A051418 -id:A051418 - cp's OEIS frontend

A119937 Triangle of numbers related to the spectrum of the hydrogen (H) atom.

3, 32, 5, 135, 27, 7, 3456, 756, 256, 81, 3500, 800, 300, 125, 44, 172800, 40500, 16000, 7425, 3456, 1300, 694575, 165375, 67375, 33075, 17199, 8575, 3375, 6272000, 1509200, 627200, 318500, 175616, 98000, 51200

Offset: 2

Wolfdieter Lang, Jul 20 2006

The rational number triangle r(m,n):=A120072(m,n)/A120073(m,n), used to compute the spectral series of the hydrogen atom, is mapped to this nonnegative number triangle by multiplying the least common multiples (LCM) for each row m.

			[3]; [32,5]; [135,27,7]; [3456,756,256,81]; [3500,800,300,125,44]; ...

W. Lang: First ten rows.

The LCM sequence which has been used here is [4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 6350400, 768398400, ...] = A051418(m) = (A003418(m))^2 = (2*A025555(m-1))^2, m >= 2.

The row sums give A119938.

a(m,n) = r(m,n)*lcm_{k=1..m-1} seq(r(m,k)) with r(m,n) = 1/n^2 - 1/m^2 = A120072(m,n)/A120073(m,n), m >= 2, n = 1..m-1.

A120078 Coefficient triangle of numerator polynomials appearing in certain column o.g.f.s related to the H-atom spectrum.

Original entry on oeis.org

1, 4, -3, 36, -27, -5, 144, -108, -20, -7, 3600, -2700, -500, -175, -81, 3600, -2700, -500, -175, -81, -44, 176400, -132300, -24500, -8575, -3969, -2156, -1300, 705600, -529200, -98000, -34300, -15876, -8624, -5200, -3375, 6350400, -4762800, -882000, -308700, -142884, -77616, -46800, -30375, -20825

Offset: 1

Wolfdieter Lang, Jul 20 2006

The row polynomials P(n,x) = Sum_{k=1..n-1} a(n,k)*x^k, n >= 1, appear in the numerator of the o.g.f. for column n of the triangle of rationals A120072(m,n)/A120073(m,n), m >= 2, n = 1..m-1. P(n,x) has degree n-1.

See the W. Lang link under A120072 for the precise form of the o.g.f.s: G(x,n) = -dilog(1-x) + x*P(n,4)/*(A(n)*(n^2)*(1-x)), with A(n) = [1, 1, 4, 9, 144, 100, 3600, 11025, 78400, 63504, ...] = conjectured to be A027451(n), n >= 1.

			For n=2 the o.g.f. of A120072(m,2)/A120073(m,2) (=[5/36, 3/16, 21/100, 2/9, ...]) is G(x,2) = -dilog(1-x) + x*P(2,x)/(1*4*(1-x)) = -dilog(1-x) + x*(4-3*x)/(4*(1-x)).
Triangle begins:
       1;
       4,      -3;
      36,     -27,     -5;
     144,    -108,    -20,    -7;
    3600,   -2700,   -500,  -175,   -81;
    3600,   -2700,   -500,  -175,   -81,   -44;
  176400, -132300, -24500, -8575, -3969, -2156, -1300;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Wolfdieter Lang, First ten rows

Row sums (unsigned) give A120079.
Signed row sums conjectured to coincide with A027451.
Cf. A027451, A051418, A120072, A120073, A120079.

f:= func< n | n eq 1 select 1 else 1/n^2 -1/(n-1)^2 >;
A120078:= func< n,k | (Lcm([1..n]))^2*f(k) >;
[A120078(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 26 2023

Table[(Apply[LCM, Range[n]])^2*If[k==1, 1, (1-2*k)/(k*(k-1))^2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 26 2023 *)

def f(k): return 1 if (k==1) else 1/k^2 - 1/(k-1)^2
def A120078(n,k): return (lcm(range(1, n+1)))^2*f(k)
flatten([[A120078(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Apr 26 2023

T(n, k) = A051418(n) * (1 if k = 1 otherwise 1/k^2 - 1/(k-1)^2). - G. C. Greubel, Apr 26 2023

A120079 Unsigned row sums of triangle A120078.

Original entry on oeis.org

1, 7, 68, 279, 7056, 7100, 349200, 1400175, 12622400, 12637296, 1530446400, 1531460700, 258950260800, 259056111600, 259141506624, 1036845584775, 299715332716800, 299771444772800, 108234634597689600, 108249271042728816, 108261866776377600, 108272784263716800

Offset: 1

Wolfdieter Lang, Jul 20 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000

Signed row sums conjectured to be A027451(n), which also appears in the denominator of o.g.f.s. G(x, n) given in A120078 as numbers A(n).

Cf. A000290, A051418, A056220.

[(2-1/n^2)*(Lcm([1..n]))^2: n in [1..40]]; // G. C. Greubel, Apr 26 2023

Table[(2-1/n^2)*(Apply[LCM, Range[n]])^2, {n, 40}] (* G. C. Greubel, Apr 26 2023 *)

def A120079(n): return (2 - 1/n^2)*(lcm(range(1, n+1)))^2
[A120079(n) for n in range(1,41)] # G. C. Greubel, Apr 26 2023

a(n) = Sum_{k=1..n} abs(A120078(n,k)), n >= 1.
From G. C. Greubel, Apr 26 2023: (Start)
a(n) = (2 - 1/n^2)*A051418(n).
a(n) = A056220(n)*A051418(n)/A000290(n). (End)

Terms a(11) onward added by G. C. Greubel, Apr 26 2023

A053608 Numbers x = LCM(1,2,...,k) such that x^2 + 1 is prime.

Original entry on oeis.org

1, 2, 6, 420, 360360, 718766754945489455304472257065075294400

Offset: 1

Labos Elemer, Feb 09 2000

The next term has k > 10^4, if it exists. - Amiram Eldar, Aug 23 2024

Cf. A003418, A049536, A046029, A051418, A051451, A051739, A053609.

Select[FoldList[LCM, 1, Select[Range[100], PrimePowerQ]], PrimeQ[#^2 + 1] &] (* Amiram Eldar, Aug 23 2024 *)

A053609 Primes of form x^2+1 where x = LCM(1,2,...,k) for some k.

Original entry on oeis.org

2, 5, 37, 176401, 129859329601, 516625648014869290354797521879383114125823989794742396526049715541246671360001

Offset: 1

Labos Elemer, Feb 09 2000

The next term has k > 10^4, if it exists. - Amiram Eldar, Aug 23 2024

Cf. A049536, A046029, A051739, A051418, A051451, A053608.

Select[FoldList[LCM, 1, Select[Range[100], PrimePowerQ]]^2 + 1, PrimeQ] (* Amiram Eldar, Aug 23 2024 *)

a(n) = A053608(A053608(n)) = A053608(n)^2 + 1. - Amiram Eldar, Aug 23 2024

A257895 Square array read by ascending antidiagonals where T(n,k) is the mean number of maxima in a set of n random k-dimensional real vectors (denominators).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 4, 1, 1, 12, 36, 8, 1, 1, 60, 144, 216, 16, 1, 1, 20, 3600, 1728, 1296, 32, 1, 1, 140, 3600, 216000, 20736, 7776, 64, 1, 1, 280, 176400, 72000, 12960000, 248832, 46656, 128, 1, 1, 2520, 705600, 24696000, 12960000, 777600000

Offset: 1

Jean-François Alcover, May 12 2015

			Array of fractions begins:
1,      1,          1,             1,                 1,                    1, ...
1,    3/2,        7/4,          15/8,             31/16,                63/32, ...
1,   11/6,      85/36,       575/216,         3661/1296,           22631/7776, ...
1,  25/12,    415/144,     5845/1728,       76111/20736,        952525/248832, ...
1, 137/60, 12019/3600, 874853/216000, 58067611/12960000, 3673451957/777600000, ...
1,  49/20, 13489/3600,  336581/72000, 68165041/12960000,   483900263/86400000, ...
...
Row 2 (denominators) is A000079 (powers of 2),
Row 3 is A000400 (powers of 6),
Row 4 is A001021 (powers of 12),
Row 5 is A159991,
Row 6 is not in the OEIS.
Column 2 (denominators) is A002805 (denominators of harmonic numbers),
Column 3 is A051418 (lcm(1..n)^2),
Column 4 is not in the OEIS.

Zhi-Dong Bai, Chern-Ching Chao, Hsien-Kuei Hwang and Wen-Qi Liang, On the variance of the number of maxima in random vectors and its applications, The Annals of Applied Probability 1998, Vol. 8, No. 3, 886-895.
O. E. Barndorff-Nielsen and M. Sobel, On the distribution of the number of admissible points in a vector random sample. Theory Probab. Appl. 11 249-269.

Cf. A257894 (numerators).

T[n_, k_] := Sum[(-1)^(j - 1)*j^(1 - k)*Binomial[n, j], {j, 1, n}]; Table[T[n - k + 1, k] // Denominator, {n, 1, 12}, {k, 1, n}] // Flatten

T(n,k) = Sum_{j=1..n} (-1)^(j-1)*j^(1-k)*binomial(n,j).

A309556 Composite numbers m such that m divides Sum_{k=1..m-1} (lcm(1,2,...,(m-1)) / k)^2.

Original entry on oeis.org

52781, 782957, 1395353, 2602439

Offset: 1

Amiram Eldar and Thomas Ordowski, Aug 07 2019

Composites m such that m | H_2(m-1) * lcm(1^2,2^2,...,(m-1)^2), where H_2(m) = 1/1^2 + 1/2^2 + ... + 1/m^2.
By Wolstenholme's theorem, if p > 3 is a prime, then p divides the numerator of H_2(p-1) and thus H_2(p-1) * lcm(1,2^2,...,(p-1)^2) == 0 (mod p). This sequence is formed by the pseudoprimes that are solutions of this congruence.
a(5) > 10^7 if it exists.

Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.
Wikipedia, Wolstenholme's theorem.

Cf. A007406, A007407, A025529 (see our comment), A051418.

s = 0; c = 1; n = 1; seq = {}; Do[s += 1/n^2; c = LCM[c, n^2]; n++; If[CompositeQ[n] && Divisible[s*c, n], AppendTo[seq, n]], {2 * 10^6}]; seq

cp's OEIS Frontend

A119937 Triangle of numbers related to the spectrum of the hydrogen (H) atom.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A120078 Coefficient triangle of numerator polynomials appearing in certain column o.g.f.s related to the H-atom spectrum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A120079 Unsigned row sums of triangle A120078.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A053608 Numbers x = LCM(1,2,...,k) such that x^2 + 1 is prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A053609 Primes of form x^2+1 where x = LCM(1,2,...,k) for some k.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A257895 Square array read by ascending antidiagonals where T(n,k) is the mean number of maxima in a set of n random k-dimensional real vectors (denominators).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A309556 Composite numbers m such that m divides Sum_{k=1..m-1} (lcm(1,2,...,(m-1)) / k)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica