A073941 -id:A073941 - cp's OEIS frontend

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

N. J. A. Sloane, Apr 23 2003

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)

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

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

Cf. A005428, A073941, A081615, A061419.

/* 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

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

More terms from Hans Havermann, Apr 23 2003

A081615 Subsequence of A005428 where state = 2.

Original entry on oeis.org

1, 2, 3, 14, 21, 47, 158, 237, 533, 1199, 4046, 6069, 13655, 46085, 103691, 1181102, 1771653, 3986219, 102162425, 229865456, 344798184, 517197276, 775795914, 1163693871, 3927466814, 5891200221, 13255200497, 29824201118, 44736301677, 100656678773, 226477527239, 764361654431, 2579720583704, 3869580875556, 5804371313334, 8706556970001, 19589753182502, 29384629773753, 66115416990944, 99173125486416

Offset: 0

N. J. A. Sloane, Apr 23 2003

Excluding the initial 1, the values of n such that A054995(n) = 2. - Ryan Brooks, Jul 17 2020
From Petros Hadjicostas, Jul 20 2020: (Start)
From a(1) = 2 to a(22) = 775795914, the values appear in Table 18 (p. 374) in Schuh (1968) under the Survivor No. 2 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(23) here is 1163693871 but 1063693871 in Schuh (1968). Burde (1987) agrees with Schuh (1968). See the table on p. 207 of the paper (with q = 1).
It seems Schuh (1968) made a calculation error and Burde (1987) copied it. See my comment for A073941 for more details. (End)

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

David A. Corneth, Table of n, a(n) for n = 0..2862
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

Cf. A005428, A073941, A081614.

/* 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 2. See A073941 or Schuh (1968) for more details. */
first(n) = {my(res = vector(n), t = 1, wn = wo = gn = go = 2); res[1] = 1; for(i = 1, oo, c = wo % 2; if(go == 2, 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 21 2020

More terms from Hans Havermann, Apr 23 2003

A082125 Smallest difference>1 between d and p/d for any divisor d of the partial product p of the sequence, starting with 4.

Original entry on oeis.org

4, 3, 4, 2, 4, 8, 16, 64, 512, 16384, 2097152, 2147483648, 140737488355328, 1180591620717411303424, 40564819207303340847894502572032, 365375409332725729550921208179070754913983135744

Offset: 0

Ralf Stephan, Apr 04 2003

a(n) is a power of two for n>1 and log_2(a(n))=A073941(n) for n>2. - Ralf Stephan, Apr 16 2003

Cf. A082120, A003681 (starts with 2, 3), A082126.

Cf. A029744.

p=4; print1(p, ", "); for(n=1, 50, v=divisors(p); r=sqrt(p); t=0; for(k=1, matsize(v)[2], if(v[k]>=r, t=k; break)); if(v[t]^2==p, u=t, u=t-1); if(v[t]-v[u]<2, u=u-1; t=t+1); print1(v[t]-v[u]", "); p=p*(v[t]-v[u]))

A120169 a(n) = 12 + floor((1 + Sum_{j=1..n-1} a(j))/4).

Original entry on oeis.org

12, 15, 19, 23, 29, 36, 45, 57, 71, 89, 111, 139, 173, 217, 271, 339, 423, 529, 661, 827, 1033, 1292, 1615, 2018, 2523, 3154, 3942, 4928, 6160, 7700, 9625, 12031, 15039, 18798, 23498, 29372, 36715, 45894, 57368, 71710

Offset: 1

Graeme McRae, Jun 10 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A072493, A073941, A112088.

Cf. A120149, A120160 - A120168, A120171.

function f(n, a, b)
  t:=0;
    for k in [1..n-1] do
      t+:= a+Floor((b+t)/4);
    end for;
  return t;
end function;
g:= func< n, a, b | f(n+1, a, b)-f(n, a, b) >;
A120169:= func< n | g(n, 12, 1) >;
[A120169(n): n in [1..60]]; // G. C. Greubel, Sep 09 2023

nxt[{t_,a_}]:=Module[{c=Floor[(t+49)/4]},{t+c,c}]; NestList[nxt,{12,12},40][[All,2]] (* Harvey P. Dale, Jun 21 2017 *)

@CachedFunction
def f(n, p, q): return p + (q +sum(f(k, p, q) for k in range(1, n)))//4
def A120169(n): return f(n, 12, 1)
[A120169(n) for n in range(1, 61)] # G. C. Greubel, Sep 09 2023

A304273 The concatenation of the first n terms is the smallest positive even number with n digits when written in base 3/2 (cf. A024629).

Original entry on oeis.org

2, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 09 2018

This sequence exists since the smallest even integers (see A303500) are prefixes of each other.

Apparently a variant of A205083. - R. J. Mathar, Jun 09 2018

			The number 5 in base 3/2 is 22, and the number 6 is 210. Therefore 210 is the smallest even integer with 3 digits in base 3/2. Its prefix 21 is 4: the smallest even integer with 2 digits in base 3/2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.

Cf. A005428, A070885, A073941, A081848, A024629, A246435, A304024, A304025, A303500, A304272, A304274.

A120135 a(n) = 5 + floor((1 + Sum_{j=1..n-1} a(j)) / 2).

Original entry on oeis.org

5, 8, 12, 18, 27, 40, 60, 90, 135, 203, 304, 456, 684, 1026, 1539, 2309, 3463, 5195, 7792, 11688, 17532, 26298, 39447, 59171, 88756, 133134, 199701, 299552, 449328, 673992, 1010988, 1516482, 2274723, 3412084, 5118126, 7677189, 11515784

Offset: 1

Graeme McRae, Jun 10 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A073941, A072493, A112088, A120134, A120136 - A120209.

a[n_]:= a[n]= 5 +Floor[(1+Sum[a[k], {k,n-1}])/2];
Table[a[n], {n,60}] (* G. C. Greubel, May 07 2023 *)

@CachedFunction
def A120135(n): return 5 + (1 + sum(A120135(k) for k in range(1,n)))//2
[A120135(n) for n in range(1,61)] # G. C. Greubel, May 07 2023

a(n) ~ c * (3/2)^n, where c = 3.514931952760438754899508881646642282344325354834703833076259269449577... - Vaclav Kotesovec, May 07 2023

A120136 a(n) = 7 + floor(Sum_{j=1..n-1} a(j) / 2).

Original entry on oeis.org

7, 10, 15, 23, 34, 51, 77, 115, 173, 259, 389, 583, 875, 1312, 1968, 2952, 4428, 6642, 9963, 14945, 22417, 33626, 50439, 75658, 113487, 170231, 255346, 383019, 574529, 861793, 1292690, 1939035, 2908552, 4362828, 6544242, 9816363, 14724545

Offset: 1

Graeme McRae, Jun 10 2006

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A073941, A072493, A112088, A120134, A120135, A120137 - A120209.

nxt[{t_,a_}] := Module[{c=7+Floor[t/2]},{t+c,c}];
NestList[nxt,{7,7},40][[All,2]] (* Harvey P. Dale, Jan 13 2017 *)

@CachedFunction
def A120136(n): return 7 +sum(A120136(k) for k in range(1,n))//2
[A120136(n) for n in range(1,60)] # G. C. Greubel, May 08 2023

A120137 a(n) = 8 + floor( (1 + Sum_{j=1..n-1} a(j)) / 2).

Original entry on oeis.org

8, 12, 18, 27, 41, 61, 92, 138, 207, 310, 465, 698, 1047, 1570, 2355, 3533, 5299, 7949, 11923, 17885, 26827, 40241, 60361, 90542, 135813, 203719, 305579, 458368, 687552, 1031328, 1546992, 2320488, 3480732, 5221098, 7831647, 11747471, 17621206

Offset: 1

Graeme McRae, Jun 10 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A073941, A072493, A112088, A120134 - A120136, A120138 - A120209.

a[n_]:= a[n]= 8 +Floor[(1 +Sum[a[k], {k,n-1}])/2];
Table[a[n], {n,60}] (* G. C. Greubel, May 08 2023 *)
nxt[{t_,a_}]:=Module[{c=8+Floor[(1+t)/2]},{t+c,c}]; NestList[nxt,{8,8},40][[;;,2]] (* Harvey P. Dale, Sep 10 2023 *)

@CachedFunction
def A120137(n): return 8 +(1 +sum(A120137(k) for k in range(1,n)))//2
[A120137(n) for n in range(1,60)] # G. C. Greubel, May 08 2023

A120138 a(n) = 10 + floor(Sum_{j=1..n-1} a(j) / 2).

Original entry on oeis.org

10, 15, 22, 33, 50, 75, 112, 168, 252, 378, 567, 851, 1276, 1914, 2871, 4307, 6460, 9690, 14535, 21803, 32704, 49056, 73584, 110376, 165564, 248346, 372519, 558779, 838168, 1257252, 1885878, 2828817, 4243226, 6364839, 9547258, 14320887

Offset: 1

Graeme McRae, Jun 10 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A073941, A072493, A112088, A120134 - A120137, A120139 - A120209.

a[n_]:= a[n]= 10 +Quotient[Sum[a[k], {k,n-1}],2];
Table[a[n], {n,60}] (* G. C. Greubel, May 08 2023 *)

@CachedFunction
def A120138(n): return 10 +sum(A120138(k) for k in range(1,n))//2
[A120138(n) for n in range(1,60)] # G. C. Greubel, May 08 2023

A120139 a(n) = 11 + floor( (1 + Sum_{j=1..n-1} a(j)) / 2).

Original entry on oeis.org

11, 17, 25, 38, 57, 85, 128, 192, 288, 432, 648, 972, 1458, 2187, 3280, 4920, 7380, 11070, 16605, 24908, 37362, 56043, 84064, 126096, 189144, 283716, 425574, 638361, 957542, 1436313, 2154469, 3231704, 4847556, 7271334, 10907001, 16360501

Offset: 1

Graeme McRae, Jun 10 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A073941, A072493, A112088, A120134 - A120138, A120140 - A120209.

a[n_]:= a[n]= 11 +Quotient[1 + Sum[a[k], {k,n-1}], 2];
Table[a[n], {n,60}] (* G. C. Greubel, May 08 2023 *)

@CachedFunction
def A120139(n): return 11 +(1 +sum(A120139(k) for k in range(1,n)))//2
[A120139(n) for n in range(1,60)]  # G. C. Greubel, May 08 2023

cp's OEIS Frontend

A081614 Subsequence of A005428 with state = 1.

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A081615 Subsequence of A005428 where state = 2.

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A082125 Smallest difference>1 between d and p/d for any divisor d of the partial product p of the sequence, starting with 4.

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A120169 a(n) = 12 + floor((1 + Sum_{j=1..n-1} a(j))/4).

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Magma

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SageMath

A304273 The concatenation of the first n terms is the smallest positive even number with n digits when written in base 3/2 (cf. A024629).

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Maple

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A120135 a(n) = 5 + floor((1 + Sum_{j=1..n-1} a(j)) / 2).

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A120136 a(n) = 7 + floor(Sum_{j=1..n-1} a(j) / 2).

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Mathematica

SageMath

A120137 a(n) = 8 + floor( (1 + Sum_{j=1..n-1} a(j)) / 2).

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