A280443 a(n) = A280442(n)/A223067(n) = A067624(n)*A046161(n)/A223068(n)

1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 11, 17, 1, 23, 1, 11, 1, 1, 1, 17, 11, 1, 1, 1, 23, 11, 43, 17, 1, 1, 121, 1, 1, 1, 1, 4301, 1, 1, 1, 73, 11, 1, 1, 17, 1, 11, 23, 43, 1, 1, 11, 17, 1, 1, 1, 11, 101, 23, 89, 17, 11, 1, 1, 83, 1, 11, 1

Offset

0,8

Comments

This sequence is related in a peculiar way to A223067 and A223068, sequences related to the complete elliptic integral of the first kind K(k), and to A280442 and A046161, sequences related to the unsigned Euler numbers A000364.

In this sequence certain prime numbers appear on a regular basis, either by itself or as a factor of a composite number, i.e., a(n)=11 if n=7+5*k, a(n)=17 if n=13+8*k, a(n)=23 if n=15+11*k, a(n)=43 if n=28+21*k, a(n)=73 if n=41+36*k, a(n)=101 if n=58+50*k, a(n)=89 if n=60+44*k, a(n)=83 if n=65+41*k, in all cases k >= 0. We observe that the period T of each prime is apparently T = (prime-1)/2.

Conjecture: The sequence A280443 will not have a(n)=1 after some point.

Links

Table of n, a(n) for n=0..68.

Formula

a(n) = A280442(n)/A223067(n).

a(n) = A067624(n)*A046161(n)/A223068(n).

a(n) = A280442(n)/numer((A280442(n)/A046161(n))/A067624(n)).

Maple

nmax:=68: A067624 := n -> 2^(2*n)*(2*n)!: f := series((exp(add((-1)^n*euler(2*n) * x^n/(2*n), n=1..nmax+1))), x=0, nmax+1): for n from 0 to nmax do b(n) := coeff(f, x, n); a(n) := numer(b(n))/numer(b(n)/A067624(n)) od: seq(a(n), n=0..nmax);

Prog

(Sage)
def A280443_list(prec):
    P.<x> = PowerSeriesRing(QQ, default_prec=2*prec)
    g = lambda x: exp(sum((-1)^k*euler_number(2*k)*x^k/(2*k) for k in (1..prec+1)))
    R = P(g(x)).coefficients()
    d = lambda n: 2*n - sum(n.digits(2))
    return [(2^d(n)*R[n]/(numerator(R[n]/factorial(2*n)))) for n in (0..prec)]
print(A280443_list(68)) # Peter Luschny, Jan 05 2017

Crossrefs

Cf. A000364 (Euler numbers), A046161, A067624, A223067, A223068, A280442.

Sequence in context: A321800 A130837 A010194 * A107045 A351449 A145140

Adjacent sequences: A280440 A280441 A280442 * A280444 A280445 A280446

Keyword

nonn,easy

Author

Johannes W. Meijer and Joseph Abate, Jan 03 2017

Status

approved