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A112563 Sieve performed by successive iterations of steps where step m is: keep m terms, remove the next 5 and repeat; as m = 1,2,3,.. the remaining terms form this sequence.

%I A112563 #4 Mar 30 2012 18:36:51
%S A112563 1,7,43,169,505,1051,2527,5083,7729,11635,22681,33937,55483,90889,
%T A112563 132595,152251,238057,327643,451249,543355,776161,997927,1258993,
%U A112563 1441609,1924315,2397571,3221737,4036033,4900399,5438665,6691651
%N A112563 Sieve performed by successive iterations of steps where step m is: keep m terms, remove the next 5 and repeat; as m = 1,2,3,.. the remaining terms form this sequence.
%C A112563 Formula: a(n) = 1 + [..[[[[n*2/1]3/2]4/3]5/4]...(k+1)/k]...] where denominators k of the fractions used in the product vary over all natural numbers not congruent to 0 (mod 6); thus the product will eventually reach a maximum value of a(n).
%F A112563 a(n) = 1 + 6*A073363(n).
%e A112563 Sieve starts with the natural numbers:
%e A112563 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,...
%e A112563 Step 1: keep 1 term, remove the next 5, repeat; giving
%e A112563 1,7,13,19,25,31,37,43,49,55,61,67,73,79,...
%e A112563 Step 2: keep 2 terms, remove the next 5, repeat; giving
%e A112563 1,7,43,49,85,91,127,133,169,175,211,217,...
%e A112563 Step 3: keep 3 terms, remove the next 5, repeat; giving
%e A112563 1,7,43,169,175,211,337,343,379,505,511,547,...
%e A112563 Continuing in this way, we obtain this sequence.
%e A112563 Using the floor function product formula:
%e A112563 a(2) = 1+[..[(2)*2/1]*3/2]*4/3]*5/4]*7/6]*8/7]*9/8]*10/9]*
%e A112563 12/11]*13/12]*14/13]*15/14]*17/16]*18/17]*19/18]*20/19]*
%e A112563 22/21]*23/22]*24/23]*25/24]*27/26]*28/27]*29/28]*30/29] =43.
%o A112563 (PARI) {a(n)=local(A=n,B=0,k=0); until(A==B,k=k+1;if(k%6==0,k=k+1);B=A;A=floor(A*(k+1)/k));1+A}
%Y A112563 Cf. A073363, A112560, A112561, A112562, A112564, A112565, A112566, A112567, A112568, A112569.
%K A112563 nonn
%O A112563 0,2
%A A112563 _Paul D. Hanna_, Oct 14 2005