A103951 Procedure "Remove every 10th term!" executed 10 times.

1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 23, 25, 27, 29, 32, 35, 38, 42, 46, 49, 51, 54, 56, 59, 62, 65, 68, 72, 75, 76, 79, 83, 84, 87, 92, 93, 96, 102, 103, 105, 106, 113, 114, 116, 117, 125, 126, 128, 129, 137, 138, 139, 142, 143, 152, 153, 154, 157, 158, 162, 168, 169, 171

Offset: 1

Author

Zak Seidov, Feb 22 2005

Keywords

• nonn

Comments

Procedure A067251, "Every 10th number has been omitted", executed 10 times.

0. Start with the sequence of natural numbers:

s0=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...

1. Remove each 10th term:

s1=1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21,22,...

2. Remove each 10th term:

s2=1,2,3,4,5,6,7,8,9,12,13,14,15,16,17,18,19,21,23,24,...

After 10 executions we have:

s10=1,2,3,4,5,6,7,8,9,21,23,25,27,29,32,35,38,42,46,49,...

This algorithm may be generalized to

"Remove each k-th term k times!". We have then

k=2: s2=1,5,9,13,17,21,25,29,33,37,41,45,... a(n)=4n-3 (n=1,2,...)

k=3: s3=1,2,7,10,14,16,20,23,28,29,34,37,41,43,47,50,55, (no simple rule here and further?)

k=4: s4=1,2,3, 9,11,14,18,22,23,29,30,33,38,39,43,46,50,51

k=5, s5=1,2,3, 4,11,13,16,19,23,26,28,32,34,39,41,42,48,51,52

k=6, s6=1,2,3, 4, 5,13,15,17,20,23,27,31,32,37,38,44,45,47,52,53

k=7, s7=1,2,3, 4, 5, 6,15,17,19,22,25,29,33,34,38,39,44,45,51,52,54

k=8, s8=1,2,3, 4, 5, 6, 7,17,19,21,23,26,29,33,37,39,42,44,47,50,53

k=9, s9=1,2,3, 4, 5, 6, 7, 8,19,21,23,25,28,31,34,38,42,43,47,48,52,

Is there any general rule for a_k(n)?

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A067251.

Programs

Mathematica

ra=Range[1000];k=10;Do[ra=Drop[ra, {k, Length[ra], k}], {i, k}];ra
Nest[Flatten[Most/@Partition[#,10]]&,Range[300],10] (* Harvey P. Dale, Mar 17 2015 *)