A163252 -id:A163252 - cp's OEIS frontend

A317018 Sequence of distinct signed integers such that a(1) = 0 and for any n > 0, the negabinary representation of a(n+1) differ by exactly one digit from the negabinary representation of a(n) and has the smallest possible absolute value (in case of a tie, choose the integer with the rightmost difference).

0, 1, -1, -2, 2, 3, 5, -3, -4, 4, 20, 12, 8, 6, 7, 9, -7, -8, -10, -6, -5, -9, -41, 23, 22, 24, 25, 29, 27, 26, 28, 36, -28, -12, -11, -13, -14, -18, 14, 15, 17, -15, -16, 16, 80, 48, 32, 30, 31, 33, -31, -27, -29, -30, -34, -32, -40, -24, -20, -19, 13, 11, 10

Offset: 1

Rémy Sigrist, Jul 19 2018

This sequence has similarities with A316995; in both sequences, the absolute value of the difference of two consecutive terms is a power of 2.

This sequence also has similarities with A163252.

			The first terms, alongside their negabinary representation, are:
  n   a(n)  nega(a(n))
  --  ----  ----------
   1     0        0
   2     1        1
   3    -1       11
   4    -2       10
   5     2      110
   6     3      111
   7     5      101
   8    -3     1101
   9    -4     1100
  10     4      100
  11    20    10100
  12    12    11100
  13     8    11000
  14     6    11010
  15     7    11011
  16     9    11001
  17    -7     1001
  18    -8     1000
  19   -10     1010
  20    -6     1110

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Line plot of the first 10000 terms
Rémy Sigrist, PARI program for A317018
Eric Weisstein's World of Mathematics, Negabinary.
Wikipedia, Negative base.

Cf. A039724, A163252, A212529, A316995.

PARI
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A330919 Lexicographically earliest sequence of distinct squarefree numbers such that for any n > 0, either a(n)/a(n+1) or a(n+1)/a(n) is a prime number.

Original entry on oeis.org

1, 2, 6, 3, 15, 5, 10, 30, 210, 42, 14, 7, 21, 105, 35, 70, 770, 110, 22, 11, 33, 66, 330, 165, 55, 385, 77, 154, 462, 231, 1155, 2310, 30030, 2730, 390, 78, 26, 13, 39, 195, 65, 130, 910, 182, 91, 273, 546, 6006, 858, 286, 143, 429, 2145, 715, 1430, 4290

Offset: 1

Rémy Sigrist, May 02 2020

In other words, consecutive terms differ exactly by one prime factor.
This sequence has strong connections with A163252:
- here consecutive terms differ by one prime factor, there by one binary digit,
- for any n > 0, A163252(n-1) encodes in binary form the prime numbers appearing in a(n).
Odd indexed terms have an even number of prime factors and vice versa.
For any prime number p: as there are only finitely many squarefree numbers with greatest prime factor < p, eventually the sequence contains a multiple of p.

			The first terms, alongside their prime factors, are:
  n   a(n)  prime factors
  --  ----  -------------
   1     1
   2     2  2
   3     6  2, 3
   4     3     3
   5    15     3, 5
   6     5        5
   7    10  2,    5
   8    30  2, 3, 5
   9   210  2, 3, 5, 7
  10    42  2, 3,    7

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A330919

Cf. A019565, A087207, A163252.

PARI
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See Links section.
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a(n) = A019565(A163252(n-1)).

A087207(a(n)) = A163252(n-1).

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