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Maximum starting value of X such that repeated replacement of X with X-ceiling(X/5) requires n steps to reach 0.
+10
6
0, 1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 37, 47, 59, 74, 93, 117, 147, 184, 231, 289, 362, 453, 567, 709, 887, 1109, 1387, 1734, 2168, 2711, 3389, 4237, 5297, 6622, 8278, 10348, 12936, 16171, 20214, 25268, 31586, 39483, 49354, 61693, 77117, 96397, 120497
COMMENTS
Limit_{n->oo} a(n)/(5/4)^n = 2.68723058270145442816383476567331957329199286146873...
FORMULA
a(n) = floor(a(n-1)*5/4) + 1.
EXAMPLE
8 -> 8-ceiling(8/5) = 6,
6 -> 6-ceiling(6/5) = 4,
4 -> 4-ceiling(4/5) = 3,
3 -> 3-ceiling(3/5) = 2,
2 -> 2-ceiling(2/5) = 1,
1 -> 1-ceiling(1/5) = 0,
so reaching 0 from 8 requires 6 steps;
9 -> 9-ceiling(9/5) = 7,
7 -> 7-ceiling(7/5) = 5,
5 -> 5-ceiling(5/5) = 4,
4 -> 4-ceiling(4/5) = 3,
3 -> 3-ceiling(3/5) = 2,
2 -> 2-ceiling(2/5) = 1,
1 -> 1-ceiling(1/5) = 0,
so reaching 0 from 9 (or more) requires 7 (or more) steps;
thus, 8 is the largest starting value from which 0 can be reached in 6 steps, so a(6) = 8.
MATHEMATICA
RecurrenceTable[{a[1] == 0, a[n] == Floor[a[n-1] 5/4] + 1}, a, {n, 50}] (* Vincenzo Librandi, Dec 06 2016 *)
PROG
(Magma) a:=[0]; aCurr:=0; for n in [1..48] do aCurr:=Floor(aCurr*5/4)+1; a[#a+1]:=aCurr; end for; a;
(Magma) [n eq 1 select n-1 else Floor(Self(n-1)*5/4)+1: n in [1..70]]; // Vincenzo Librandi, Dec 06 2016
CROSSREFS
See the following sequences for maximum starting value of X such that repeated replacement of X with X-ceiling(X/k) requires n steps to reach 0: A000225 (k=2), A006999 (k=3), A155167 (k=4, apparently; see Formula entry there), (this sequence) (k=5), A279076 (k=6), A279077 (k=7), A279078 (k=8), A279079 (k=9), A279080 (k=10). For each of these values of k, is the sequence the L-sieve transform of {k-1, 2k-1, 3k-1, ...}?
Maximum starting value of X such that repeated replacement of X with X-ceiling(X/10) requires n steps to reach 0.
+10
6
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 22, 25, 28, 32, 36, 41, 46, 52, 58, 65, 73, 82, 92, 103, 115, 128, 143, 159, 177, 197, 219, 244, 272, 303, 337, 375, 417, 464, 516, 574, 638, 709, 788, 876, 974, 1083, 1204, 1338, 1487, 1653, 1837, 2042, 2269
COMMENTS
Limit_{n->oo} a(n)/(10/9)^n = 5.60655601136196116133057876294687807265035051745268...
FORMULA
a(n) = floor(a(n-1)*10/9) + 1.
EXAMPLE
13 -> 13-ceiling(13/10) = 11,
11 -> 11-ceiling(11/10) = 9,
9 -> 9-ceiling(9/10) = 8,
8 -> 8-ceiling(8/10) = 7,
...
1 -> 1-ceiling(1/10) = 0,
so reaching 0 from 13 requires 11 steps;
14 -> 14-ceiling(14/10) = 12,
12 -> 12-ceiling(12/10) = 10,
10 -> 10-ceiling(10/10) = 9,
9 -> 9-ceiling(9/10) = 8,
8 -> 8-ceiling(8/10) = 7,
...
1 -> 1-ceiling(1/10) = 0,
so reaching 0 from 14 (or more) requires 12 (or more) steps;
thus, 13 is the largest starting value from which 0 can be reached in 11 steps, so a(11) = 13.
MAPLE
H:= proc(y) local u, v;
v:= -y-1 mod 9+1;
(10*y+v)/9
end proc:
A:= Array(0..100):
A[0]:= 0:
for i from 1 to 100 do A[i]:= H(A[i-1]) od:
MATHEMATICA
With[{s = Array[-1 + Length@ NestWhileList[# - Ceiling[#/10] &, #, # > 0 &] &, 2400, 0]}, Array[-1 + Position[s, #][[-1, 1]] &, Max@ s, 0]] (* Michael De Vlieger, Jun 23 2020 *)
PROG
(Magma) a:=[0]; aCurr:=0; for n in [1..57] do aCurr:=Floor(aCurr*10/9)+1; a[#a+1]:=aCurr; end for; a;
CROSSREFS
See the following sequences for maximum starting value of X such that repeated replacement of X with X-ceiling(X/k) requires n steps to reach 0: A000225 (k=2), A006999 (k=3), A155167 (k=4, apparently; see Formula entry there), A279075 (k=5), A279076 (k=6), A279077 (k=7), A279078 (k=8), A279079 (k=9), (this sequence) (k=10). For each of these values of k, is the sequence the L-sieve transform of {k-1, 2k-1, 3k-1, ...}?
Maximum starting value of X such that repeated replacement of X with X-ceiling(X/6) requires n steps to reach 0.
+10
5
0, 1, 2, 3, 4, 5, 7, 9, 11, 14, 17, 21, 26, 32, 39, 47, 57, 69, 83, 100, 121, 146, 176, 212, 255, 307, 369, 443, 532, 639, 767, 921, 1106, 1328, 1594, 1913, 2296, 2756, 3308, 3970, 4765, 5719, 6863, 8236, 9884, 11861, 14234, 17081, 20498, 24598, 29518, 35422
COMMENTS
Limit_{n->oo} a(n)/(6/5)^n = 3.24387249751177521384734853905517802618171089570674...
FORMULA
a(n) = floor(a(n-1)*6/5) + 1.
EXAMPLE
7 -> 7-ceiling(7/6) = 5,
5 -> 5-ceiling(5/6) = 4,
4 -> 4-ceiling(4/6) = 3,
3 -> 3-ceiling(3/6) = 2,
2 -> 2-ceiling(2/6) = 1,
1 -> 1-ceiling(1/6) = 0,
so reaching 0 from 7 requires 6 steps;
8 -> 8-ceiling(8/6) = 6,
6 -> 6-ceiling(6/6) = 5,
5 -> 5-ceiling(5/6) = 4,
4 -> 4-ceiling(4/6) = 3,
3 -> 3-ceiling(3/6) = 2,
2 -> 2-ceiling(2/6) = 1,
1 -> 1-ceiling(1/6) = 0,
so reaching 0 from 8 (or more) requires 7 (or more) steps;
thus, 7 is the largest starting value from which 0 can be reached in 6 steps, so a(6) = 7.
PROG
(Magma) a:=[0]; aCurr:=0; for n in [1..51] do aCurr:=Floor(aCurr*6/5)+1; a[#a+1]:=aCurr; end for; a;
CROSSREFS
See the following sequences for maximum starting value of X such that repeated replacement of X with X-ceiling(X/k) requires n steps to reach 0: A000225 (k=2), A006999 (k=3), A155167 (k=4, apparently; see Formula entry there), A279075 (k=5), (this sequence) (k=6), A279077 (k=7), A279078 (k=8), A279079 (k=9), A279080 (k=10). For each of these values of k, is the sequence the L-sieve transform of {k-1, 2k-1, 3k-1, ...}?
Maximum starting value of X such that repeated replacement of X with X-ceiling(X/7) requires n steps to reach 0.
+10
5
0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 26, 31, 37, 44, 52, 61, 72, 85, 100, 117, 137, 160, 187, 219, 256, 299, 349, 408, 477, 557, 650, 759, 886, 1034, 1207, 1409, 1644, 1919, 2239, 2613, 3049, 3558, 4152, 4845, 5653, 6596, 7696, 8979, 10476, 12223, 14261
COMMENTS
Limit_{n->oo} a(n)/(7/6)^n = 4.03710211215303193642791458111196922950551168987041...
FORMULA
a(n) = floor(a(n-1)*7/6) + 1.
EXAMPLE
10 -> 10-ceiling(10/7) = 8,
8 -> 8-ceiling(8/7) = 6,
6 -> 6-ceiling(6/7) = 5,
5 -> 5-ceiling(5/7) = 4,
4 -> 4-ceiling(4/7) = 3,
3 -> 3-ceiling(3/7) = 2,
2 -> 2-ceiling(2/7) = 1,
1 -> 1-ceiling(1/7) = 0,
so reaching 0 from 10 requires 8 steps;
11 -> 11-ceiling(11/7) = 9,
9 -> 9-ceiling(9/7) = 7,
7 -> 7-ceiling(7/7) = 6,
6 -> 6-ceiling(6/7) = 5,
5 -> 5-ceiling(5/7) = 4,
4 -> 4-ceiling(4/7) = 3,
3 -> 3-ceiling(3/7) = 2,
2 -> 2-ceiling(2/7) = 1,
1 -> 1-ceiling(1/7) = 0,
so reaching 0 from 11 (or more) requires 9 (or more) steps;
thus, 10 is the largest starting value from which 0 can be reached in 8 steps, so a(8) = 10.
PROG
(Magma) a:=[0]; aCurr:=0; for n in [1..53] do aCurr:=Floor(aCurr*7/6)+1; a[#a+1]:=aCurr; end for; a;
CROSSREFS
See the following sequences for maximum starting value of X such that repeated replacement of X with X-ceiling(X/k) requires n steps to reach 0: A000225 (k=2), A006999 (k=3), A155167 (k=4, apparently; see Formula entry there), A279075 (k=5), A279076 (k=6), (this sequence) (k=7), A279078 (k=8), A279079 (k=9), A279080 (k=10). For each of these values of k, is the sequence the L-sieve transform of {k-1, 2k-1, 3k-1, ...}?
Maximum starting value of X such that repeated replacement of X with X-ceiling(X/8) requires n steps to reach 0.
+10
5
0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 18, 21, 25, 29, 34, 39, 45, 52, 60, 69, 79, 91, 105, 121, 139, 159, 182, 209, 239, 274, 314, 359, 411, 470, 538, 615, 703, 804, 919, 1051, 1202, 1374, 1571, 1796, 2053, 2347, 2683, 3067, 3506, 4007, 4580, 5235, 5983, 6838
COMMENTS
Limit_{n->oo} a(n)/(8/7)^n = 4.42210347959393228709604412445802201220907917744900...
FORMULA
a(n) = floor(a(n-1)*8/7) + 1.
EXAMPLE
11 -> 11-ceiling(11/8) = 9,
9 -> 9-ceiling(9/8) = 7,
7 -> 7-ceiling(7/8) = 6,
6 -> 6-ceiling(6/8) = 5,
...
1 -> 1-ceiling(1/8) = 0,
so reaching 0 from 11 requires 9 steps;
12 -> 12-ceiling(12/8) = 10,
10 -> 10-ceiling(10/8) = 8,
8 -> 8-ceiling(8/8) = 7,
7 -> 7-ceiling(7/8) = 6,
...
1 -> 1-ceiling(1/8) = 0,
so reaching 0 from 12 (or more) requires 10 (or more) steps;
thus, 11 is the largest starting value from which 0 can be reached in 9 steps, so a(9) = 11.
PROG
(Magma) a:=[0]; aCurr:=0; for n in [1..55] do aCurr:=Floor(aCurr*8/7)+1; a[#a+1]:=aCurr; end for; a;
CROSSREFS
See the following sequences for maximum starting value of X such that repeated replacement of X with X-ceiling(X/k) requires n steps to reach 0: A000225 (k=2), A006999 (k=3), A155167 (k=4, apparently; see Formula entry there), A279075 (k=5), A279076 (k=6), A279077 (k=7), (this sequence) (k=8), A279079 (k=9), A279080 (k=10). For each of these values of k, is the sequence the L-sieve transform of {k-1, 2k-1, 3k-1, ...}?
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