A112561 Sieve performed by successive iterations of steps where step m is: keep m terms, remove the next 3 and repeat; as m = 1,2,3,.. the remaining terms form this sequence.

1, 5, 21, 61, 125, 261, 421, 605, 1101, 1681, 2525, 2781, 4201, 5645, 6741, 9541, 11765, 13701, 17641, 21305, 27981, 29401, 37265, 43521, 51541, 59945, 65781, 78121, 89345, 99981, 121381, 124445, 144321, 173041, 189965, 212361, 229381

Offset

0,2

Comments

Formula: a(n) = 1 + [..[[[[n*2/1]3/2]4/3]6/5]...(k+1)/k]...] where denominators k of the fractions used in the product vary over all natural numbers not congruent to 0 (mod 4); thus the product will eventually reach a maximum value of a(n).

Links

Table of n, a(n) for n=0..36.

Formula

a(n) = 1 + 4*A073361(n).

Example

Sieve starts with the natural numbers:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15...
Step 1: keep 1 term, remove the next 3, repeat; giving
1,5,9,13,17,21,25,29,33,37,41,45,49,53,57,61,65,...
Step 2: keep 2 terms, remove the next 3, repeat; giving
1,5,21,25,41,45,61,65,81,85,101,105,121,125,141,...
Step 3: keep 3 terms, remove the next 3, repeat; giving
1,5,21,61,65,81,121,125,141,181,185,201,241,245,261,...
Continuing in this way, we obtain this sequence.
Using the floor function product formula:
a(2)=1+[..[(2)*2/1]*3/2]*4/3]*6/5]*7/6]*8/7]*10/9]*11/10]*
12/11]*14/13]*15/14]*16/15]*18/17]*19/18]*20/19] = 21.

Prog

(PARI) {a(n)=local(A=n, B=0, k=0); until(A==B, k=k+1; if(k%4==0, k=k+1); B=A; A=floor(A*(k+1)/k)); 1+A}

Crossrefs

Cf. A073361, A112560, A112562, A112563, A112564, A112565, A112566, A112567, A112568, A112569.

Sequence in context: A245240 A370839 A203233 * A303170 A287617 A354551

Adjacent sequences: A112558 A112559 A112560 * A112562 A112563 A112564

Keyword

nonn

Author

Paul D. Hanna, Oct 14 2005

Status

approved