A297076 Lexicographically earliest sequence of distinct positive numbers such that, for any n > 1, the prime factorizations of a(n) and of a(n+1) share a prime power > 1.

1, 2, 6, 3, 12, 4, 20, 5, 10, 14, 7, 21, 15, 24, 8, 40, 30, 18, 9, 36, 28, 35, 42, 22, 11, 33, 39, 13, 26, 34, 17, 51, 48, 16, 80, 45, 55, 44, 52, 60, 57, 19, 38, 46, 23, 69, 66, 50, 25, 75, 78, 54, 27, 108, 68, 76, 84, 56, 63, 70, 58, 29, 87, 93, 31, 62, 74

Offset

1,2

Comments

For any n > 1, there is a prime number p that divides a(n) and such that the p-adic valuation of a(n) equals the p-adic valuation of a(n+1).

This sequence has similarities with the EKG sequence A064413: here consecutive terms share a prime power, there consecutive terms share a prime factor.

The scatterplots of this sequence and of A064413 show three similar lines starting from the origin; however here we also have some isolated points.

This sequence and A064413 match for n = 1, 2, 9, 28, 78, 113, 245, 419, 420, 421, 580, 581, 582, 644, 816, 937, 969, 1105, 1117, 1118, etc.

This sequence is likely to be a permutation of the natural numbers.

Links

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

Rémy Sigrist, PARI program for A297076

Example

The first terms, alongside their prime factorizations, are:
  n      a(n)     Prime factorization of a(n)
  --     ----     ---------------------------
   1       1      1
   2       2      2
   3       6      2 * 3
   4       3      3
   5      12      2^2 * 3
   6       4      2^2
   7      20      2^2 * 5
   8       5      5
   9      10      2 * 5
  10      14      2 * 7
  11       7      7
  12      21      3 * 7
  13      15      3 * 5
  14      24      2^3 * 3
  15       8      2^3
  16      40      2^3 * 5
  17      30      2 * 3 * 5
  18      18      2 * 3^2
  19       9      3^2
  20      36      2^2 * 3^2

Prog

(PARI) See Links section.

Crossrefs

Cf. A064413.

Sequence in context: A370501 A253279 A355212 * A304752 A113552 A282291

Adjacent sequences: A297073 A297074 A297075 * A297077 A297078 A297079

Keyword

nonn

Author

Rémy Sigrist, Dec 25 2017

Status

approved