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

1, 41, 71, 169, 209, 239, 281, 391, 449, 559, 601, 631, 671, 769, 799, 839, 841, 881, 911, 1009, 1049, 1079, 1121, 1231, 1289, 1399, 1441, 1471, 1511, 1609, 1639, 1679, 1681, 1721, 1751, 1849, 1889, 1919, 1961, 2071, 2129, 2239, 2281, 2311, 2351, 2449

Offset

1,2

Links

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)

A. Feist, On the Density of Birthday Sets, The Pentagon, 60 (No. 1, Fall 2000), 31-35.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Formula

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

Example

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.

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A007310, A057538, A057539 and A057541 are also birthday sets.

Sequence in context: A269807 A289982 A054806 * A362592 A105126 A214643

Adjacent sequences: A057537 A057538 A057539 * A057541 A057542 A057543

Keyword

nonn,easy

Author

Andrew R. Feist (andrewf(AT)math.duke.edu), Sep 06 2000

Extensions

Offset corrected to 1 by Ray Chandler, Jul 29 2019

Status

approved