A292608 -id:A292608 - cp's OEIS frontend

A371638 a(n) = 2*n + valuation(n, 3) with valuation(n, 3) = A007949(n).

2, 4, 7, 8, 10, 13, 14, 16, 20, 20, 22, 25, 26, 28, 31, 32, 34, 38, 38, 40, 43, 44, 46, 49, 50, 52, 57, 56, 58, 61, 62, 64, 67, 68, 70, 74, 74, 76, 79, 80, 82, 85, 86, 88, 92, 92, 94, 97, 98, 100, 103, 104, 106, 111, 110, 112, 115, 116, 118, 121, 122, 124, 128, 128

Offset: 1

Peter Luschny, Mar 30 2024

See A371639 for the connection with Voronoi's congruence.

Cf. A007949, A371639, A292608 (c=2).

A371638 := n -> 2*n + padic:-ordp(n, 3): seq(A371638(n), n = 1..64);

Array[2 # + IntegerExponent[#, 3] &, 64] (* Michael De Vlieger, Mar 31 2024 *)

def A371638(n): return 2 * n + valuation(n, 3)
print([A371638(n) for n in range(1, 65)])

a(n) = valuation(denominator(Voronoi(3, n))) where Voronoi(c, n) = ((c^n - 1) * bernoulli(n)) / (n * c^(n - 1)).

A255932 a(n) is the denominator of Gamma(n+1/2)^2/(2nPi), the value of an integral with sinh in the denominator.

Original entry on oeis.org

8, 64, 128, 2048, 2048, 16384, 32768, 1048576, 524288, 4194304, 8388608, 134217728, 134217728, 1073741824, 2147483648, 137438953472, 34359738368, 274877906944, 549755813888, 8796093022208, 8796093022208, 70368744177664, 140737488355328

Offset: 1

Jean-François Alcover, Mar 11 2015

Conjecture: a(n) <= 2^(3*n). - Vaclav Kotesovec, Mar 11 2015

			1/8, 9/64, 75/128, 11025/2048, 178605/2048, 36018675/16384, 2608781175/32768, ...

MathOverflow, An identity involving an infinite integral with a sinh in the denominator

Cf. A255931 (numerators), A292608.

seq(2^A292608(n), n=1..23); # Peter Luschny, Sep 23 2017

a[n_] := Gamma[n+1/2]^2/(2*n*Pi) // Denominator; Array[a, 30]
Table[(2*n)!^2 / (n * 2^(4*n+1) * n!^2), {n, 1, 20}] // Denominator (* Vaclav Kotesovec, Mar 11 2015 *)
b[n_] := 2*n + 1 + IntegerExponent[n,2]; Table[2^b[n], {n,1,23}] (* Peter Luschny, Sep 23 2017 *)

The n-th fraction also equals the n-th coefficient in the expansion of 2F1(1/2,1/2; 1; x) * n!*(n-1)!/2.

a(n) = 2^(2*n + 1 + valuation(n, 2)) = 2^A292608(n). - Peter Luschny, Sep 23 2017

A371639 a(n) = numerator(Voronoi(3, 2n)) where Voronoi(c, n) = ((c^n - 1) Bernoulli(n)) / (n * c^(n - 1)).

Original entry on oeis.org

2, -2, 26, -82, 1342, -100886, 1195742, -57242642, 31945440878, -276741323122, 26497552755742, -9169807783193206, 418093081574417342, -66910282127782482482, 37158050152167281792026, -2626016090388858294953362, 632184834985453539204543742, -1543534415494449734887808117378

Offset: 1

Peter Luschny, Mar 30 2024

To begin with, we observe that if c = 2, then the numerator of Voronoi(2, 2*n) is the same as the numerator of Euler(2*n - 1, 1), which is equal to (-1)^n*A002425(n). Similarly, the denominator of Voronoi(2, 2*n) is A255932(n), which is equal to 2^A292608(n). The rational sequence r(n) = a(n) / A371640(n) examines the corresponding relationships in the case c = 3.

The function Voronoi, which is defined in the Name, was inspired by Voronoi's congruence. This congruence states that for any even integer k >= 2 and all positive coprime integers c, n: (c^k - 1)*N(k) == k*c^(k-1)*D(k)*Sum_{m=1..n-1} m^(k-1)* floor(m*c / n) mod n, where N(k) = numerator(Bernoulli(k)), D(k) = denominator( Bernoulli(k)) and gcd(N(k), D(k)) = 1.

			r(n) = 2/9, -2/81, 26/2187, -82/6561, 1342/59049, -100886/1594323, ...

Emma Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. Math. 39 (1938), 350-360.
Štefan Porubský, Further Congruences Involving Bernoulli Numbers, Journal of Number Theory 16, 87-94 (1983).
Georgy Feodosevich Voronyi, On Bernoulli numbers, Comm. Charkou Math. Sot. 2, 129-148 (1890) (in Russian).

Digital Library of Mathematical Functions, Voronoi's congruence.

Cf. A371640 (denominator), A371638.

Cf. A002425, A255932, A292608.

Voronoi := (a, k) -> ((a^k - 1) * bernoulli(k)) / (k * a^(k - 1)):
VoronoiList := (a, len) -> local k; [seq(Voronoi(a, 2*k), k = 1..len)]:
numer(VoronoiList(3, 18));

a(n) = Voronoi(3, 2*n) * 3^(2*n + valuation(n, 3)).

cp's OEIS Frontend

A371638 a(n) = 2*n + valuation(n, 3) with valuation(n, 3) = A007949(n).

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A255932 a(n) is the denominator of Gamma(n+1/2)^2/(2nPi), the value of an integral with sinh in the denominator.

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A371639 a(n) = numerator(Voronoi(3, 2n)) where Voronoi(c, n) = ((c^n - 1) Bernoulli(n)) / (n * c^(n - 1)).

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cp's OEIS Frontend

A371638 a(n) = 2*n + valuation(n, 3) with valuation(n, 3) = A007949(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A255932 a(n) is the denominator of Gamma(n+1/2)^2/(2*n*Pi), the value of an integral with sinh in the denominator.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A371639 a(n) = numerator(Voronoi(3, 2*n)) where Voronoi(c, n) = ((c^n - 1) * Bernoulli(n)) / (n * c^(n - 1)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Formula

A255932 a(n) is the denominator of Gamma(n+1/2)^2/(2nPi), the value of an integral with sinh in the denominator.

A371639 a(n) = numerator(Voronoi(3, 2n)) where Voronoi(c, n) = ((c^n - 1) Bernoulli(n)) / (n * c^(n - 1)).