A291897 -id:A291897 - cp's OEIS frontend

A292911 Numbers n such that A291897(n) is divisible by (2*n-1)^3.

1, 3, 7, 9, 15, 19, 21, 27, 31, 37, 45, 49, 51, 55, 57, 69, 75, 79, 87, 91, 97, 99, 115, 117, 121, 129, 135, 139, 141, 147, 157, 159, 169, 175, 177, 187, 195, 199, 201, 205, 211, 217, 225, 229, 231, 255, 261, 271, 279, 285, 289, 297, 301, 307, 309, 321, 327

Offset: 1

Vladimir Shevelev, Sep 26 2017

nonn

Conjecture: Every prime of the form 4*k+1 (A002144) is contained in the sequence {2*a(n)-1}.
The author's former conjecture that, for n>=2 the numbers {2*a(n)-1} are consecutive primes of the form 4*k+1, was disproved at n = 553 by Peter J. C. Moses. (553*2 - 1 = 1105 is the smallest term which is a product of three distinct (4*k+1)-primes). - Vladimir Shevelev, Sep 27 2017
553 is also (after 1) the smallest number which is missing from A119681 but is present here. - R. J. Mathar, Sep 29 2017

Michael De Vlieger, Table of n, a(n) for n = 1..500
Michael De Vlieger, Comparison of A292911 and A002144

Cf. A002144, A291897.

Select[Array[{2^IntegerExponent[2 #, 2] EulerE[2 # - 1, #], #} &, 330], Divisible[#1, (2 #2 - 1)^3] & @@ # &][[All, -1]] (* Michael De Vlieger, Sep 27 2017, after Peter Luschny at A291897 *)

If the conjecture is true, then for n>=2, a(n) <= (A002144(n-1) + 1)/2 (the equality holds up to 90).

More terms from Peter J. C. Moses, Sep 26 2017

A291982 a(n) = Euler(n, n+1) * 2^valuation(n+1, 2), where Euler(n, x) denotes the Euler polynomial.

Original entry on oeis.org

1, 3, 6, 161, 380, 9251, 68922, 9718545, 24721272, 1140755269, 14712346550, 1678097074579, 13104139232340, 889926827467887, 16319429252249970, 10286621696853755681, 27076409740571217392, 2427916115944458451025, 57728302956904672126062

Offset: 0

Peter Luschny, Sep 22 2017

nonn

Conjecture: If n >= 2 is even then n*(n+1) divides a(n).

This conjecture was inspired by Vladimir Shevelev's conjecture in A291897.

Cf. A007814, A291897.

A291982 := n -> euler(n, n+1)*2^(padic[ordp](n+1, 2)):
seq(A291982(n), n=0..18);

Table[2^IntegerExponent[n+1, 2] EulerE[n, n+1], {n, 1, 15}]

from sympy import euler
def A291982(n): return euler(n,n+1)*(n+1 & -n-1) # Chai Wah Wu, Jul 07 2022

a(n) = Euler(n, n+1)*2^A007814(n+1).

A292791 The numerator of the real part of E(2n-1, i), where E(n, x) is the Euler polynomial.

Original entry on oeis.org

-1, 7, -11, 199, -361, 8017, -63311, 10775663, -37120861, 2572609327, -54738555011, 11225458402189, -170606509547761, 24269619087650437, -998364772178081111, 1505193846304099711711, -10065529459831250937061, 2427246234079407797537347, -163790353311268893725697611

Offset: 1

Robert G. Wilson v, Sep 23 2017

sign

The imaginary part is +-i.
The denominators are powers of two; A171977(n) = 2^A001511(n).
For E(2n, i) see A292792.
a(4n) == +-1 (mod 6),
a(4n+1) == 5 (mod 6),
a(4n+2) == 1 (mod 6),
a(4n+3) == 1 (mod 6).
Inspired by A291897.

			a(3) = -11 since E(5, i) = -11/2 + i.

Robert G. Wilson v, Table of n, a(n) for n = 1..276

Cf. A292792.

f[n_] := Numerator[ EulerE[2n -1, I] - I^(2n -1)]; Array[f, 19]

A292792 The imaginary part of E(2n, i)/i, where E(n, x) is the Euler polynomial.

Original entry on oeis.org

-1, 3, -11, 63, -571, 7623, -140531, 3417183, -105946891, 4079170743, -190947551651, 10679561580303, -703342782666811, 53875223570677863, -4749070236390684371, 477331966210918815423, -54264228263241408276331, 6927625954707653092264983, -986892857756676619338994691

Offset: 1

Robert G. Wilson v, Sep 23 2017

sign

The real part is +- 1.
For E(2n-1, i) see A292791.
a(2n) == 3 (mod 6),
a(4n+1) == 5 (mod 6),
a(4n+3) == 1 (mod 6).
Inspired by A291897.
Also the negative imaginary part of E(2n, 1+i) or the imaginary part of E(2n, 1-i).

			a(3) = -11 since E(6, i) = -1 - 11i.

Robert G. Wilson v, Table of n, a(n) for n = 1..276

Cf. A291897, A292791.

f[n_] := (EulerE[2n, I] - I^(2n))/I; Array[f, 19]

A292782 a(n) = E(2n,n)/2, where E(n,x) is the Euler polynomial.

Original entry on oeis.org

0, 1, 63, 6306, 990550, 227890755, 72524317341, 30560156566660, 16483798503292716, 11080974333713379525, 9085235508141504416155, 8924963654575108415598246, 10349560274697013067017980738, 13989200573862071630368836403591, 21802322447828101388917112243376825

Offset: 1

Vladimir Shevelev, Sep 23 2017

nonn

Conjecture. For n >= 2, a(n) is divisible by n(n-1)/2, moreover, for odd n, a(n) is divisible by n^2(n-1)/2.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 1972, Ch. 23.

Cf. A291897, A291982.

Table[EulerE[2 n, n]/2, {n, 15}] (* Michael De Vlieger, Sep 23 2017 *)

a(n) = (-1)^n*(1^(2*n) - 2^(2*n) + ... +(-1)^n*(n-1)^(2*n)).

a(n) ~ c * n^(2*n), where c = A349003/2 = 1/(1 + exp(2)) = 0.1192029220221175559402708586976... - Vaclav Kotesovec, Nov 05 2021

More terms from Peter J. C. Moses, Sep 23 2017

A292830 a(1) = 1, for n>=2, a(n) = B(2*n-1, n), where B(n, x) is the Bernoulli polynomial.

Original entry on oeis.org

1, 3, 85, 5558, 651186, 119617025, 31697064295, 11444459810700, 5400661033684164, 3227014932144214335, 2381276769035483594793, 2126703075527239956801538, 2260781626706432961741917750, 2820945601365221814523529200893, 4082702018096881373945823658830923

Offset: 1

Vladimir Shevelev, Sep 24 2017

nonn

Note that B(2*n-1,n) is an integer for all positive integer n, except for n=1, for which B(1,1) = 1/2, so for all n>=1, a(n) is the numerator of B(2*n-1,n). Also note that a(n) is always divisible by (2*n-1) (cf. formula).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 1972, Ch. 23.

Cf. A291897.

a := n -> `if`(n=1, 1, bernoulli(2*n-1, n)): # Peter Luschny, Sep 25 2017

Array[Ceiling@ BernoulliB[2 # - 1, #] &, 15] (* Michael De Vlieger, Sep 24 2017 *)

a(n) = if (n==1, 1, subst(bernpol(2*n-1), x, n)); \\ Michel Marcus, Sep 25 2017

From [Abramowitz and Stegun] for n >= 2 we have a(n) = (2*n - 1) * (1^(2*n - 2) + 2^(2*n - 2) + ... + (n-1)^(2*n - 2)).

More terms from Peter J. C. Moses, Sep 24 2017

cp's OEIS Frontend

A292911 Numbers n such that A291897(n) is divisible by (2*n-1)^3.

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A291982 a(n) = Euler(n, n+1) * 2^valuation(n+1, 2), where Euler(n, x) denotes the Euler polynomial.

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A292791 The numerator of the real part of E(2n-1, i), where E(n, x) is the Euler polynomial.

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A292792 The imaginary part of E(2n, i)/i, where E(n, x) is the Euler polynomial.

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A292782 a(n) = E(2n,n)/2, where E(n,x) is the Euler polynomial.

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A292830 a(1) = 1, for n>=2, a(n) = B(2*n-1, n), where B(n, x) is the Bernoulli polynomial.

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