A118858 -id:A118858 - cp's OEIS frontend

A249750 Number of primitive Pythagorean triples with perimeter < 10^n.

0, 7, 70, 703, 7026, 70229, 702309, 7023027, 70230484, 702304875, 7023049293, 70230492763, 702304928492, 7023049277919, 70230492773355, 702304927727064, 7023049277265686, 70230492772676557

Offset: 1

Frank M Jackson, Nov 04 2014

The ratio a(n)/10^n as n->inf is log(2)/Pi^2 = 0.70230... (Lehmer - A118858).

			a(2)=7 because there are 7 primitive solutions (a,b,c) with a+b+c<10^2.
(3,4,5),(5,12,13),(8,15,17),(7,24,25),(20,21,29),(12,35,37),(9,40,41)

D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.
Eric Weisstein's World of Mathematics, Primitive Pythagorean Triple

Cf. A118858.

Mathematica
```
lst1 = {}; Do[If[GCD[m, n]==1&&m
				
```

a(8)-a(18) from Hiroaki Yamanouchi, Nov 17 2014

A284983 Decimal expansion of log(2)/Pi.

Original entry on oeis.org

2, 2, 0, 6, 3, 5, 6, 0, 0, 1, 5, 2, 6, 5, 1, 5, 9, 3, 3, 9, 6, 4, 5, 6, 4, 3, 2, 1, 1, 7, 9, 9, 7, 6, 9, 0, 9, 8, 2, 6, 8, 9, 7, 4, 8, 7, 2, 9, 6, 5, 5, 4, 7, 0, 4, 8, 9, 2, 6, 3, 2, 3, 3, 7, 0, 7, 1, 2, 1, 7, 6, 7, 0, 4, 6, 6, 8, 6, 6, 8, 2, 4, 9, 7, 9, 9, 3, 1, 1, 1, 8, 5, 3, 9, 6, 7, 5

Offset: 0

Terry D. Grant, Apr 06 2017

			0.2206356001526515933964564321...

Cf. A118858 (log(2)/(Pi^2)), A163973 (Pi/log(2)), A131223 (2*Pi/log(2)).

Cf. A000796 (Pi), A002162 (log(2)), A185361 (2^(1/Pi)).

RealDigits[Log[2]/Pi,10,120][[1]] (* Harvey P. Dale, Feb 04 2025 *)

PARI
```
log(2)/Pi \\ Michel Marcus, Apr 09 2017
```

Equals A002162/A000796.
Equals 1/A163973.
Equals log(A185361). - Amiram Eldar, Nov 24 2020
Equals Integral_{z>=0} Pi*z*sech(Pi*z)^2. - Peter Luschny, Aug 03 2021

Previous Mathematica program replaced by Harvey P. Dale, Feb 04 2025

A242439 Decimal expansion of the coefficient C used in the asymptotic evaluation of the number of primitive Pythagorean triangles with area less than n, as C*sqrt(n).

Original entry on oeis.org

5, 3, 1, 3, 3, 9, 9, 4, 9, 9, 5, 8, 4, 2, 1, 8, 2, 5, 5, 9, 1, 2, 4, 5, 0, 9, 7, 5, 5, 2, 2, 5, 6, 0, 4, 5, 5, 3, 1, 2, 7, 4, 9, 5, 9, 1, 6, 3, 0, 1, 0, 3, 4, 7, 1, 2, 0, 5, 2, 7, 4, 3, 7, 0, 3, 6, 6, 9, 8, 0, 5, 5, 6, 3, 5, 9, 1, 1, 4, 1, 8, 3, 7, 0, 1, 4, 9, 5, 4, 6, 8, 6, 0, 2, 0, 3, 1, 8, 7, 2

Offset: 0

Jean-François Alcover, May 14 2014

			0.531339949958421825591245097552256...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 277.

Eric Weisstein's MathWorld, Primitive Pythagorean Triple

Cf. A118858.

RealDigits[1/Sqrt[2*Pi^5]*Gamma[1/4]^2 , 10, 100] // First

(1/sqrt(2*Pi^5))*GAMMA(1/4)^2.

A362149 Decimal expansion of K, a constant arising in the analysis of the binary Euclidean algorithm.

Original entry on oeis.org

7, 0, 5, 9, 7, 1, 2, 4, 6, 1, 0, 1, 9, 1, 6, 3, 9, 1, 5, 2, 9, 3, 1, 4, 1, 3, 5, 8, 5, 2, 8, 8, 1, 7, 6, 6, 6, 6, 7, 7

Offset: 0

Paolo Xausa, Apr 09 2023

Corresponds to the 2/b constant reported in Knuth (1998), p. 352.

Vallée (1998) conjectured that this constant times A362150 equals 4*log(2)/Pi^2; Brent (1999) supported the conjecture with numerical computations and Morris (2016) proved the conjecture.

			0.7059712461019163915293141358528817666677...

Richard P. Brent, Further analysis of the binary Euclidean algorithm, Programming Research Group technical report TR-7-99, Oxford University (1999) (see also the arXiv link).
Steven R. Finch, Mathematical Constants, Cambridge University Press, New York, NY, 2003, p. 158.
Donald E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd edition, Addison-Wesley, 1998, Sect. 4.5.2, pp. 348-353.

Richard P. Brent, Further analysis of the binary Euclidean algorithm, arXiv:1303.2772 [cs.DS], 1999, p. 12.
Ian D. Morris, A rigorous version of R. P. Brent's model for the binary Euclidean algorithm, Advances in Mathematics, Vol. 290, 26 Feb. 2016, pp. 73-143.
Brigitte Vallée, Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators, Algorithmica 22 (1998), pp. 660-685.
Wikipedia, Binary GCD algorithm.

Cf. A118858, A345987, A362150.

Equals (4*log(2)/Pi^2)/A362150 = 4*A118858/A362150.

A362150 Decimal expansion of lambda, a constant arising in the analysis of the binary Euclidean algorithm.

Original entry on oeis.org

3, 9, 7, 9, 2, 2, 6, 8, 1, 1, 8, 8, 3, 1, 6, 6, 4, 4, 0, 7, 6, 7, 0, 7, 1, 6, 1, 1, 4, 2, 6, 5, 4, 9, 8, 2, 3, 0, 9, 8

Offset: 0

Paolo Xausa, Apr 09 2023

See Brent (1999), p. 12, and Morris (2016), p. 79, where this constant is called zeta(1).

See A362149 for additional comments, references and links.

			0.3979226811883166440767071611426549823098...

Richard P. Brent, Further analysis of the binary Euclidean algorithm, arXiv:1303.2772 [cs.DS], 1999, p. 12.
Ian D. Morris, A rigorous version of R. P. Brent's model for the binary Euclidean algorithm, Advances in Mathematics, Vol. 290, 26 Feb. 2016, p. 79.

Cf. A118858, A345987, A362149.

Equals (4*log(2)/Pi^2)/A362149 = 4*A118858/A362149.

A259679 Lampard's constant, decimal expansion of log(2)/(4*Pi^2).

Original entry on oeis.org

0, 1, 7, 5, 5, 7, 6, 2, 3, 1, 9, 3, 1, 7, 0, 7, 1, 9, 1, 0, 2, 2, 3, 4, 6, 4, 9, 8, 7, 4, 2, 4, 9, 2, 5, 2, 4, 0, 8, 2, 1, 9, 1, 3, 3, 1, 1, 0, 8, 1, 5, 6, 3, 5, 3, 4, 4, 3, 5, 8, 5, 9, 4, 5, 5, 7, 0, 6, 2, 4, 1, 0, 3, 3, 4, 2, 4, 2, 1, 3, 3, 5, 0, 3, 5, 5, 0, 4, 2, 3, 3, 9, 5, 1, 8, 3, 3, 5, 0, 2, 3, 5, 8, 1, 9

Offset: 0

Johannes W. Meijer, Jul 03 2015

Lampard dealt in a paper, see the links, with the calculation of internal cross capacitances of cylinders under certain conditions of symmetry. Van der Pauw generalized Lampard's results with the formula exp(-4*Pi^2*Cab,cd) + exp(-4*Pi^2*Cbc,da) = 1, see the links. Van der Pauw observed that in Lampard's case of symmetry, the two capacitances Cab,cd and Cbc,da are mutually equal, and hence are both equal to C = log(2)/(4*Pi^2) independently of the size or shape of the cross-section, which is Lampard's theorem.
Lampard's constant is closely related to Van der Pauw's constant A163973.
This constant was named after the Australian professor of electrical engineering Douglas Geoffrey Lampard (1927 - 1994). - Amiram Eldar, Dec 03 2020

			0.0175576231931707191...

D. G. Lampard, A new theorem in electrostatics with applications to calculable standards of capacitance, Proceedings of the IEE, Vol. 104, No. 6, pp. 271-280, September 1957.
L. J. van der Pauw, A method of measuring specific resistivity and Hall effect of disc of arbitrary shape, Philips Research Reports, Vol. 13. no. 1, pp 1-9, February 1958.
Index entries for transcendental numbers

Cf. A163973 (Pi/log(2)), A118858 (log(2)/Pi^2), A000796 (Pi), A002162 (log(2)), A002388 (Pi^2), A092742 (1/Pi^2).

log(2)/(4*Pi^2) \\ Michel Marcus, Jul 04 2015

C = log(2)/(4*Pi^2).

A328499 The number of primitive Pythagorean triangles with perimeter less than n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Mo Li, Oct 17 2019

D. N. Lehmer has proved that the asymptotic density of a(n) is a(n)/n = log(2)/Pi^2 = 0.07023049... See A118858.

			For n=90, the triples are
   {3,  4,  5},  3 +  4 +  5 = 12 < 90
   {5, 12, 13},  5 + 12 + 13 = 30 < 90
   {7, 24, 25},  7 + 24 + 25 = 56 < 90
   {8, 15, 17},  8 + 15 + 17 = 40 < 90
   {9, 40, 41},  9 + 40 + 41 = 90
  {12, 35, 37}, 12 + 35 + 37 = 84 < 90
  {20, 21, 29}, 20 + 21 + 29 = 70 < 90
so a(90)=7.

Ron Knott, Pythagorean Triples and Online Calculators
D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.

Cf. A008846, A024364, A118858, A249750.

A379600 a(n) is the semiperimeter of the primitive Pythagorean triangle (x(n), y(n), z(n)) with x(n) < y(n) < z(n) and x(n) > x(n-1), y(n) > y(n-1), z(n) > z(n-1), which has the smallest perimeter (if there are several triangles with smallest perimeter: the one of these with the smallest area), starting from a(1) = (3 + 4 + 5)/2 = 6.

Original entry on oeis.org

6, 15, 20, 35, 63, 77, 99, 104, 130, 165, 204, 247, 266, 336, 345, 391, 425, 450, 513, 580, 609, 651, 713, 805, 825, 888, 945, 1036, 1107, 1204, 1271, 1376, 1457, 1530, 1617, 1645, 1764, 1887, 1961, 2014, 2090, 2280, 2337, 2419, 2537, 2562, 2684, 2772, 2990, 3149

Offset: 1

Felix Huber, Feb 15 2025

nonn

Conjecture: There is no primitive Pythagorean triangle that has a smaller semiperimeter than a(n) that can be drawn around the primitive Pythagorean triangle (x(n-1), y(n-1), z(n-1)) without touching it.

Subsequence of A020886.

			(8, 15, 17) is the primitive Pythagorean triangle with semiperimeter a(3) = 20. (20, 21, 29) is the primitive Pythagorean triangle with semiperimeter a(4) = 35 because 20 > 8, 21 > 15, 29 > 17 and there is no other primitive Pythagorean triangle with perimeter <= 70 satisfying this criterium. For example, the primitive Pythagorean triangle (7, 24, 25) has the perimeter 56 but 7 < 8.

Felix Huber, Table of n, a(n) for n = 1..3257
Eric Weisstein's World of Mathematics, Primitive Pythagorean Triple
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A010814, A020886, A118858.

A379600:=proc(S) # to get all terms <= S
    local p,q,i,L,M;
    L:=[];
    M:=[[3,4,5,6,6]];
    for p from 3 to floor((sqrt(4*S+1)-1)/2) do
        for q to min(p-1,S/p-p) do
            if gcd(p,q)=1 and is(p-q,odd) then
                L:=[op(L),[min(p^2-q^2,2*p*q),max(p^2-q^2,2*p*q),p^2+q^2,p*(p+q),(p^2-q^2)*p*q]];
            fi
        od
    od;
    L:=sort(sort(L,(x,y)->x[5]<=y[5]),(x,y)->x[4]<=y[4]);
    for i in L do
        if i[1]>M[nops(M),1] and i[2]>M[nops(M),2] and i[3]>M[nops(M),3] then
            M:=[op(M),i]
        fi
    od;
    return seq(M[i,4],i=1..nops(M))
end proc;
A379600(3149);
# 3 lines above: change 4 to 3 for hypotenuses, to 2 for long legs and to 1 for short legs, to 5 for areas

cp's OEIS Frontend

A249750 Number of primitive Pythagorean triples with perimeter < 10^n.

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A284983 Decimal expansion of log(2)/Pi.

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A242439 Decimal expansion of the coefficient C used in the asymptotic evaluation of the number of primitive Pythagorean triangles with area less than n, as C*sqrt(n).

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A362149 Decimal expansion of K, a constant arising in the analysis of the binary Euclidean algorithm.

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A362150 Decimal expansion of lambda, a constant arising in the analysis of the binary Euclidean algorithm.

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Comments

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Formula

A259679 Lampard's constant, decimal expansion of log(2)/(4*Pi^2).

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PARI

Formula

A328499 The number of primitive Pythagorean triangles with perimeter less than n.

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