A379068 -id:A379068 - cp's OEIS frontend

A379067 Smallest missing positive number after A377091(n) has been found.

1, 2, 3, 3, 3, 4, 5, 6, 6, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11, 12, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 22, 22, 24, 24, 24, 24, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 42, 42, 42, 42, 42, 42, 42, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 51, 51, 51, 51, 51, 51, 51

Offset: 0

N. J. A. Sloane, Dec 28 2024

nonn

Conjecture: lim sup |a(n) - n/2|/sqrt(n) = 1/2 (see links). This would imply that after A377091(n) has been found, A377091 contains all numbers in the range [0,f(n)], where lim sup f(n) = (n-sqrt(n))/2. There is a corresponding conjecture for A379068. - N. J. A. Sloane and Paolo Xausa, Feb 03 2025

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Table of n, A379067(n), A379068(n) for n = 0..999995.
Paolo Xausa, Scatterplot of |a(n) - n/2|/sqrt(n) for 1 <= n <= 10^6. The orange line is y = 1/2.
Paolo Xausa, Scatterplot of |a(n) - n/2|/sqrt(n) for 900000 <= n <= 10^6. The orange line is y = 1/2.

Cf. A377091, A379068.

(* A377091list is defined at A377091 *)
Module[{s, a}, s[A377091list%5B100%5D%5D%5D%5D%20(*%20_Paolo%20Xausa">] := False; FoldList[(s[#2] = True; a = #; While[s[a], a++]; a) &, 1, Rest[A377091list[100]]]] (* _Paolo Xausa, Apr 08 2025 *)

A380174 a(n) is the least integer (in absolute value) not among the n initial terms of A377091; in case of a tie, preference is given to the positive value.

Original entry on oeis.org

0, 1, -1, -1, -1, 3, -3, -3, -3, -3, -5, -5, -5, -5, -5, -5, -5, 9, 9, 9, 9, 9, 9, 10, 11, 12, -13, 14, 14, 14, 14, 14, 14, 14, 15, 16, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 22, 22, 24, 24, 24, 24, -25, -26, -27, -28, 29, 29, 29, 29, 29, 29, 29, 29, 33

Offset: 0

Rémy Sigrist, Jan 15 2025

sign

If abs(A377091(n-1) - a(n)) is a square number for some n > 0, then A377091(n) = a(n).

			The initial terms are:
  n   a(n)  n initial terms of A377091
  --  ----  ----------------------------------
   0     0  {}
   1     1  {0}
   2    -1  {0, 1}
   3    -1  {0, 1, 2}
   4    -1  {-2, 0, 1, 2}
   5     3  {-2, -1, 0, 1, 2}
   6    -3  {-2, -1, 0, 1, 2, 3}
   7    -3  {-2, -1, 0, 1, 2, 3, 4}
   8    -3  {-2, -1, 0, 1, 2, 3, 4, 5}
   9    -3  {-4, -2, -1, 0, 1, 2, 3, 4, 5}
  10    -5  {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

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

See A379067 and A379068 for similar sequences.

Cf. A377091.

PARI
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\\ See Links section.
```

abs(a(n)) = min(A379067(n-1), A379068(n-1)) for any n > 0.

cp's OEIS Frontend

A379067 Smallest missing positive number after A377091(n) has been found.

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