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Search: a171947 -id:a171947
Displaying 1-5 of 5 results found.
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A249034
Odd numbers missing from A171947.
+20
5
5, 13, 17, 21, 29, 37, 45, 49, 53, 61, 65, 69, 77, 81, 85, 93, 101, 109, 113, 117, 125, 133, 141, 145, 149, 157, 165, 173, 177, 181, 189, 193, 197, 205, 209, 213, 221, 229, 237, 241, 245, 253, 257, 261, 269, 273, 277, 285, 293, 301, 305, 309, 317, 321, 325
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
These are the odd terms in A171946.
Consider the sequence of first differences, divided by 4: 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, ... This is, almost certainly, A026465 without its leading 1. - N. J. A. Sloane, Oct 30 2014
This sequence appears to be the same as A260191, Numbers n such that there exists no square whose base-n digit sum is binomial(n,2), without that sequence's leading 3. - Jon E. Schoenfield, Jul 19 2015
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
MATHEMATICA
f[n_] := Block[{a = {1}, b = {}, k}, Do[k = 2; While[MemberQ[a, k] || MemberQ[b, k], k++]; AppendTo[a, 2 k - 1]; AppendTo[b, k], {i, 2, n}]; a]; Complement[Range[1, Max@ #, 2], #] &@ f@ 120 (* Michael De Vlieger, Jul 20 2015 *)
PROG
(Haskell)
a249034 n = a249034_list !! (n-1)
a249034_list = filter odd a171946_list
-- Reinhard Zumkeller, Oct 26 2014
CROSSREFS
Cf. A026465, A171946, A171947, A260191.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 26 2014
STATUS
approved
A171946
N-positions for game of UpMark.
+10
6
0, 2, 4, 5, 6, 8, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 26, 28, 29, 30, 32, 34, 36, 37, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 92, 93, 94, 96, 98
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
It appears that this is the sequence of positions of 0 in the 1-limiting word of the morphism 0 -> 10, 1 -> 00; see A284948. - Clark Kimberling, Apr 18 2017
It appears that this sequence gives the positions of 1 in the limiting 0-word of the morphism 0->11, 1-> 01. See A285383. - Clark Kimberling, Apr 26 2017
Apparently a(n) = 1+A003159(n-1). - R. J. Mathar, Jun 24 2021
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.
PROG
(Haskell)
import Data.List (delete)
a171946 n = a171946_list !! (n-1)
a171946_list = 0 : f [2..] where
f (w:ws) = w : f (delete (2 * w - 1) ws)
-- Reinhard Zumkeller, Oct 26 2014
CROSSREFS
Complement of A171947.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Oct 29 2010
STATUS
approved
A284948
1-limiting word of the morphism 0 -> 10, 1 -> 00
+10
4
1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
1
COMMENTS
Consider iterations of the morphism defined by 0 -> 10, 1 -> 00: 0 -> 10 -> 0010 -> 10100010 -> 0010001010100010 -> ... There are two limiting words, one of which has initial term 1 and the other, 0. These are fixed points of the morphism squared: 0-> 0010, 1->1010. [Corrected by Michel Dekking, Jan 06 2019]
The 0-limiting word is 0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,... (A328979). It is the characteristic sequence of those natural numbers whose binary representation ends in an odd numbers of zeros, sequence A036554, but with offset 0 (easy to see from the fact that if the binary representation of N is equal to w, then the binary representations of 4N, 4N+1, 4N+2 and 4N+3 are w00, w01, w10 and w11). - Michel Dekking, Jan 06 2019
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
MAPLE
f(0):= (0, 0, 1, 0): f(1):= (1, 0, 1, 0):
A:= [0]: # if start at 0 get A328979, if start at 1 get the present sequence
for i from 1 to 8 do A:= map(f, A) od:
A; # N. J. A. Sloane, Nov 05 2019
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {0, 0}}] &, {0}, 7] (* A284948 *)
u = Flatten[Position[s, 0]] (* A171946 *)
v = Flatten[Position[s, 1]] (* A171947 *)
CROSSREFS
Cf. A036554, A171946, A171947, A328979.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 18 2017
STATUS
approved
A285383
Limiting 0-word of the morphism 0 -> 11, 1 -> 01.
+10
4
0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
1
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
Index entries for sequences that are fixed points of mappings
FORMULA
a(n) = A285384(n) for n>=2.
Conjecture: a(n) = A035263(n-1). - R. J. Mathar, May 08 2017
EXAMPLE
0 -> 11-> 0101 -> 11011101 -> 0101110101011101 ->
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {0, 1}}] &, {0}, 10] (* A285383 *)
Flatten[Position[s, 0]] (* A171947 *)
Flatten[Position[s, 1]] (* A171946 *)
CROSSREFS
Cf. A171947, 171946, A285384.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 26 2017
STATUS
approved
A340780
Losing positions n (P-positions) in the following game: two players take turns dividing the current value of n by either a prime power > 1 or by A007947(n) to obtain the new value of n. The winner is the player whose division results in 1.
+10
0
1, 12, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 120, 124, 147, 148, 153, 164, 168, 171, 172, 175, 188, 207, 212, 216, 236, 242, 244, 245, 261, 264, 268, 270, 275, 279, 280, 284, 292, 312, 316, 325, 332, 333, 338, 356, 363, 369, 378, 387, 388
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The game is equivalent to the game of Nim with the additional allowed move consisting of removing one object from each pile.
LINKS
Table of n, a(n) for n=1..57.
MATHEMATICA
Clear[moves, los]; A003557[n_]:= {Module[{aux = FactorInteger[n], L=Length[FactorInteger[n]]}, Product[aux[[i, 1]]^(aux[[i, 2]]-1), {i, L}]]};
moves[n_] :=moves[n] = Module[{aux = FactorInteger[n], L=Length[ FactorInteger [n]]}, Union[Flatten[Table[n/aux[[i, 1]]^j, {i, 1, L}, {j, 1, aux[[i, 2]]}], 1], A003557[n]]]; los[1]=True; los[m_] := los[m] = If[PrimeQ[m], False, Union@Flatten@Table[los[moves[m][[i]]], {i, 1, Length[moves[m]]}] == {False}]; Select[Range[400], los]
CROSSREFS
Cf. A003557, A227691, A227763, A227764, A171947, A005240, A081691, A099352, A171945, A171949, A285304, A120442, A137295, A275432.
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Jan 21 2021
STATUS
approved
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