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

Author

Amiram Eldar, Dec 12 2019

Keywords

• nonn

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.

Examples

			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).
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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