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A057540 Birthday set of order 8: i.e., numbers congruent to +- 1 modulo 2, 3, 4, 5, 6, 7 and 8.

%I A057540 #21 Mar 18 2021 13:34:36
%S A057540 1,41,71,169,209,239,281,391,449,559,601,631,671,769,799,839,841,881,
%T A057540 911,1009,1049,1079,1121,1231,1289,1399,1441,1471,1511,1609,1639,1679,
%U A057540 1681,1721,1751,1849,1889,1919,1961,2071,2129,2239,2281,2311,2351,2449
%N A057540 Birthday set of order 8: i.e., numbers congruent to +- 1 modulo 2, 3, 4, 5, 6, 7 and 8.
%H A057540 Ray Chandler, <a href="/A057540/b057540.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Colin Barker)
%H A057540 A. Feist, <a href="http://www.kappamuepsilon.org/pages/a/Pentagon/Vol_60_Num_1_Fall_2000.pdf">On the Density of Birthday Sets</a>, The Pentagon, 60 (No. 1, Fall 2000), 31-35.
%H A057540 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).
%F A057540 G.f.: x*(x^16 +40*x^15 +30*x^14 +98*x^13 +40*x^12 +30*x^11 +42*x^10 +110*x^9 +58*x^8 +110*x^7 +42*x^6 +30*x^5 +40*x^4 +98*x^3 +30*x^2 +40*x +1) / ((x -1)^2*(x +1)*(x^2 +1)*(x^4 +1)*(x^8 +1)). - _Colin Barker_, Mar 16 2015
%e A057540 2129 is on the list because it is congruent to 1 mod 2, -1 mod 3, 1 mod 4, -1 mod 5, -1 mod 6, 1 mod 7 and 1 mod 8.
%t A057540 bso8Q[n_]:=Module[{s1=Mod[n,Range[2,8]],s2},s2=Abs[s1-Range[2,8]];AllTrue[ Thread[{s1,s2}],MemberQ[#,1]&]]; Select[Range[2500],bso8Q] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 18 2021 *)
%o A057540 (PARI) Vec(x*(x^16 +40*x^15 +30*x^14 +98*x^13 +40*x^12 +30*x^11 +42*x^10 +110*x^9 +58*x^8 +110*x^7 +42*x^6 +30*x^5 +40*x^4 +98*x^3 +30*x^2 +40*x +1) / ((x -1)^2*(x +1)*(x^2 +1)*(x^4 +1)*(x^8 +1)) + O(x^100)) \\ _Colin Barker_, Mar 16 2015
%Y A057540 Cf. A007310, A057538, A057539 and A057541 are also birthday sets.
%K A057540 nonn,easy
%O A057540 1,2
%A A057540 Andrew R. Feist (andrewf(AT)math.duke.edu), Sep 06 2000
%E A057540 Offset corrected to 1 by _Ray Chandler_, Jul 29 2019