A057541 -id:A057541 - cp's OEIS frontend

A057538 Birthday set of order 5: numbers congruent to +-1 modulo 2, 3, 4 and 5.

1, 11, 19, 29, 31, 41, 49, 59, 61, 71, 79, 89, 91, 101, 109, 119, 121, 131, 139, 149, 151, 161, 169, 179, 181, 191, 199, 209, 211, 221, 229, 239, 241, 251, 259, 269, 271, 281, 289, 299, 301, 311, 319, 329, 331, 341, 349, 359, 361, 371, 379, 389, 391, 401, 409

Offset: 1

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

Also numbers congruent to +-1 or +-11 modulo 30 and numbers k where (k^2 - 1)/120 is an integer; all but the first two prime legs of Pythagorean triangles which also have prime hypotenuses appear within in this sequence (A048161). - Henry Bottomley, Jan 31 2002
Numbers k such that k^2 == 1 (mod 30). - Gary Detlefs, Apr 16 2012
Subsequence of primes gives A045468. - Ray Chandler, Jul 29 2019

			229 is congruent to 1 (mod 2), 1 (mod 3), 1 (mod 4) and -1 (mod 5).
x+ 11*x^2 + 19*x^3 + 29*x^4 + 31*x^5 + 41*x^6 + 49*x^7 + 59*x^8 + 61*x^9 + ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
A. Feist, On the Density of Birthday Sets, The Pentagon, 60 (No. 1, Fall 2000), 31-35.
A. Feist, Maple source for birthday sets. [Broken link]
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A007310, A045468, A057539, A057540, A057541, A093722.

for n from 1 to 409 do if (n^2 mod 30 =1) then print(n) fi od; # Gary Detlefs, Apr 17 2012

a057538[n_] := Block[{f},
  f[x_] :=
   If[Mod[x, #] == 1 || Mod[x, #] == # - 1, True, False] & /@
    Range[2, 5];
Select[Range[n], DeleteDuplicates[f[#]] == {True} &]]; a057538[409] (* Michael De Vlieger, Dec 26 2014 *)

{a(n+1) = (n\4*3 + n%4)*10 + (-1)^(n\2)} /* Michael Somos, Oct 17 2006 */

A093722(n) = (a(n)^2 - 1)/120.
G.f.: x * (1 + 10*x + 8*x^2 + 10*x^3 + x^4) / ((1 - x) * (1 - x^4)). a(-1 - n) = -a(n). - Michael Somos, Jan 21 2012
4*a(n) = 30*(n+1) - 45 + 5*(-1)^n + 6*(-1)^floor((n+1)/2). - R. J. Mathar, Jul 30 2019

Corrected by Henry Bottomley, Jan 31 2002

Offset corrected to 1 by Ray Chandler, Jul 29 2019

A057539 Birthday set of order 7, i.e., numbers congruent to +- 1 modulo 2, 3, 4, 5, 6 and 7.

Original entry on oeis.org

1, 29, 41, 71, 139, 169, 181, 209, 211, 239, 251, 281, 349, 379, 391, 419, 421, 449, 461, 491, 559, 589, 601, 629, 631, 659, 671, 701, 769, 799, 811, 839, 841, 869, 881, 911, 979, 1009, 1021, 1049, 1051, 1079, 1091, 1121, 1189, 1219, 1231, 1259, 1261, 1289

Offset: 1

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

Integers of the form sqrt(840*k+1) for k >= 0. - Boyd Blundell, Jul 10 2021

Ray Chandler, Table of n, a(n) for n = 1..10000
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 (1,0,0,0,0,0,0,1,-1).

Cf. A057538, A057540, A057541.

LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{1,29,41,71,139,169,181,209,211},50] (* Harvey P. Dale, Sep 24 2014 *)

is_A057539(n,m=[2,3,4,5,6,7])=!for(i=1,#m,abs((n+1)%m[i]-1)==1||return)

is(n)=for(i=4,7,if(abs(centerlift(Mod(n,i)))!=1, return(0))); 1 \\ Charles R Greathouse IV, Oct 20 2014

def ok(n): return all(n%d in [1, d-1] for d in range(2, 8))
def aupto(nn): return [m for m in range(1, nn+1) if ok(m)]
print(aupto(1300)) # Michael S. Branicky, Jan 29 2021

G.f.: x*(1 + 28*x + 12*x^2 + 30*x^3 + 68*x^4 + 30*x^5 + 12*x^6 + 28*x^7 + x^8) / ((1+x)*(x^2+1)*(x^4+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
a(n) = a(n-8) + 210 = a(n-1) + a(n-8) - a(n-9). - Charles R Greathouse IV, Oct 20 2014
a(n) = 105n/4 + O(1). - Charles R Greathouse IV, Oct 20 2014

Offset corrected to 1 by Ray Chandler, Jul 29 2019

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

Original entry on oeis.org

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

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

			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.

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).

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

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 *)

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

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

Offset corrected to 1 by Ray Chandler, Jul 29 2019

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A057538 Birthday set of order 5: numbers congruent to +-1 modulo 2, 3, 4 and 5.

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A057539 Birthday set of order 7, i.e., numbers congruent to +- 1 modulo 2, 3, 4, 5, 6 and 7.

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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.

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Mathematica

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