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A096023 Numbers congruent to {63, 123, 183, 243, 303, 363} mod 420. 6
63, 123, 183, 243, 303, 363, 483, 543, 603, 663, 723, 783, 903, 963, 1023, 1083, 1143, 1203, 1323, 1383, 1443, 1503, 1563, 1623, 1743, 1803, 1863, 1923, 1983, 2043, 2163, 2223, 2283, 2343, 2403, 2463, 2583, 2643, 2703, 2763, 2823, 2883, 3003, 3063, 3123 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Numbers n such that (n+j) mod (2+j) = 1 for j from 0 to 4 and (n+5) mod 7 <> 1.

Numbers n such that n mod 60 = 3 and n mod 420 <> 3.

LINKS

Table of n, a(n) for n=1..45.

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

FORMULA

G.f.: 3*x*(21+20*x+20*x^2+20*x^3+20*x^4+20*x^5+19*x^6) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

From Wesley Ivan Hurt, Jul 22 2016: (Start)

a(n) = a(n-1) + a(n-6) - a(n-7) for n>7; a(n) = a(n-6) + 420 for n>6.

a(n) = (210*n - 96 - 30*cos(n*Pi/3) - 30*cos(2*n*Pi/3) - 15*cos(n*Pi) + 30*sqrt(3)*sin(n*Pi/3) + 10*sqrt(3)*sin(2*n*Pi/3))/3.

a(6k) = 420k-57, a(6k-1) = 420k-117, a(6k-2) = 420k-177, a(6k-3) = 420k-237, a(6k-4) = 420k-297, a(6k-5) = 420k-357. (End)

EXAMPLE

63 mod 2 = 64 mod 3 = 65 mod 4 = 66 mod 5 = 67 mod 6 = 1 and 68 mod 7 = 5, hence 63 is in the sequence.

MAPLE

A096023:=n->420*floor(n/6)+[63, 123, 183, 243, 303, 363][(n mod 6)+1]: seq(A096023(n), n=0..80); # Wesley Ivan Hurt, Jul 22 2016

MATHEMATICA

Select[Range[0, 5*10^3], MemberQ[{63, 123, 183, 243, 303, 363}, Mod[#, 420]] &] (* Wesley Ivan Hurt, Jul 22 2016 *)

PROG

(PARI) {k=5; m=3150; for(n=1, m, j=0; b=1; while(b&&j<k, if((n+j)%(2+j)==1, j++, b=0)); if(b&&(n+k)%(2+k)!=1, print1(n, ", ")))}

(MAGMA) [ n : n in [1..3500] | n mod 420 in [63, 123, 183, 243, 303, 363] ] // Vincenzo Librandi, Mar 24 2011

(MAGMA) /* Alternatively:*/ &cat[ [ 60*n+3, 60*n+63 ]: n in [1..52] | n mod 7 in [1, 3, 5] ]; // Bruno Berselli, Mar 25 2011

CROSSREFS

Cf. A007310, A017629, A096022, A096024, A096025, A096026, A096027.

Cf. A047391 (see MAGMA code). - Bruno Berselli, Mar 25 2011

Sequence in context: A038865 A274460 A181556 * A256813 A324488 A080947

Adjacent sequences:  A096020 A096021 A096022 * A096024 A096025 A096026

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Jun 15 2004

EXTENSIONS

New definition from Ralf Stephan, Dec 01 2004

STATUS

approved

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Last modified May 28 15:04 EDT 2022. Contains 354115 sequences. (Running on oeis4.)