x <= y <= z, then the triangle is obtuse iff n > (x+y-z)/2.
The only Heronian triangle with inradius 1 is the right triangle (3, 4, 5).
The number of right integer triangles with inradius n is given by A078644, the number of acute Heronian triangles with inradius n is given by A386980 and the total number of Heronian triangles with inradius n is given by A120062.

Examples

			a(2) = 3, and the 3 obtuse Heronian triangles with inradius 2 have sides (6, 25, 29), (7, 15, 20), (9, 10, 17).

Links

Crossrefs

Cf. A078644, A120062, A386980

Programs

  • Mathematica
    (* See link above. *)
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