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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.
4
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
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