A160144 -id:A160144 - cp's OEIS frontend

A160145 a(n) = the odd number 2n+1 minus the numerator of (2n+1)/(2^(2n+1)-1).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 90, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 150

Offset: 0

Peter Luschny, May 03 2009

nonn

Explains the similarity of the sequences A009843 and A160143. (Cf. also the pair A036279 and A156769.) The first nonzero values occur at n = 10, 31, 52 and 73.

Previous name was: Odd numbers 2n+1 minus the numerators of (2n+1)/(4^(2n+1)-2^(2n+1)), (A005408 - A160144). - Altug Alkan, Apr 21 2018

Altug Alkan, Table of n, a(n) for n = 0..10000

Cf. A005408, A009843, A160143, A160144.

seq((2*n+1)-numer((2*n+1)/(4^(2*n+1)-2^(2*n+1))),n=0..77);
seq((2*n+1)-numer((2*n+1)/(2^(2*n+1)-1)),n=0..100); # Altug Alkan, Apr 21 2018

Array[# - Numerator[#/(2^# - 1)] &[2 # + 1] &, 78, 0] (* Michael De Vlieger, Apr 21 2018 *)

forstep(k=1, 1e2, 2, print1(k - numerator(k/(2^k-1)), ", ")); \\ Altug Alkan, Apr 21 2018

a(n) = A005408(n) - A160144(n).

Name simplified by Altug Alkan, Apr 21 2018

A160143 a(n) = Numerator((-1)^nEuler(2n)(2n+1)/(4^(2n+1)-2^(2n+1))), where Euler(n) = A122045(n).

Original entry on oeis.org

1, 3, 25, 427, 12465, 555731, 35135945, 2990414715, 329655706465, 45692713833379, 1111113564712575, 1595024111042171723, 387863354088927172625, 110350957750914345093747

Offset: 0

Peter Luschny, May 03 2009

Resembles the coefficients of the series for x/cos(x).
The first difference with sequence A009843 (expansion of x/cos(x)) occurs at a(10). An explanation can be found in the similarity of the numerators of (2*n+1)/(2^(2*n+1)-1) and the odd numbers 2n+1 (cf. A160144).
Similarly, A156769 resembles A036279 (from the expansion of tan(x)).

Cf. A009843, A122045, A160144, A160145, A036279, A156769.

a := n -> (-1)^iquo(n,2)*euler(n)*(n+1)/(4^(n+1)-2^(n+1));
seq(numer(a(2*n)),n=0..13);

A193476 The denominators of the Bernoulli secant numbers at odd indices.

Original entry on oeis.org

2, 56, 992, 16256, 261632, 4192256, 67100672, 1073709056, 17179738112, 274877382656, 628292059136, 70368735789056, 1125899873288192, 18014398375264256, 288230375614840832, 4611686016279904256, 73786976286248271872, 1180591620683051565056

Offset: 0

Peter Luschny, Aug 17 2011

Denominator of the coefficient [x^(2n)] of sec(x)*(2*n+1)!/(4*16^n-2*4^n), that is, a(n) is the denominator of A000364(n)*(2*n+1)/(4*16^n-2*4^n). [Edited by Altug Alkan, Apr 22 2018]
Numerators are A160143. [Corrected by Peter Luschny, Mar 18 2021]
A193475(n) = 4*16^n-2*4^n is similar, but differs at n = 10, 31, 52, 73, 77, 94, ...

Peter Luschny, The lost Bernoulli numbers.

Cf. A000364, A160143, A160144, A193475.

gf := (f,n) -> coeff(series(f(x),x,n+4),x,n):
A193476 := n -> denom(gf(sec,2*n)*(2*n+1)!/(4*16^n - 2*4^n)):
seq(A193476(n), n = 0..17); # Altug Alkan, Apr 23 2018

a[n_] := Sum[Sum[Binomial[k, m] (-1)^(n+k)/(2^(m-1)) Sum[Binomial[m, j]*(2j - m)^(2n), {j, 0, m/2}]*(-1)^(k-m), {m, 0, k}], {k, 1, 2n}] (2n+1)/ (4*16^n - 2*4^n) // Denominator; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jun 26 2019, after Vladimir Kruchinin in A000364 *)

a(n) = denominator(subst(bernpol(2*n+1), 'x, 1/4)*2^(2*n+1)/(2^(2*n+1)-1)); \\ Altug Alkan, Apr 22 2018 after Charles R Greathouse IV at A000364

A303449 Denominator of (2n+1)/(2^(2n+1)-1).

Original entry on oeis.org

1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 299593, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311, 2251799813685247

Offset: 0

Altug Alkan, Apr 24 2018

If A160145(n) = 0, then a(n) = A083420(n).

Least values of k such that a(k) = A083420(k)/A036259(n) are 0, 10, 126, 77, 540, 73, 1242, 328, 1540, 489 for 1 <= n <= 10.

Cf. A005408, A036259, A083420, A160144 (numerators), A160145.

seq(denom((2*n+1)/(2^(2*n+1)-1)), n=0..25);

a(n) = denominator((2*n+1)/(2^(2*n+1)-1));

forstep(k=1, 1e2, 2, print1(denominator(k/(2^k-1)), ", "));

cp's OEIS Frontend

A160145 a(n) = the odd number 2n+1 minus the numerator of (2n+1)/(2^(2n+1)-1).

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A160143 a(n) = Numerator((-1)^n*Euler(2*n)*(2*n+1)/(4^(2*n+1)-2^(2*n+1))), where Euler(n) = A122045(n).

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A193476 The denominators of the Bernoulli secant numbers at odd indices.

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Maple

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PARI

A303449 Denominator of (2*n+1)/(2^(2*n+1)-1).

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Maple

PARI

PARI

A160143 a(n) = Numerator((-1)^nEuler(2n)(2n+1)/(4^(2n+1)-2^(2n+1))), where Euler(n) = A122045(n).

A303449 Denominator of (2n+1)/(2^(2n+1)-1).