login
A057538
Birthday set of order 5: numbers congruent to +-1 modulo 2, 3, 4 and 5.
11
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
1,2
COMMENTS
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
LINKS
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]
FORMULA
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
EXAMPLE
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 + ...
MAPLE
for n from 1 to 409 do if (n^2 mod 30 =1) then print(n) fi od; # Gary Detlefs, Apr 17 2012
MATHEMATICA
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 *)
PROG
(PARI) {a(n+1) = (n\4*3 + n%4)*10 + (-1)^(n\2)} /* Michael Somos, Oct 17 2006 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Andrew R. Feist (andrewf(AT)math.duke.edu), Sep 06 2000
EXTENSIONS
Corrected by Henry Bottomley, Jan 31 2002
Offset corrected to 1 by Ray Chandler, Jul 29 2019
STATUS
approved