A120077 -id:A120077 - cp's OEIS frontend

A309829 Numbers k for which A120077(k) != A007407(k).

20, 21, 110, 136, 156, 930, 44310

Offset: 1

Jeppe Stig Nielsen, Aug 18 2019

The sequence of rationals related to A120077 is f(k) = Sum_{j=1..k-1} (1/j^2 - 1/k^2), motivated by each term's interpretation as the energy difference between shells k and j in a hydrogen atom model. This can easily be seen to be equal to f(k) = (Sum_{j=1..k} 1/j^2) - 1/k. Compare this with g(k) = Sum_{j=1..k} 1/j^2 which is the starting point for A007407. The question is, when does the final subtraction of 1/k change the denominator (in lowest term)? In one case (k=21), the denominator belonging to f(k) is greater than that belonging to g(k). In cases k=20, 110, 136, 156, 930, 44310, the opposite is true.
Will gcd(A120077(k), A007407(k)) always be one of the numbers A120077(k) and A007407(k)?
Should this sequence be infinite?

Cf. A120072, A120077, A007407.

s=1; for(n=2, +oo, s += 1/n^2; denominator(s)!=denominator(s-1/n) && print1(n, ", "))

A120072 Numerator triangle for hydrogen spectrum rationals.

Original entry on oeis.org

3, 8, 5, 15, 3, 7, 24, 21, 16, 9, 35, 2, 1, 5, 11, 48, 45, 40, 33, 24, 13, 63, 15, 55, 3, 39, 7, 15, 80, 77, 8, 65, 56, 5, 32, 17, 99, 6, 91, 21, 3, 4, 51, 9, 19, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21, 143, 35, 5, 1, 119, 1, 95, 5, 7, 11, 23

Offset: 2

Wolfdieter Lang, Jul 20 2006

Frequencies or energies of the spectral lines of the hydrogen (H) atom are given, according to quantum theory, by r(m,n)*3.287*PHz (1 Peta Hertz= 10^15 s^{-1}) or r(m,n)*13.599 eV (electron Volts), respectively. The wave lengths are lambda(m,n) = (1/r(m,n))* 91.196 nm (all decimals rounded). See the W. Lang link for more details.
The spectral series for n=1,2,...,7, m>=n+1, are named after Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys, Hansen-Strong, respectively.
The corresponding denominator triangle is A120073.
The rationals are r(m,n):= a(m,n)/A120073(m,n) = A120070(m,n)/(m^2*n^2) = 1/ n^2 - 1/m^2 and they are given in lowest terms.

			For the rational triangle see W. Lang link.
Numerator triangle begins as:
   3;
   8,  5;
  15,  3,  7;
  24, 21, 16,  9;
  35,  2,  1,  5, 11;
  48, 45, 40, 33, 24, 13;
  63, 15, 55,  3, 39,  7, 15;
  80, 77,  8, 65, 56,  5, 32, 17;
  99,  6, 91, 21,  3,  4, 51,  9, 19;

G. C. Greubel, Rows n = 2..50 of the triangle, flattened
Wolfdieter Lang, First ten rows, rationals and more.
T. Lyman, The Spectrum of Hydrogen in the Region of Extremely Short Wave-Lengths, The Astrophysical Journal, 23 (April 1906), 181-210. - _Paul Curtz_, May 30 2017

Row sums give A120074.
Row sums of r(m, n) triangle give A120076(m)/A120077(m), m>=2.
Cf. A120070, A120073, A120075, A126252.

[Numerator(1/k^2 - 1/n^2): k in [1..n-1], n in [2..18]]; // G. C. Greubel, Apr 24 2023

Table[1/n^2 - 1/m^2, {m,2,12}, {n,m-1}]//Flatten//Numerator (* Jean-François Alcover, Sep 16 2013 *)

def A120072(n,k): return numerator(1/k^2 - 1/n^2)
flatten([[A120072(n,k) for k in range(1,n)] for n in range(2,19)]) # G. C. Greubel, Apr 24 2023

a(m,n) = numerator(r(m,n)) with r(m,n) = 1/n^2 - 1/m^2, m>=2, n=1..m-1.

The g.f.s for the columns n=1,..,10 of triangle r(m,n) = a(m, n) / A120073(m, n), m >= 2, 1 <= n <= m-1, are given in the W. Lang link.

A120073 Denominator triangle for hydrogen spectrum rationals.

Original entry on oeis.org

4, 9, 36, 16, 16, 144, 25, 100, 225, 400, 36, 9, 12, 144, 900, 49, 196, 441, 784, 1225, 1764, 64, 64, 576, 64, 1600, 576, 3136, 81, 324, 81, 1296, 2025, 324, 3969, 5184, 100, 25, 900, 400, 100, 225, 4900, 1600, 8100, 121, 484, 1089, 1936, 3025, 4356, 5929, 7744, 9801, 12100

Offset: 2

Wolfdieter Lang, Jul 20 2006

The corresponding numerator triangle is A120072.

See A120072 and A120070 for more details.

			For the rational triangle see W. Lang link.
Denominator triangle begins as:
    4;
    9,  36;
   16,  16, 144;
   25, 100, 225,  400;
   36,   9,  12,  144,  900;
   49, 196, 441,  784, 1225, 1764;
   64,  64, 576,   64, 1600,  576, 3136;
   81, 324,  81, 1296, 2025,  324, 3969, 5184;
  100,  25, 900,  400,  100,  225, 4900, 1600, 8100;

G. C. Greubel, Rows n = 2..50 of the triangle, flattened
Wofdieter Lang, First ten rows, rationals and more.

Row sums give A120075.

Cf. A120070, A120072, A120074, A120076, A120077, A126252.

[Denominator(1/k^2 - 1/n^2): k in [1..n-1], n in [2..18]]; // G. C. Greubel, Apr 24 2023

Table[(1/n^2 - 1/m^2)//Denominator, {m,2,15}, {n,m-1}]//Flatten (* Jean-François Alcover, Sep 16 2013 *)

def A120073(n,k): return denominator(1/k^2 - 1/n^2)
flatten([[A120073(n,k) for k in range(1,n)] for n in range(2,19)]) # G. C. Greubel, Apr 24 2023

a(m,n) = denominator(r(m,n)) with r(m,n) = 1/n^2 - 1/m^2, m>=2, n=1..m-1.

A120074 Row sums of triangle A120072 (numerator triangle for H atom spectrum).

Original entry on oeis.org

3, 13, 25, 70, 54, 203, 197, 340, 303, 825, 445, 1378, 892, 1221, 1565, 3128, 1545, 4389, 2427, 3592, 3688, 7843, 3589, 8420, 6191, 9097, 7135, 15834, 5774, 19375, 12493, 14814, 14147, 19647, 12264, 33078

Offset: 2

Wolfdieter Lang, Jul 20 2006

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A120070, A120072, A120073, A120075, A120076, A120077.

A120072:= func< n,k | Numerator(1/k^2 - 1/n^2) >;
[(&+[A120072(n,k): k in [1..n-1]]): n in [2..50]]; // G. C. Greubel, Apr 24 2023

Table[Sum[1/n^2 - 1/m^2 //Numerator, {n,m-1}], {m,2,40}]  (* Jean-François Alcover, Sep 16 2013 *)

def A120072(n,k): return numerator(1/k^2 - 1/n^2)
[sum(A120072(n,k) for k in range(1,n)) for n in range(2,51)] # G. C. Greubel, Apr 24 2023

a(n) = Sum_{k=1..n-1} A120072(n,k) for n >= 2.

A120075 Row sums of triangle A120073 (denominator triangle for H atom spectrum).

Original entry on oeis.org

4, 45, 176, 750, 1101, 4459, 6080, 13284, 16350, 46585, 33954, 109850, 92463, 142705, 198400, 432344, 255096, 761349, 500355, 824866, 925529, 2007555, 1044616, 2612500, 2158130, 3301641, 2848741

Offset: 2

Wolfdieter Lang, Jul 20 2006

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A120070, A120072, A120073, A120074, A120076, A120077.

A120073:= func< n,k | Denominator(1/k^2 - 1/n^2) >;
[(&+[A120073(n,k): k in [1..n-1]]): n in [2..50]]; // G. C. Greubel, Apr 24 2023

A120075[n_]:= Sum[Denominator[1/k^2 -1/n^2], {k,n-1}];
Table[A120075[n], {n,2,50}] (* G. C. Greubel, Apr 24 2023 *)

def A120073(n,k): return denominator(1/k^2 - 1/n^2)
[sum(A120073(n,k) for k in range(1,n)) for n in range(2,51)] # G. C. Greubel, Apr 24 2023

a(n) = Sum_{k=1..n-1} A120073(n,k), for n >= 2.

A120076 Numerators of row sums of rational triangle A120072/A120073.

Original entry on oeis.org

3, 37, 169, 4549, 4769, 241481, 989549, 9072541, 1841321, 225467009, 227698469, 38801207261, 39076419341, 196577627041, 790503882349, 229526961468061, 230480866420061, 83512167402400421, 3351610394325821

Offset: 2

Wolfdieter Lang, Jul 20 2006

The corresponding denominators are given by A120077.

See the W. Lang link under A120072 for more details.

			The rationals a(m)/A120077(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ...).

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A120070, A120072, A120073, A120074, A120075, A120077.

A120076:= func< n | Numerator( (&+[1/k^2: k in [1..n]]) -1/n) >;
[A120076(n): n in [2..30]]; // G. C. Greubel, Apr 24 2023

Table[Numerator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 24 2023 *)

def A120076(n): return numerator(harmonic_number(n,2) - 1/n)
[A120076(n) for n in range(2,31)] # G. C. Greubel, Apr 24 2023

a(n) = numerator(r(m)), with the rationals r(m) = Sum_{n=1..m-1} A120072(m,n)/A120073(m,n), m >= 2.
The rationals are r(m) = Zeta(2; m-1) - (m-1)/m^2, m >= 2, with the partial sums Zeta(2; n) = Sum_{k=1..n} 1/k^2. See the W. Lang link in A103345.
O.g.f. for the rationals r(m), m>=2: log(1-x) + polylog(2,x)/(1-x).

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A309829 Numbers k for which A120077(k) != A007407(k).

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A120072 Numerator triangle for hydrogen spectrum rationals.

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A120073 Denominator triangle for hydrogen spectrum rationals.

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A120074 Row sums of triangle A120072 (numerator triangle for H atom spectrum).

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A120075 Row sums of triangle A120073 (denominator triangle for H atom spectrum).

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A120076 Numerators of row sums of rational triangle A120072/A120073.

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