A081614 Subsequence of A005428 with state = 1.

1, 4, 6, 9, 31, 70, 105, 355, 799, 1798, 2697, 9103, 20482, 30723, 69127, 155536, 233304, 349956, 524934, 787401, 2657479, 5979328, 8968992, 13453488, 20180232, 30270348, 45405522, 68108283, 153243637, 1745540806, 2618311209, 8836800331, 19882800745, 67104452515, 150985018159, 339716290858, 509574436287, 1146542481646, 1719813722469, 13059835455001, 44076944660629, 753095921662471, 1694465823740560

Offset: 0

Author

N. J. A. Sloane, Apr 23 2003

Keywords

• nonn
• easy

Comments

Values of n such that A054995(n) = 1. - Ryan Brooks, Jul 17 2020

From Petros Hadjicostas, Jul 20 2020: (Start)

From a(1) = 4 to a(28) = 153243637, the values appear in Table 18 (p. 374) in Schuh (1968) under the Survivor No. 1 column (in a variation of Josephus's counting off game where m people on a circle are labeled 1 through m and every third person drops out).

a(29) here is 1745540806 but 1595540806 in Schuh (1968). Burde (1987) agrees with Schuh (1968). See the table on p. 207 of the paper (with q = 0). Actually, 1595540806 is the last number on the table with q = 0.

It seems Schuh (1968) made a calculation error and Burde (1987) copied it. See my comment for A073941 for more details. (End)

References

• Fred Schuh, The Master Book of Mathematical Recreations, Dover, New York, 1968. [See Table 18, p. 374.]

Links

• David A. Corneth, Table of n, a(n) for n = 0..2816
• K. Burde, Das Problem der Abzählreime und Zahlentwicklungen mit gebrochenen Basen [The problem of counting rhymes and number expansions with fractional bases], J. Number Theory 26(2) (1987), 192-209. [The author deals with the representation of n in fractional bases k/(k-1) and its relation to counting-off games. Here k = 3. See the table on p. 207. See also the review in MathSciNet (MR0889384) by R. G. Stoneham.]
• Index entries for sequences related to the Josephus Problem

Crossrefs

Cf. A005428, A073941, A081615, A061419.

Programs

PARI

/* In the program below, we use a truncated version of either A005428 or A073941 and choose those terms that correspond to "state" or "number of last survivor" equal to 1. See A073941 or Schuh (1968) for more details. */
first(n) = {my(res = vector(n), t = 1, wn = wo = 4, go = gn = 1); res[1] = 1; for(i = 1, oo, c = wo % 2; if(go == 1, t++; res[t] = wo; if(t >= n, return(res) ) ); wn = floor(wo*3/2) + c * (2 - go); gn = 3 * c + go * (-1)^c; wo = wn; go = gn; ) } \\ David A. Corneth and Petros Hadjicostas, Jul 20 2020

Formula

a(n) = [(n+1)-th even number of A061419]/2. - John-Vincent Saddic, May 29 2021

Extensions

More terms from Hans Havermann, Apr 23 2003