A330530 Lexicographically earliest sequence of distinct positive integers such that the product of two consecutive terms is always divisible by 4.

1, 4, 2, 6, 8, 3, 12, 5, 16, 7, 20, 9, 24, 10, 14, 18, 22, 26, 28, 11, 32, 13, 36, 15, 40, 17, 44, 19, 48, 21, 52, 23, 56, 25, 60, 27, 64, 29, 68, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 72, 31, 76, 33, 80, 35, 84, 37, 88, 39, 92, 41, 96, 43, 100, 45, 104

Offset

1,2

Comments

For any k > 0, let f_k be the lexicographically earliest sequence of distinct positive integers such that the product of two consecutive terms is always divisible by k:

- in particular:

f_1 = f_2 = A000027,

f_3 = A006368,

f_4 = a (this sequence),

f_6 = A330531,

- f_k is a permutation of the natural numbers,

- f_k(1) = 1, f_k(2) = max(2, k),

- if k is prime, then f_k corresponds to the integers that are not multiple of k interspersed with the integers that are multiple of k.

Apparently:

- for m > 0, the m-th run of consecutive terms such that gcd(a(n), 4) = 2 has A153893(m+1) terms,

- for m > 1, the m-th run of consecutive terms such that gcd(a(n), 4) = 1 or 4 has A068156(m+1) terms.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Colored scatterplot of the first 10000 terms

Index entries for sequences that are permutations of the natural numbers

Example

The first terms, alongside their product with the next term, are:
  n   a(n)  a(n)*a(n+1)
  --  ----  -----------
   1     1            4
   2     4            8
   3     2           12
   4     6           48
   5     8           24
   6     3           36
   7    12           60
   8     5           80
   9    16          112
  10     7          140

Prog

(PARI) s=0; v=1; for (n=1, 10 000, print (n " " v); s+=2^v; for (w=1, oo, if (!bittest(s, w) && (v*w)%4==0, v=w; break)))

Crossrefs

Cf. A006368, A068156, A153893, A330531 (f_6), A330576 (inverse).

Sequence in context: A306874 A114478 A367882 * A302659 A363705 A134239

Adjacent sequences: A330527 A330528 A330529 * A330531 A330532 A330533

Keyword

nonn,easy,look

Author

Rémy Sigrist, Dec 17 2019

Status

approved