A222640 Hyper-lucky numbers.

1, 3, 9, 15, 21, 27, 33, 39, 63, 69, 75, 81, 87, 93, 141, 147, 171, 177, 183, 189, 285, 291, 297, 303, 309, 333, 417, 441, 447, 453, 507, 555, 621, 627, 633, 681, 687, 711, 951, 957, 981, 1035, 1041, 1047, 1095, 1101, 1107, 1113, 1119, 1143, 1377, 1383, 1389

Offset

1,2

Comments

The hyper-lucky numbers are a generalization of the lucky numbers obtained by replacing the strike-out numbers by the strike-out sets of numbers whose length is successively 1, 2, 3, ...

First, as a brief review, the algorithm for generating lucky numbers consists of the following steps:

First step: write out all the odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...

Second step: the first odd number greater than 1 is 3, so strike out every third number from the list; the survivors are 1, 3, 7, 9, 13, 15, 19, ...

Third step: the first odd number greater than 3 in the list is 7, so strike out every seventh number; the survivors are 1, 3, 7, 9, 13, 15, 21, 25, 31, ...

Fourth step: the first odd number greater than 7 in the list is 9, so strike out every ninth number; the survivors are 1, 3, 7, 9, 13, 15, 21, 25, 31, ...; etc.

After this procedure has been carried out completely, the numbers remaining are called lucky numbers. The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, ... (A000959).

A similar algorithm may be employed for the development of a generalization of the lucky numbers (the hyper-lucky numbers) by the following process:

First step: write out all the odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, ...

Second step: the first odd number greater than 1 is 3, so strike out every set of two numbers with the period 3 from the list by elimination of the sets {5, 7}, {11, 13}, {17, 19} ...; the survivors are 1, 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, ...

Third step: the first odd number greater than 3 in the list is 9, so strike out every set of three numbers with the period 9 from the list by elimination of the sets {45, 51, 57}, {99, 105, 111}, {153, 159, 165} ...; the survivors are 1, 3, 9, 15, 21, 27, 33, 39, 63, 69, 75, ...

After this procedure has been carried out completely, the numbers remaining are called hyper-lucky numbers. The first numbers are 1, 3, 9, 15, 21, 27, 33, 39, 63, 69, 75, ...

Property of this sequence: for a(n) > 1, a(n) == 0 (mod 3) and a(n)/3 = 1, 3, 5, 7, 9, 11, 13, 21, 23, 25, 27, 29, 31, 47, 49, ...; of these, the prime numbers are 3, 5, 7, 11, 13, 23, 29, 31, 47, 59, 61, 97, 101, 103, 139, 149, ...

Links

Table of n, a(n) for n=1..53.

Maple

it:=7:nn:=2000:lst :={} :lst1:={}: lst2:={}:
   for i from 0 to 10*nn do:
      lst:=lst union {2*i+1}:
   od:
    for n from 1 to it do:
          mm :=nn-(n+1)*floor(nn/ lst[n+1]):
       for j from 0 to mm do:
         for jj from 0 to n do:
         m:= lst[n+1]+j*lst[n+1]+ jj: m1:=lst[m]:lst1 :=lst1 union {m1} :
        od:
    od:
   lst:=lst minus lst1: d:=nops(lst):lst1:={}: nn :=mm :
od:
          for a from 1 to nn do: m2:=lst[a]:if m2 < 1400 then
          lst2 :=lst2 union {m2} :
         else fi:
od:
print(lst2):

Crossrefs

Cf. A000959.

Sequence in context: A366849 A228935 A016945 * A110108 A162843 A102954

Adjacent sequences: A222637 A222638 A222639 * A222641 A222642 A222643

Keyword

nonn

Author

Michel Lagneau, Feb 27 2013

Extensions

Edited by Jon E. Schoenfield, Jun 04 2017

Status

approved