A330359 Race of lucky numbers of the form 4*k - 1 vs. 4*k + 1 is tied at the a(n)-th lucky number.

2, 4, 6, 16, 20, 22, 24, 2684, 2686, 2688, 2696, 2710, 2712, 109978, 110026, 110028, 110030, 110052, 110056, 110060, 110068, 110070, 110154, 110156, 110158, 110160, 118048, 118050, 118126, 118128, 118130, 118132, 118134, 118136, 118138, 118152, 118154, 118156

Offset

1,1

Comments

All the terms are even by definition. For each term m, there are m/2 lucky numbers of the form 4*k - 1 and m/2 lucky numbers of the form 4*k + 1 up to the m-th lucky number.

Gardiner et al. (1956) noted that the ratio between the numbers of lucky numbers of the form 4*k - 1 and 4*k + 1 seems to tend to 1, with a preponderance, at first, of the lucky numbers of the form 4*k + 3.

Links

Table of n, a(n) for n=1..38.

Vema Gardiner, Roger Lazarus, Nicholas Metropolis and Stanislaw Ulam, On certain sequences of integers defined by sieves, Mathematics Magazine, Vol. 29, No. 3 (1956), pp. 117-122. See Table IV, p. 121.

Example

6 is in the sequence since the first 6 lucky numbers are 1, 3, 7, 9, 13, 15, half of them are of the form 4*k-1 (3, 7, 15) and half of the form 4*k+1 (1, 9, 13).

Mathematica

lucky = Import["b000959.txt", "Table"][[;; , 2]]; Flatten[Position[Accumulate[ Mod[lucky, 4] - 2], 0]] (* use the b-file from A000959 *)

Crossrefs

Cf. A000959, A038691, A137168, A137170, A330360, A330361.

Sequence in context: A100361 A259939 A069654 * A000068 A067662 A248334

Adjacent sequences: A330356 A330357 A330358 * A330360 A330361 A330362

Keyword

nonn

Author

Amiram Eldar, Dec 12 2019

Status

approved