A160014 -id:A160014 - cp's OEIS frontend

A337945 Numbers m with a solution (s,t,k) such that s^2 + t^2 = k*m, s + t = m, 1 <= s <= t and 1 <= k <= m - 1.

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 108, 110, 112, 114, 116, 117, 118, 120

Offset: 1

Wesley Ivan Hurt, Oct 01 2020

nonn

			8 is in the sequence since it has the solutions (s,t,k) = (4,4,4) and (2,6,5) such that s^2 + t^2 = k*m, s + t = m, 1 <= s <= t and 1 <= k <= m - 1.
9 is in the sequence since it has the solution (s,t,k) = (3,6,5) such that s^2 + t^2 = k*m, s + t = m, 1 <= s <= t and 1 <= k <= m - 1.

Cf. A337101, A160014, A039956.

# Quite inefficient compared to the conjectured formula.
KD := (n, k) -> Physics:-KroneckerDelta[n, k]:
S := k -> local i, j; add(add(KD((i^2 + (k - i)^2)/j , k), j = 1..k-1),
i = 1..floor(k/2)): select(k -> S(k) > 0, [seq(k, k = 1..40)]); # Peter Luschny, Jun 08 2023

Table[If[Sum[Sum[KroneckerDelta[(i^2 + (n - i)^2)/k, n], {k, n - 1}], {i, Floor[n/2]}] > 0, n, {}], {n, 120}] // Flatten

k is a term <=> Sum_{i=1..floor(k/2)} Sum_{j=1..k-1} KroneckerDelta((i^2 + (k - i)^2)/j, k) > 0.

Conjecture: k is a term <=> k * Clausen(k, 1) <> 2 * Clausen(k, 0), (Clausen = A160014). In other words: k is in this sequence iff it is not an odd squarefree number. - Peter Luschny, Jun 08 2023

A047817 Denominators of Hurwitz numbers H_n (coefficients in expansion of Weierstrass P-function).

Original entry on oeis.org

10, 10, 130, 170, 10, 130, 290, 170, 4810, 410, 10, 2210, 530, 290, 7930, 170, 10, 351130, 10, 6970, 3770, 890, 10, 214370, 1010, 530, 524290, 557090, 10, 325130, 10, 170, 130, 1370, 290, 5969210, 1490, 10, 1081730, 6970, 10, 3770

Offset: 1

N. J. A. Sloane

Hurwitz showed (see Katz, eqn. 9) that a(n) = product of the prime p = 2 and the primes p of the form 4*k + 1 such that p - 1 divides 4*n. It follows that a(n) is a divisibility sequence, that is, if n | m then a(n) | a(m). - Peter Bala, Jan 08 2014

			Hurwitz numbers H_1, H_2, ... = 1/10, 3/10, 567/130, 43659/170, 392931/10, ...

F. Lemmermeyer, Reciprocity Laws, Springer-Verlag, 2000; see p. 276.

T. D. Noe, Table of n, a(n) for n = 1..1000
L. Carlitz, The coefficients of the lemniscate function, Math. Comp., 16 (1962), 475-478.
A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII.
A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII. [Annotated scanned copy]
N. M. Katz, The Congruences of Clausen - von Staudt and Kummer for Bernoulli-Hurwitz Numbers, Mathematische Annalen 216, 1-4 (1975)
Alexei Pantchichkine, Constructions of p-adic L-functions and admissible measures for Hermitian modular forms, Number Theory [math.NT], 2018.
Alexei Pantchichkine, Algebraic differential operators on arithmetic automorphic forms, modular distributions, p-adic interpolation of their critical l values via BGG modules and Hecke algebras, J. Math. Math. Sci., Thang Long Univ. (Viet Nam, 2022) Vol. 1, No. 4, 1-26.
Index to divisibility sequences

For numerators see A002306.

Cf. A160014.

H := proc(n) local k; option remember; if n = 1 then 1/10 else 3*add((4*k - 1)*(4*n - 4*k - 1)*binomial(4*n, 4*k)*H(k)*H(n - k), k = 1 .. n - 1)/( (2*n - 3)*(16*n^2 - 1)) fi; end;
a := n -> denom(H(n));
# Implementation based on Hurwitz's extension of Clausen's theorem:
GenClausen := proc(n) local k,S; map(k->k+1, numtheory[divisors](n));
    S := select(p-> isprime(p) and p mod 4 = 1, %);
    if S <> {} then 2*mul(k,k=S) else NULL fi end:
A047817_list := proc(n) local i; seq(GenClausen(i),i=1..4*n) end;
A047817_list(42); # Peter Luschny, Oct 03 2011
# Implementation based on Weierstrass's P-function:
c := n -> (n*(4*n-2)!/(2^(4*n-2)))*coeff(series(WeierstrassP(z,4,0),z, 4*n+2),z,4*n-2); a := n -> denom(c(n)); seq(a(n), n=1..42); # Peter Luschny, Aug 18 2014

a[1] = 1/10; a[n_] := a[n] = (3/(2*n - 3)/(16*n^2 - 1))* Sum[(4*k - 1)*(4*n - 4*k - 1)*Binomial[4*n, 4*k]*a[k]* a[n - k], {k, 1, n - 1}]; Denominator[ Table[a[n], {n, 1, 42}]] (* Jean-François Alcover, Oct 18 2011, after PARI *)
a[ n_] := If[ n < 1, 0, Denominator[ 2^(-4 n) (4 n)! SeriesCoefficient[ 1 - x WeierstrassZeta[ x, {4, 0}], {x, 0, 4 n}]]]; (* Michael Somos, Mar 05 2015 *)

do(lim)=v=vector(lim); v[1]=1/10; for(n=2,lim,v[n]=3/(2*n-3)/(16*n^2-1)*sum(k=1,n-1,(4*k-1)*(4*n-4*k-1)*binomial(4*n,4*k)*v[k]*v[n-k])) \\ Henri Cohen, Mar 18 2002

Let P be the Weierstrass P-function satisfying P'^2 = 4*P^3 - 4*P. Then P(z) = 1/z^2 + Sum_{n>=1} 2^(4n)*H_n*z^(4n-2)/(4n*(4n-2)!).
Sum_{ (r, s) != (0, 0) } 1/(r+si)^(4n) = (2w)^(4n)*H_n/(4n)! where w = 2 * Integral_{0..1} dx/(sqrt(1-x^4)).
See PARI line for recurrence.

A166120 a(n) = A027642(n-1) / A089026(n).

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 6, 1, 30, 1, 6, 1, 210, 1, 6, 1, 30, 1, 42, 1, 330, 1, 6, 1, 2730, 1, 6, 1, 30, 1, 462, 1, 510, 1, 6, 1, 51870, 1, 6, 1, 330, 1, 42, 1, 690, 1, 6, 1, 46410, 1, 66, 1, 30, 1, 798, 1, 870, 1, 6, 1, 930930, 1, 6, 1, 510, 1, 966, 1, 30, 1, 66, 1, 1919190, 1, 6, 1, 30, 1, 42, 1

Offset: 1

Paul Curtz, Oct 07 2009

nonn

As in A166062, the offset is rather arbitrary.
The sequence contains numbers like 210 which are not in A006954.
One could also consider dividing by the largest prime divisor of A027642 instead of A089026, which yields 1, 1, 2, 1, 6, 1, 6, 1, 6, 1, 6, 1, 210, 1, 2, 1, 30, 1, 42, 1, 30, ... as an alternative version.
These are the Clausen numbers based on the proper divisors of n whereas the classical Clausen numbers A160014 are based on all divisors of n. (The proper divisors are the divisors of n that are less than n.) - Peter Luschny, Aug 20 2022

Cf. A027642, A089026, A160014.

A027642 := proc(n) denom(bernoulli(n)) ; end:
A089026 := proc(n) if isprime(n) then n; else 1; end if; end proc:
A166120 := proc(n) A027642(n-1)/A089026(n) ; end proc: seq(A166120(n), n=1..80) ; # R. J. Mathar, Mar 25 2010
# Second program, assuming offset 0:
clausen := proc(n) if irem(n,2)=1 then 1 else numtheory[divisors](n) minus {n};
map(i -> i+1, %); select(isprime, %); mul(i, i=%) fi end:
seq(clausen(n), n = 0..79); # Peter Luschny, Aug 20 2022

Extended by R. J. Mathar, Mar 25 2010

A346463 a(n) = 6 * GaussBinomial(2n, 2, 2) / denominator(Bernoulli(2n, 1)).

Original entry on oeis.org

0, 1, 7, 93, 2159, 15841, 6141, 44731051, 8421119, 86113647, 3331843885, 127479517837, 103104368637, 750599904340651, 82824819807611, 80500035008073, 36170086393773823, 49191317521302203051, 2460603943675971, 12592977287514948283051, 89351501819019263845

Offset: 0

Peter Luschny, Jul 19 2021

nonn

Cf. A006095, A002445, A160014, A346464.

a := n -> (4^n - 2)*(4^n - 1) / mul(i, i=select(isprime, map(i->i+1, numtheory[divisors] (2*n)))): seq(a(n), n = 0..20);

Table[6 QBinomial[2 n, 2, 2] / Denominator[BernoulliB[2 n, 1]], {n, 0, 20}]

a(n) = (4^n - 2)*(4^n - 1)/Clausen(2*n, 1), where Clausen(n, k) = A160014(n, k).

A363395 a(n) = n * Clausen(n, 1) / Clausen(n, 0).

Original entry on oeis.org

0, 2, 6, 2, 60, 2, 42, 2, 120, 6, 66, 2, 5460, 2, 6, 2, 4080, 2, 2394, 2, 660, 2, 138, 2, 10920, 10, 6, 18, 1740, 2, 14322, 2, 8160, 2, 6, 2, 11515140, 2, 6, 2, 54120, 2, 1806, 2, 1380, 6, 282, 2, 371280, 14, 330, 2, 3180, 2, 7182, 2, 3480, 2, 354, 2, 113573460

Offset: 0

Peter Luschny, Jun 08 2023

nonn

Cf. A160014, A363402, A363403.

# Using function 'Clausen' from A160014.
a := n -> n * Clausen(n, 1) / Clausen(n, 0):
seq(a(n), n = 0..60);

a(n) = n * A160014(n, 1) / A160014(n, 0).

A363402 a(n) = n * (4^n - 2^n) / Clausen(n, 0).

Original entry on oeis.org

0, 2, 12, 56, 480, 992, 4032, 16256, 261120, 784896, 1047552, 4192256, 33546240, 67100672, 268419072, 1073709056, 34359214080, 17179738112, 206157643776, 274877382656, 2199021158400, 4398044413952, 17592181850112, 70368735789056, 1125899839733760, 5629499366440960

Offset: 0

Peter Luschny, Jun 08 2023

nonn

Cf. A160014, A363403, A363395.

# Using function 'Clausen' from A160014.
a := n -> n * (4^n - 2^n) / Clausen(n, 0):
seq(a(n), n = 0..25);

a(n) = n * A020522(n) / A160014(n, 0).

A363403 a(n) = (4^n - 2^n) / Clausen(n, 1).

Original entry on oeis.org

0, 1, 2, 28, 8, 496, 96, 8128, 2176, 130816, 15872, 2096128, 6144, 33550336, 44736512, 536854528, 8421376, 8589869056, 86114304, 137438691328, 3331850240, 2199022206976, 127479578624, 35184367894528, 103104380928, 562949936644096, 750599926710272, 9007199187632128

Offset: 0

Peter Luschny, Jun 08 2023

nonn

Cf. A160014, A363402, A363395.

# Using function 'Clausen' from A160014.
a := n -> (4^n - 2^n) / Clausen(n, 1):
seq(a(n), n = 0..25);

a(n) = A020522(n) / A160014(n, 1).

A368093 Cumulative products of the generalized Clausen numbers. Array read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 12, 6, 1, 1, 9, 24, 12, 1, 5, 5, 135, 720, 60, 1, 1, 25, 5, 405, 1440, 360, 1, 7, 7, 875, 175, 8505, 60480, 2520, 1, 1, 49, 7, 4375, 175, 127575, 120960, 5040, 1, 1, 1, 343, 49, 21875, 875, 382725, 3628800, 15120

Offset: 0

Peter Luschny, Dec 12 2023

A160014 are the generalized Clausen numbers, for m = 0 the formula computes the cumulative radical A048803, and for m = 1 the Hirzebruch numbers A091137.

			Array A(m, n) starts:
  [0] 1, 1,  2,   6,   12,     60,     360,      2520, ...  A048803
  [1] 1, 2, 12,  24,  720,   1440,   60480,    120960, ...  A091137
  [2] 1, 3,  9, 135,  405,   8505,  127575,    382725, ...  A368092
  [3] 1, 1,  5,   5,  175,    175,     875,       875, ...
  [4] 1, 5, 25, 875, 4375,  21875,  765625,  42109375, ...
  [5] 1, 1,  7,   7,   49,     49,    3773,      3773, ...
  [6] 1, 7, 49, 343, 2401, 184877, 1294139, 117766649, ...
  [7] 1, 1,  1,   1,   11,     11,     143,       143, ...
  [8] 1, 1,  1,  11,   11,    143,    1573,      1573, ...
  [9] 1, 1, 11,  11, 1573,   1573,   17303,     17303, ...

Cf. A160014, A048803 (m=0), A091137 (m=1), A368092 (m=2).

Cf. A171080, A238963, A368116, A368048.

from functools import cache
def Clausen(n, k):
    return mul(s for s in map(lambda i: i+n, divisors(k)) if is_prime(s))
@cache
def CumProdClausen(m, n):
    return Clausen(m, n) * CumProdClausen(m, n - 1) if n > 0 else 1
for m in range(10): print([m], [CumProdClausen(m, n) for n in range(8)])

A(m, n) = A160014(m, n) * A(m, n - 1) for n > 0 and A(m, 0) = 1.

A165823 Large denominators of Bernoulli numbers. Mix A002445, 2*A141421 .

Original entry on oeis.org

1, 2, 6, 24, 30, 1440, 42, 120960, 30, 7257600, 66, 958003200, 2730, 5230697472000, 6, 62768369664000, 510, 64023737057280000

Offset: 0

Paul Curtz, Sep 28 2009

b(n)=a(2n+1)/a(2n) =2,4,48,2880,241920,145152,= 2*(1,2,24,1440,=1,2*A141421). Among other denominators, A027642,A141056,A164020. 2*A141421 is second bisection of A091137 which is linked to Bernoulli via A027760. See A160014,von Staudt-Clausen theorem.

A213621 The denominator of the Bernoulli polynomial B(n,x) divided by the Clausen number C(n), A144845(n)/A141056(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 1, 105, 5, 3, 1, 15, 5, 105, 7, 165, 5, 15, 1, 273, 7, 7, 1, 15, 1, 231, 77, 1785, 35, 3, 1, 25935, 455, 105, 7, 1155, 55, 1155, 7, 2415, 35, 105, 1, 3315, 221, 429, 11, 165, 55, 399, 19, 435, 5, 15, 1, 465465, 5005, 2145

Offset: 0

Peter Luschny, Jun 16 2012

nonn

Cf. A213623.

# Clausen(n,k) defined in A160014.
seq(denom(bernoulli(i,x))/Clausen(i,1), i=0..63);

c[0, ] = 1; c[n, k_] := Times @@ (Select[Divisors[n], PrimeQ[#+k]&] + k);
Table[Denominator[BernoulliB[i, x] // Together]/c[i, 1], {i, 0, 63}] (* Jean-François Alcover, Aug 02 2019 *)

cp's OEIS Frontend

A337945 Numbers m with a solution (s,t,k) such that s^2 + t^2 = k*m, s + t = m, 1 <= s <= t and 1 <= k <= m - 1.

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A047817 Denominators of Hurwitz numbers H_n (coefficients in expansion of Weierstrass P-function).

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PARI

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A166120 a(n) = A027642(n-1) / A089026(n).

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A346463 a(n) = 6 * GaussBinomial(2*n, 2, 2) / denominator(Bernoulli(2*n, 1)).

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A363395 a(n) = n * Clausen(n, 1) / Clausen(n, 0).

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A363402 a(n) = n * (4^n - 2^n) / Clausen(n, 0).

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A363403 a(n) = (4^n - 2^n) / Clausen(n, 1).

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A368093 Cumulative products of the generalized Clausen numbers. Array read by ascending antidiagonals.

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A165823 Large denominators of Bernoulli numbers. Mix A002445, 2*A141421 .

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A346463 a(n) = 6 * GaussBinomial(2n, 2, 2) / denominator(Bernoulli(2n, 1)).