A377225 a(0) = 0, a(1) = 1, a(2) = -1, a(3) = 2, and for n > 3, a(n) is the least integer (in absolute value) not yet in the sequence sharing a factor with a(n-1); in case of a tie, preference is given to the positive value.

0, 1, -1, 2, -2, 4, -4, 6, 3, -3, -6, 8, -8, 10, 5, -5, -10, 12, 9, -9, -12, 14, 7, -7, -14, 16, -16, 18, 15, -15, -18, 20, -20, 22, 11, -11, -22, 24, 21, -21, -24, 26, 13, -13, -26, 28, -28, 30, 25, -25, -30, 27, -27, 33, -33, 36, 32, -32, 34, 17, -17, -34

Offset

0,4

Comments

This sequence is a variant of the EKG sequence (A064413) allowing negative values; both sequences share similar features.

For any pair of prime numbers p and q such that p < q, a multiple of q is always preceded by a multiple of p.

For any odd prime number p, the first multiple of p is followed by p, -p and 2*p or -2*p, in that order.

All prime numbers appear in the sequence:

- by contradiction: if the sequence is p-smooth for some prime number p appearing in the sequence,

- there are infinitely many multiples of some prime number q <= p in the sequence,

- so all multiples of q will show up, including q*r some some prime number r > p, a contradiction.

Hence, the sequence contains infinitely many even numbers, and so all even numbers will show up.

For any nonzero integer v, as there are infinitely many even multiples of v in the sequence, and these are all chances for v to be chosen, so v will eventually appear: this sequence is a bijection from the nonnegative integers (N) to the integers (Z).

Partials sums are nonnegative (for any n > 0, among the first n terms, a negative term, say -v, will always be cancelled by the corresponding positive term v appearing prior, but some positive term might be present without the corresponding negative term).

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Rémy Sigrist, PARI program

Index entries for sequences related to EKG sequence

Formula

Sum_{k = 0..n} a(k) >= 0 for any n >= 0.

Example

The first terms are:
  n   a(n)  gcd(a(n-1), a(n))
  --  ----  -----------------
   0     0  N/A
   1     1                  1
   2    -1                  1
   3     2                  1
   4    -2                  2
   5     4                  2
   6    -4                  4
   7     6                  2
   8     3                  3
   9    -3                  3
  10    -6                  3
  11     8                  2
  12    -8                  8
  13    10                  2
  14     5                  5
  15    -5                  5
  16   -10                  5

Prog

(PARI) \\ See Links section.
(Python)
from math import gcd
from itertools import count, islice, product
def agen(): # generator of terms
    a, m = [0, 1, -1, 2], 2
    yield from a
    an, aset = a[-1], set(a)
    for n in count(4):
        an = next(s*k for k in count(m) for s in [1, -1] if s*k not in aset and gcd(k, an) > 1)
        yield an
        aset.add(an)
        while m in aset and -m in aset: m += 1
print(list(islice(agen(), 62))) # Michael S. Branicky, Oct 21 2024

Crossrefs

Cf. A064413.

Sequence in context: A324104 A278233 A354753 * A354755 A354858 A128923

Adjacent sequences: A377222 A377223 A377224 * A377226 A377227 A377228

Keyword

sign

Author

Rémy Sigrist, Oct 20 2024

Status

approved