A047621 - OEIS

%I #63 Nov 22 2024 09:06:45

%S 3,5,11,13,19,21,27,29,35,37,43,45,51,53,59,61,67,69,75,77,83,85,91,

%T 93,99,101,107,109,115,117,123,125,131,133,139,141,147,149,155,157,

%U 163,165,171,173,179,181,187,189,195,197,203,205,211,213,219,221,227,229

%N Numbers that are congruent to {3, 5} mod 8.

%C Numbers k for which Jacobi symbol J(2,k) = -1, so 2 (as well as 2^k) is not a square mod k. - _Antti Karttunen_, Aug 27 2005, corrected by _Jianing Song_, Nov 05 2019, see also A329095.

%C Numbers n whose multiplicative order modulo 2^k is 2^(k - 2) for k >= 4. For k = 3, the numbers whose multiplicative order modulo 8 is 2 are in sequence A047484. - _Jianing Song_, Apr 29 2018

%H Muniru A Asiru, <a href="/A047621/b047621.txt">Table of n, a(n) for n = 1..5000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F a(n) = 8*n - a(n-1) - 8 (with a(1) = 3). - _Vincenzo Librandi_, Aug 06 2010

%F G.f.: x*(3 + 2*x + 3*x^2) / ( (1 + x)*(x - 1)^2 ). - _R. J. Mathar_, Oct 08 2011

%F A089911(3*a(n)) = 10. - _Reinhard Zumkeller_, Jul 05 2013

%F a(n) = 8*floor((n - 1)/2) + 4 + (-1)^n. - _Gary Detlefs_, Dec 03 2018

%F From _Franck Maminirina Ramaharo_, Dec 03 2018: (Start)

%F a(n) = 4*n - 2 - (-1)^n.

%F E.g.f.: 3 - (2 - 4*x)*exp(x) - exp(-x). (End)

%F a(n + 2) = a(n) + 8. - _David A. Corneth_, Dec 03 2018

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8. - _Amiram Eldar_, Dec 11 2021

%F From _Amiram Eldar_, Nov 22 2024: (Start)

%F Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/8) (1/A144981).

%F Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/8) (A101464). (End)

%t LinearRecurrence[{1, 1, -1}, {3, 5, 11}, 100] (* _Jean-François Alcover_, Jul 31 2018 *)

%o (Haskell)

%o a047621 n = a047621_list !! (n-1)

%o a047621_list = 3 : 5 : map (+ 8) a047621_list

%o -- _Reinhard Zumkeller_, Jul 05 2013

%o (GAP) a:=[3];; for n in [2..60] do a[n]:=8*n-a[n-1]-8; od; a; # _Muniru A Asiru_, Dec 04 2018

%Y Row 1 of A112070. Complement of A047522 relative to A005408. Primes in this sequence: A003629.

%Y Subsequence of A329095.

%Y Cf. A047522, A066507, A101464, A144981.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_