cp's OEIS Frontend

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

%I A280443 #16 Jun 02 2025 12:20:34
%S A280443 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,
%T A280443 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,
%U A280443 1,11,101,23,89,17,11,1,1,83,1,11,1
%N A280443 a(n) = A280442(n)/A223067(n) = A067624(n)*A046161(n)/A223068(n).
%C A280443 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.
%C A280443 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.
%C A280443 Conjecture: The sequence A280443 will not have a(n)=1 after some point.
%F A280443 a(n) = A280442(n)/A223067(n).
%F A280443 a(n) = A067624(n)*A046161(n)/A223068(n).
%F A280443 a(n) = A280442(n)/numer((A280442(n)/A046161(n))/A067624(n)).
%p A280443 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);
%o A280443 (Sage)
%o A280443 def A280443_list(prec):
%o A280443     P.<x> = PowerSeriesRing(QQ, default_prec=2*prec)
%o A280443     g = lambda x: exp(sum((-1)^k*euler_number(2*k)*x^k/(2*k) for k in (1..prec+1)))
%o A280443     R = P(g(x)).coefficients()
%o A280443     d = lambda n: 2*n - sum(n.digits(2))
%o A280443     return [(2^d(n)*R[n]/(numerator(R[n]/factorial(2*n)))) for n in (0..prec)]
%o A280443 print(A280443_list(68)) # _Peter Luschny_, Jan 05 2017
%Y A280443 Cf. A000364 (Euler numbers), A046161, A067624, A223067, A223068, A280442.
%K A280443 nonn,easy
%O A280443 0,8
%A A280443 _Johannes W. Meijer_ and _Joseph Abate_, Jan 03 2017