A066830 -id:A066830 - cp's OEIS frontend

A171522 Denominator of 1/n^2-1/(n+2)^2.

0, 9, 16, 225, 144, 1225, 576, 3969, 1600, 9801, 3600, 20449, 7056, 38025, 12544, 65025, 20736, 104329, 32400, 159201, 48400, 233289, 69696, 330625, 97344, 455625, 132496, 613089, 176400, 808201, 230400, 1046529, 295936, 1334025, 374544, 1677025, 467856

Offset: 0

Paul Curtz, Dec 11 2009

This is the third column in the table of denominators of the hydrogenic spectra (the main diagonal A147560):
0,   0,   0,   0,   0,   0,   0,   0... A000004
1,   4,   9,  16,  25,  36,  49,  64... A000290
1,  36,  16, 100,   9, 196,  64, 324... A061038
1, 144, 225,  12, 441, 576,  81, 900... A061040
1, 400, 144, 784,  64,1296, 400,1936... A061042
1, 900 1225,1600,2025, 100,3025,3600... A061044
1,1764, 576, 324, 225,4356,  48,6084... A061046
1,3136,3969,4900,5929,7056,8281, 196... A061048.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A105371. Bisections: A060300, A069075.

A171522 := proc(n) if n = 0 then 0 else lcm(n+2,n) ; %^2 ; end if ; end:
seq(A171522(n),n=0..70) ; # R. J. Mathar, Dec 15 2009

a[n_] := If[EvenQ[n], (n*(n+2))^2/4, (n*(n+2))^2]; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Jun 13 2017 *)

concat(0, Vec(x*(x^8+4*x^6+16*x^5+190*x^4+64*x^3+180*x^2+16*x+9) / ((x-1)^5*-(x+1)^5) + O(x^100))) \\ Colin Barker, Nov 05 2014

a(n) = (A066830(n+1))^2.
a(n) = -((-5+3*(-1)^n)*n^2*(2+n)^2)/8. - Colin Barker, Nov 05 2014
G.f.: x*(x^8+4*x^6+16*x^5+190*x^4+64*x^3+180*x^2+16*x+9) / ((x-1)^5*-(x+1)^5). - Colin Barker, Nov 05 2014

Edited and extended by R. J. Mathar, Dec 15 2009

A166010 a(n) = prime(n)^2-4.

Original entry on oeis.org

0, 5, 21, 45, 117, 165, 285, 357, 525, 837, 957, 1365, 1677, 1845, 2205, 2805, 3477, 3717, 4485, 5037, 5325, 6237, 6885, 7917, 9405, 10197, 10605, 11445, 11877, 12765, 16125, 17157, 18765, 19317, 22197, 22797, 24645, 26565, 27885, 29925, 32037

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 04 2009

Least common multiple of prime(n)-2 and prime(n)+2.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A066830, A084921, A166011.

Cf. A049001, A084920, A182200; A182174.

[NthPrime(n)^2-4: n in [1..41]]; // Bruno Berselli, Apr 17 2012

f[n_]:=LCM[n-2,n+2]; lst={};Do[p=Prime[n];AppendTo[lst,f[p]],{n,5!}]; lst
Prime[Range[5!]]^2 - 4 (* Zak Seidov, Apr 17 2012 *)

a(n)=prime(n)^2-4 \\ Charles R Greathouse IV, Apr 17 2012

a(n) = A001248(n)-4 = A040976(n)*A052147(n). [Bruno Berselli, Apr 17 2012]

Definition rewritten by Bruno Berselli, Apr 17 2012

A249860 a(n) = Least common multiple of n + 3 and n - 3.

Original entry on oeis.org

4, 5, 0, 7, 8, 9, 20, 55, 12, 91, 56, 45, 80, 187, 36, 247, 140, 105, 176, 391, 72, 475, 260, 189, 308, 667, 120, 775, 416, 297, 476, 1015, 180, 1147, 608, 429, 680, 1435, 252, 1591, 836, 585, 920, 1927, 336, 2107, 1100, 765, 1196, 2491, 432, 2695, 1400, 969

Offset: 1

Colin Barker, Nov 07 2014

The recurrence for the general case lcm(n+k, n-k) is a(n) = 3*a(n-2*k)-3*a(n-4*k)+a(n-6*k) for n>6*k.

			a(8) = 55 because lcm(8+3, 8-3) = lcm(11, 5) = 55.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,1).

Cf. A066830 (k=1), A249859 (k=2), A060789.

A249860:=n->lcm(n+3,n-3): seq(A249860(n), n=1..100); # Wesley Ivan Hurt, Feb 12 2017

CoefficientList[Series[(-10 x^19 - 8 x^18 - 3 x^17 - 4 x^16 -5 x^15 + 37 x^13 + 32 x^12 + 18 x^11 + 32 x^10 + 70 x^9 + 12 x^8 + 40 x^7 + 8 x^6 + 9 x^5 + 8 x^4 + 7 x^3 + 5 x + 4) / (- x^18 + 3 x^12 - 3 x^6 + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 08 2014 *)
Table[LCM @@ (n + {-3, 3}), {n, 54}] (* Michael De Vlieger, Feb 13 2017 *)

PARI
```
a(n) = lcm(n+3, n-3)
```

Vec(x*(-10*x^19 -8*x^18 -3*x^17 -4*x^16 -5*x^15 +37*x^13 +32*x^12 +18*x^11 +32*x^10 +70*x^9 +12*x^8 +40*x^7 +8*x^6 +9*x^5 +8*x^4 +7*x^3 +5*x +4) / (-x^18 +3*x^12 -3*x^6 +1) + O(x^100))

a(n) = 3*a(n-6)-3*a(n-12)+a(n-18) for n>18.
G.f.: x*(-10*x^19 -8*x^18 -3*x^17 -4*x^16 -5*x^15 +37*x^13 +32*x^12 +18*x^11 +32*x^10 +70*x^9 +12*x^8 +40*x^7 +8*x^6 +9*x^5 +8*x^4 +7*x^3 +5*x +4) / (-x^18 +3*x^12 -3*x^6 +1).
From Peter Bala, Feb 15 2019: (Start)
For n >= 3, a(n) = (n^2 - 9)/b(n), where (b(n)), n >= 3, is the periodic sequence [6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, ...] of period 6. a(n) is thus a quasi-polynomial in n
For n >= 4, a(n) = (n + 3)*A060789(n-3). (End)
Sum_{n>=4} 1/a(n) = 47/60. - Amiram Eldar, Aug 09 2022
Sum_{k=1..n} a(k) ~ 7*n^3/36. - Vaclav Kotesovec, Aug 09 2022

A249859 Least common multiple of n + 2 and n - 2.

Original entry on oeis.org

3, 0, 5, 6, 21, 8, 45, 30, 77, 24, 117, 70, 165, 48, 221, 126, 285, 80, 357, 198, 437, 120, 525, 286, 621, 168, 725, 390, 837, 224, 957, 510, 1085, 288, 1221, 646, 1365, 360, 1517, 798, 1677, 440, 1845, 966, 2021, 528, 2205, 1150, 2397, 624, 2597, 1350, 2805

Offset: 1

Colin Barker, Nov 07 2014

The recurrence for the general case lcm(n+k, n-k) is a(n) = 3*a(n-2*k) - 3*a(n-4*k) + a(n-6*k) for n > 6*k.

			a(8) = 30 because lcm(8 + 2, 8 - 2) = lcm(6, 10) = 30.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A066830 (k=1), A249860 (k=3), A060819.

[Lcm(n-2, n+2): n in [1..60]]; // Vincenzo Librandi, Nov 10 2014

A249859:=n-> ilcm(n+2,n-2): seq(A249859(n), n=1..100); # Wesley Ivan Hurt, Jul 09 2017

Table[LCM[n - 2, n + 2], {n, 50}] (* Alonso del Arte, Nov 07 2014 *)
CoefficientList[Series[(-6 x^12 - 2 x^11 - 3 x^10 + 23 x^8 + 12 x^7 + 30 x^6 + 8 x^5 + 12 x^4 + 6 x^3 + 5 x^2 + 3) / (-x^12 + 3 x^8 - 3 x^4 + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)
LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{3,0,5,6,21,8,45,30,77,24,117,70,165},60] (* Harvey P. Dale, Jul 11 2017 *)

PARI
```
a(n) = lcm(n+2, n-2)
```

Vec(x*(-6*x^12 -2*x^11 -3*x^10 +23*x^8 +12*x^7 +30*x^6 +8*x^5 +12*x^4 +6*x^3 +5*x^2 +3) / (-x^12 +3*x^8 -3*x^4 +1) + O(x^100))

a(n) = lcm(n - 2, n + 2).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12) for n > 12.
G.f.: x*(-6*x^12 - 2*x^11 - 3*x^10 + 23*x^8 + 12*x^7 + 30*x^6 + 8*x^5 + 12*x^4 + 6*x^3 + 5*x^2 + 3) / (-x^12 + 3*x^8 - 3*x^4 + 1).
From Peter Bala, Feb 15 2019: (Start)
For n >= 2, a(n) = (n^2 - 4)/b(n), where b(n), n >= 1, is the periodic sequence [1, 4, 1, 2, 1, 4, 1, 2, ...] of period 4. a(n) is thus a quasi-polynomial in n.
For n >= 3, a(n) = (n + 2)*A060819(n-2). (End)
Sum_{n>=3} 1/a(n) = 5/6. - Amiram Eldar, Aug 09 2022
Sum_{k=1..n} a(k) ~ 11*n^3/48. - Vaclav Kotesovec, Aug 09 2022

A166011 Least common multiple of prime(n)-3 and prime(n)+3.

Original entry on oeis.org

5, 0, 8, 20, 56, 80, 140, 176, 260, 416, 476, 680, 836, 920, 1100, 1400, 1736, 1856, 2240, 2516, 2660, 3116, 3440, 3956, 4700, 5096, 5300, 5720, 5936, 6380, 8060, 8576, 9380, 9656, 11096, 11396, 12320, 13280, 13940, 14960, 16016, 16376, 18236, 18620

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 04 2009

From Altug Alkan, Apr 22 2016: (Start)
For n > 1, a(n) is (p-3)*(p+3)/2 where p is the n-th prime. The reason is that the greatest common divisor of p-3 and p+3 is always 2 where p is the n-th prime and n > 2.
Proof: Let us assume that q is the greatest common divisor of p-3 and p+3. Because of the fact that any divisor of a and b must divide a-b, we know that q must divide 6. Note that q cannot be a multiple of 3 because p is prime, that is, q must be 1 or 2. Since we know that p-3 and p+3 are always even numbers for odd prime p, q must be 2 because we define it as the greatest common divisor.
If the greatest common divisor of p-3 and p+3 is always 2 where p is the n-th prime and n > 2, then the least common multiple of p-3 and p+3 must be (p-3)*(p+3)/2 where p is the n-th prime and n > 2 because of the general identity lcm(a, b) * gcd(a, b) = a*b. Note that for p = 3, (p-3)*(p+3)/t always is equal to 0 for any nonzero integer t, so it can be said that a(n) is (p-3)*(p+3)/2 where p is the n-th prime and n > 1. (End)

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A066830, A084921, A166010.

A166011:=n->lcm(ithprime(n)+3,ithprime(n)-3): seq(A166011(n), n=1..100); # Wesley Ivan Hurt, Apr 22 2016

f[n_]:=LCM[n-3,n+3]; lst={};Do[p=Prime[n];AppendTo[lst,f[p]],{n,5!}]; lst
LCM[#+3,#-3]&/@Prime[Range[50]] (* Harvey P. Dale, Aug 09 2015 *)

a(n) = lcm(prime(n)-3, prime(n)+3); \\ Michel Marcus, Apr 22 2016

A277384 Least common multiple of n + 4 and n - 4.

Original entry on oeis.org

15, 6, 7, 0, 9, 10, 33, 12, 65, 42, 105, 16, 153, 90, 209, 60, 273, 154, 345, 48, 425, 234, 513, 140, 609, 330, 713, 96, 825, 442, 945, 252, 1073, 570, 1209, 160, 1353, 714, 1505, 396, 1665, 874, 1833, 240, 2009, 1050, 2193, 572, 2385, 1242, 2585, 336, 2793

Offset: 1

Colin Barker, Oct 12 2016

The recurrence for the general case lcm(n+k, n-k) is b(n) = 3*b(n-2*k) - 3*b(n-4*k) + b(n-6*k) for n>6*k.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,1).

Cf. A066830 (k=1), A249859 (k=2), A249860 (k=3).

Cf. A277385.

A277384:=n->lcm(n+4,n-4): seq(A277384(n), n=1..100); # Wesley Ivan Hurt, Jul 09 2017

Table[LCM[n + 4, n - 4], {n, 1, 25}] (* G. C. Greubel, Oct 12 2016 *)

PARI
```
a(n) = lcm(n+4, n-4)
```

Vec(x*(15 +6*x +7*x^2 +9*x^4 +10*x^5 +33*x^6 +12*x^7 +20*x^8 +24*x^9 +84*x^10 +16*x^11 +126*x^12 +60*x^13 +110*x^14 +24*x^15 +123*x^16 +46*x^17 +51*x^18 -7*x^20 -6*x^21 -15*x^22 -4*x^23 -30*x^24 -12*x^25 -14*x^26) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^3*(1 +x^4)^3) + O(x^60))

a(n) = 3*a(n-8)-3*a(n-16)+a(n-24) for n>27.

G.f.: x*(15 +6*x +7*x^2 +9*x^4 +10*x^5 +33*x^6 +12*x^7 +20*x^8 +24*x^9 +84*x^10 +16*x^11 +126*x^12 +60*x^13 +110*x^14 +24*x^15 +123*x^16 +46*x^17 +51*x^18 -7*x^20 -6*x^21 -15*x^22 -4*x^23 -30*x^24 -12*x^25 -14*x^26) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^3*(1 +x^4)^3).

A386823 Triangle read by rows: T(n,k) = numerator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 15, 3, 7, 1, 12, 21, 8, 9, 1, 35, 4, 3, 5, 11, 1, 24, 45, 20, 33, 12, 13, 1, 63, 15, 55, 3, 39, 7, 15, 1, 40, 77, 4, 65, 28, 5, 16, 17, 1, 99, 12, 91, 21, 3, 8, 51, 9, 19, 1, 60, 117, 56, 105, 48, 85, 36, 57, 20, 21, 1, 143, 35, 15, 4, 119, 3, 95, 5, 7, 11, 23

Offset: 1

Stefano Spezia, Aug 04 2025

			The triangle of the fractions begins as:
  1/1;
  1/1,   3/5;
  1/1,   4/5,  5/13;
  1/1, 15/17,   3/5,  7/25;
  1/1, 12/13, 21/29,  8/17,  9/41;
  1/1, 35/37,   4/5,   3/5,  5/13, 11/61;
  1/1, 24/25, 45/53, 20/29, 33/65, 12/37, 13/85;
  ...

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A000012 (k=0), A000290, A005408, A066830 (k=1), A069011, A094728, A386824 (denominators).

T[n_,k_]:=Numerator[(n^2-k^2)/(n^2+k^2)]; Table[T[n,k],{n,12},{k,0,n-1}]//Flatten

T(n,n-1) = A005804(n-1).

A362218 Three-column array read by rows: row n gives the unique ordered primitive Pythagorean triple (a,b,c) with a

Original entry on oeis.org

3, 4, 5, 8, 15, 17, 5, 12, 13, 12, 35, 37, 7, 24, 25, 16, 63, 65, 9, 40, 41, 20, 99, 101, 11, 60, 61, 24, 143, 145, 13, 84, 85, 28, 195, 197, 15, 112, 113, 32, 255, 257, 17, 144, 145, 36, 323, 325, 19, 180, 181, 40, 399, 401

Offset: 3

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Author

Miguel-Ángel Pérez García-Ortega, Apr 11 2023

Keywords

nonn
tabf

Comments

Given an ordered primitive Pythagorean triple (a,b,c) with a
For n>=3 there exists a unique ordered primitive Pythagorean triple such that (b+c)/a = n.

For n odd, the triple is {n, (n^2-1)/2, (n^2+1)/2}.

For n even, the triple is { 2*n, n^2-1, n^2+1 }.

Examples

Irregular array begins: n=3: 3, 4, 5; n=4: 8, 15, 17; n=5: 5, 12, 13; n=6: 12, 35, 37; n=7: 7, 24, 25; ... Row n=3 is (3,4,5) and has (b+c)/a = (4+5)/3 = 3. Row n=4 is (8,15,17) and has (b+c)/a = (15+17)/8 = 4.

References

J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.

Links

Miguel-Ángel Pérez García-Ortega, Radial proportions (in Spanish).

Crossrefs

Cf. A022998 (short leg), A066830 (long leg), A228564 (hypotenuse).

Programs

Mathematica

k=50; ternas={{n," ",a,b,c," ",r," "," γ2 "," ",s," ",rb}};Do[If[Mod[t,2]==0,ternas=Join[ternas,{{t," ",2t,t^2-1,t^2+1," ",t-1," ",t," ",t(t+1)," ",t(t-1)}}],ternas=Join[ternas,{{t," ",t,(t^2-1)/2,(t^2+1)/2," ",(t-1)/2," ",t," ",(t(t+1))/2," ",(t(t-1))/2}}]],{t,3,k+2}] MatrixForm[Transpose[ternas]]

Formula

T(n,1) = A022998(n).

T(n,2) = A066830(n).

T(n,3) = A228564(n).

a(6*k-3) = 2*k+1;

a(6*k-2) = ((2*k+1)^2 - 1)/2;

a(6*k-1) = ((2*k+1)^2 + 1)/2;

a(6*k) = 4*(k+1);

a(6*k+1) = 4*(k+1)^2 - 1;

a(6*k+2) = 4*(k+1)^2 + 1.

Extensions

Edited by N. J. A. Sloane, Apr 30 2023

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cp's OEIS Frontend

A171522 Denominator of 1/n^2-1/(n+2)^2.

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A166010 a(n) = prime(n)^2-4.

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A249860 a(n) = Least common multiple of n + 3 and n - 3.

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A249859 Least common multiple of n + 2 and n - 2.

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A166011 Least common multiple of prime(n)-3 and prime(n)+3.

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A277384 Least common multiple of n + 4 and n - 4.

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