Randall L Rathbun - cp's OEIS frontend

A349948 Tetrahedral-sided isosceles Heron triangle pairs.

0, 10, 48, 190, 720, 2698, 10080, 37630, 140448, 524170, 1956240, 7300798, 27246960, 101687050, 379501248, 1416317950, 5285770560, 19726764298, 73621286640, 274758382270, 1025412242448, 3826890587530, 14282150107680, 53301709843198, 198924689265120

Offset: 1

Randall L Rathbun, Mar 26 2022

Isosceles Heron triangle pairs with tetrahedral sides: [t(a(n)+1), t(a(n)+1), t(a(n))] and [t(a(n)+6), t(a(n)+5), t(a(n)+5)] where t(n) = A000292(n) is a tetrahedral number, i.e., t(n) = n*(n+1)*(n+2)/6. The Heron triangle pair areas have been checked for rationality to 100 terms of {a(n)}.
Not all isosceles Heron triangles with tetrahedral sides are generated by this sequence. For example, [t(63),t(50),t(50)] is not included. Also, scalene Heron triangles with tetrahedral sides are not included. For example, [t(111),t(104),t(62)]. - Michael Somos, Mar 27 2022
Area of triangles: T1(n) = (b(n)-2)^2*(b(n)-3)^2*(b(n)-4)*c(n)/48 and T2(n) = (b(n)+2)^2*(b(n)+3)^2*(b(n)+4)*c(n)/48, where b(n) = A003500(n) and c(n) =  A052530(n). - Randall L Rathbun, Apr 01 2022
Conjecture: for k a positive integer, the sequence {a(k^n): n >= 1} is a strong divisibility sequence; that is, for n, m >= 1,  gcd(a(k^n), a(k^m)) = a(k^gcd(n,m)). - Peter Bala, Dec 03 2022

			10 is a term, so there exists one Heron isosceles triangle whose sides are the 10th, 11th, and 11th tetrahedral numbers (220, 286, 286) and another whose sides are the 15th, 15th, and 16th tetrahedral numbers (680, 680, 816). Those two triangles have areas 29040 and 221952, respectively. (See the n=2 row of the table below.)
.
             Triangle sides               Triangle sides
     k=    ------------------          --------------------
  n a(n)   T(k) T(k+1) T(k+1)  Area    T(k+5) T(k+5) T(k+6)   Area
  - ----   ---- ------ ------ ------   ------ ------ ------  ------
  1    0      0      1      1      0*      35     35     56     588
  2   10    220    286    286  29040      680    680    816  221952
*(degenerate triangle)

Eric Weisstein's World of Mathematics, Tetrahedral Number
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A000292, A001075, A003500, A052530.

a[ n_] := 2*ChebyshevT[n, 2] - 4; (* Michael Somos, Mar 27 2022 *)

Vec(2*x^2*(5 - x)/(1 - 5*x + 5*x^2 - x^3) + O(x^42))

{a(n) = 2*polchebyshev(n,1, 2) - 4}; /* Michael Somos, Mar 27 2022 */

a(n+2) = 4*a(n+1) - a(n) + 8.
From Stefano Spezia, Mar 26 2022: (Start)
G.f.: 2*x^2*(5 - x)/((1-x)*(1 - 4*x +x^2)).
a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) for n > 3.
a(n) = (2 + sqrt(3))^n + (2 - sqrt(3))^n - 4. (End)
a(n) = 2*A001075(n) - 4. - Michael Somos, Mar 27 2022

A066823 Primes in the main diagonal of the Stolarsky array (cf. A035506).

Original entry on oeis.org

157, 6011, 55717, 74636904300223, 1524920857037139836753, 533561095785570791522387803, 960826779344998235261901369163166807780807

Offset: 1

Randall L Rathbun, Jan 25 2002

Corresponding row/column numbers of the array are 6, 12, 16, 57, 91, 117, 189, 597, 707, 1729, 3933, (through 5000). The values in row/column 597, 707, ... are probable primes.

Cf. A035506.

Stolarsky(r,c) = tau=(1+sqrt(5))/2; a=floor(r*(1+tau)-tau/2); b=round(a*tau); if(c==1,a, if(c==2,b, for(i=1,c-2,d=a+b; a=b; b=d; ); d))
for(x=1,5000,if(isprime(p=Stolarsky(x,x)),print1(p,",")))

Edited by Klaus Brockhaus, May 26 2003

Edited by T. D. Noe, Oct 30 2008

A066833 Continued fraction expansion of Product_{p prime} (1 - 1/(p^4 - p^3)).

Original entry on oeis.org

0, 1, 5, 1, 33, 49, 2, 1, 3, 2, 2, 2, 8, 1, 1, 28, 1, 7, 1, 1, 34, 2, 2, 2, 10, 2, 1, 9, 1, 1, 8, 2, 3, 16, 5, 7, 2, 14, 4, 2, 3, 4, 1, 3, 2, 3, 1, 8, 6, 2, 1, 32, 1, 10, 1, 3, 11, 2, 6, 6, 2, 1, 42, 1, 1, 2, 1, 2, 5, 1, 3, 4, 1, 1, 2, 1, 2, 5, 1, 1, 1, 15, 1, 2, 1, 4, 1, 1, 2, 6, 4, 15, 1, 1, 1, 2, 3, 1

Offset: 0

Randall L Rathbun, Jan 16 2002

Cf. A065415 (decimal expansion).

contfrac(prodeulerrat(1-1/(p^4-p^3))) \\ Amiram Eldar, Jun 13 2021

A066834 Continued fraction expansion of Product_{p prime} (1 - 1/(p^5 - p^4)).

Original entry on oeis.org

0, 1, 13, 1, 1, 4, 1, 1, 1, 3, 16, 16, 1, 11, 1, 1, 1, 1, 9, 13, 1, 5, 22, 4, 2, 6, 1, 1, 1, 39, 1, 1, 3, 1, 12, 4, 1, 106, 15, 19, 7, 7, 4, 1, 5, 1, 2, 1, 1, 2, 4, 3, 1, 10, 1, 1, 1, 1, 2, 3, 1, 6, 7, 8, 1, 7, 5, 1, 2, 44, 2, 1, 5, 1, 4, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 20, 8, 2, 3, 1, 1, 1, 4, 1, 1, 2

Offset: 0

Randall L Rathbun, Jan 16 2002

G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Cf. A065416 (decimal expansion).

contfrac(prodeulerrat(1-1/(p^5-p^4))) \\ Amiram Eldar, Jun 13 2021

A066912 Fourth column of the Eulerian triangle A008292 in square array format.

Original entry on oeis.org

0, 1, 26, 302, 2416, 15619, 88234, 455192, 2203488, 10187685, 45533450, 198410786, 848090912, 3572085255, 14875399450, 61403313100, 251732291184, 1026509354985, 4168403181210, 16871482830550, 68111623139600

Offset: 0

Randall L Rathbun, Jan 22 2002

nonn

Essentially the same as A000498.

a(n)=4^(n+3)-(n+4)*3^(n+3)+1/2*(n+3)*(n+4)*2^(n+3)-1/6*(n+2)*(n+3)*(n+4)

a(n) = 4^(n+3) - (n+4)*3^(n+3) + (1/2)*(n+3)*(n+4)*2^(n+3) - (1/6)*(n+2)*(n+3)*(n+4).

More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2002

A073470 Trisection of A007294.

Original entry on oeis.org

1, 2, 4, 6, 10, 15, 22, 32, 44, 60, 81, 107, 141, 181, 233, 295, 372, 465, 575, 710, 869, 1059, 1286, 1549, 1863, 2225, 2654, 3149, 3724, 4391, 5157, 6043, 7059, 8220, 9553, 11066, 12798, 14760, 16988, 19509, 22357, 25576, 29200, 33274, 37856, 42985

Offset: 0

Randall L Rathbun, Aug 26 2002

nonn

Cf. A007294, A073471, A073472.

function a007294(n, k) return n gt k*(k+1)/2 select a007294(n-k*(k+1)/2, k) + a007294(n, k+1) else n eq k*(k+1)/2 select 1 else 0; end function; [ a007294(n, 1): n in [0..135 by 3] ]; // Klaus Brockhaus, Jan 23 2009

More terms from Klaus Brockhaus, Jan 23 2009

A073471 Trisection of A007294.

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 24, 35, 48, 66, 89, 117, 153, 197, 252, 320, 401, 501, 619, 763, 933, 1134, 1374, 1654, 1985, 2372, 2821, 3345, 3949, 4652, 5460, 6390, 7457, 8676, 10071, 11662, 13468, 15526, 17850, 20487, 23463, 26818, 30598, 34842, 39614, 44962

Offset: 0

Randall L Rathbun, Aug 26 2002

nonn

Cf. A007294, A073470, A073472.

function a007294(n, k) return n gt k*(k+1)/2 select a007294(n-k*(k+1)/2, k) + a007294(n, k+1) else n eq k*(k+1)/2 select 1 else 0; end function; [ a007294(n, 1): n in [1..136 by 3] ]; // Klaus Brockhaus, Jan 23 2009

More terms from Klaus Brockhaus, Jan 23 2009

A073472 Trisection of A007294.

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 25, 36, 50, 68, 92, 121, 159, 205, 262, 332, 417, 520, 645, 793, 971, 1180, 1430, 1722, 2065, 2467, 2934, 3478, 4109, 4835, 5676, 6638, 7747, 9013, 10459, 12108, 13980, 16110, 18522, 21247, 24333, 27800, 31715, 36106, 41039, 46567

Offset: 0

Randall L Rathbun, Aug 26 2002

nonn

Cf. A007294, A073471, A073470.

function a007294(n, k) return n gt k*(k+1)/2 select a007294(n-k*(k+1)/2, k) + a007294(n, k+1) else n eq k*(k+1)/2 select 1 else 0; end function; [ a007294(n, 2): n in [2..137 by 3] ]; // Klaus Brockhaus, Jan 23 2009

More terms from Klaus Brockhaus, Jan 23 2009

A077177 Number of primitive Pythagorean triangles with perimeter equal to A002110(n), the product of the first n primes.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 5, 8, 17, 34, 59, 111, 213, 396, 746, 1413, 2690, 5147, 9826, 18885, 36269, 69952, 134949, 260743, 504636, 978311, 1899832, 3692980, 7190329, 13994206, 27279898, 53195986

Offset: 1

Kermit Rose and Randall L Rathbun, Nov 29 2002

A Pythagorean triangle is a right triangle whose edge lengths are all integers; such a triangle is 'primitive' if the lengths are relatively prime.

Equivalently, number of divisors of s=A070826(n) in the range (sqrt(s), sqrt(2s)). More generally, for any positive integer s, the number of primitive Pythagorean triangles with perimeter 2's equals the number of odd unitary divisors of s in the range (sqrt(s), sqrt(2s)). (A divisor d of n is 'unitary' if gcd(d, n/d) = 1.)

			a(5) = 1 since there is exactly one primitive Pythagorean triangle with perimeter 2*3*5*7*11; its edge lengths are (132, 1085, 1093). a(7) = 3; the 3 triangles have edge lengths (70941, 214060, 225509), (96460, 195789, 218261) and (142428, 156485, 211597).

A. S. Anema, "Pythagorean Triangles with Equal Perimeters", Scripta Mathematica, vol. 15 (1949) p. 89.
Albert H. Beiler, "Recreations in the Theory of Numbers", chapter XIV, "The Eternal Triangle", pp. 131, 132.
F. L. Miksa, "Pythagorean Triangles with Equal Perimeters", Mathematics, vol. 24 (1950), p. 52.

Randall L. Rathbun, Equal Perimeter primitive right triangles

Cf. A002110, A070109, A070826, A078926.

a[n_] := Length[Select[Divisors[s=Times@@Prime/@Range[2, n]], s<#^2<2s&]]

semi_peri(p)= {local(q,r,ct,tot); ct=0; tot=0; pt=0; fordiv(p,q,r=p/q-q; if(r<=q&&r>0,print(q,",",r," [",gcd(q,r),"] "); if(gcd(q,r)==1,ct=ct+1; if(q*r%2==0,pt=pt+1; ); ); tot=tot+1); ); print("semiperimeter:"p," Total sets:",tot," Coprime:",ct," Primitive:",pt); } /* Lists all pairs q,r such that the triangle with edge lengths (q^2-r^2, 2qr, q^2+r^2) has semiperimeter p. */

a(n) = A070109(A002110(n)) = A078926(A070826(n)).

Edited by Dean Hickerson, Dec 18 2002

A066767 a(n) = Sum_{k=1..n} sigma(k)*2^(n-k) where sigma(k) = A000203(k) is the sum of divisors of k.

Original entry on oeis.org

1, 5, 14, 35, 76, 164, 336, 687, 1387, 2792, 5596, 11220, 22454, 44932, 89888, 179807, 359632, 719303, 1438626, 2877294, 5754620, 11509276, 23018576, 46037212, 92074455, 184148952, 368297944, 736595944, 1473191918, 2946383908

Offset: 1

Randall L Rathbun, Jan 16 2002

nonn

a(n) is the numerator of the unreduced fraction of the n-th partial sum of Sum_{k>=1} sigma(k)/2^k where the denominator of that unreduced fraction is 2^n. The partial sums converge to A066766 = 2.744033...

			a(1) = 2*(1/2);
a(2) = 4*(1/2 + (1+2)/4) since sigma(1) = 1 and sigma(2) = 1 + 2 = 3;
a(3) = 8*(1/2 + (1+2)/4 + (1+3)/8);
a(4) = 16*(1/2 + (1+2)/4 + (1+3)/8 + (1+2+4)/16).

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Steven R. Finch, Digital Search Tree Constants [Broken link]
Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]

Cf. A000203, A066766.

smv(v)= s=0; for(i=1,matsize(v)[2],s=s+v[i]); s
a(n)= sm=0; for(j=1,n,sm=sm+smv(divisors(j)/2^j)); sm*2^n

a(n) = 2^n*(sum(k=1, n, sigma(k)/2^k)); \\ Michel Marcus, Apr 25 2022

cp's OEIS Frontend

User: Randall L Rathbun

A349948 Tetrahedral-sided isosceles Heron triangle pairs.

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A066823 Primes in the main diagonal of the Stolarsky array (cf. A035506).

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A066833 Continued fraction expansion of Product_{p prime} (1 - 1/(p^4 - p^3)).

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A066834 Continued fraction expansion of Product_{p prime} (1 - 1/(p^5 - p^4)).

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A066912 Fourth column of the Eulerian triangle A008292 in square array format.

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A073470 Trisection of A007294.

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A073471 Trisection of A007294.

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A073472 Trisection of A007294.

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A077177 Number of primitive Pythagorean triangles with perimeter equal to A002110(n), the product of the first n primes.

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