A291770 -id:A291770 - cp's OEIS frontend

A291771 Filter based on runlengths of 0-digits in base-3 expansion of n: a(n) = A278222(A291770(n)).

1, 1, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 8, 4, 4, 6, 2, 2, 6, 2, 2, 4, 2, 2, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 8, 4, 4, 6, 2, 2, 6, 2, 2, 4, 2, 2, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 16, 8, 8, 12, 4, 4, 12, 4, 4, 12, 6, 6, 6, 2, 2, 6, 2, 2, 12, 6, 6, 6, 2, 2, 6

Offset: 1

Antti Karttunen, Sep 11 2017

Antti Karttunen, Table of n, a(n) for n = 1..59049

Cf. A007089, A077267, A278222, A291770, A290091, A290092, A290093, A290094.

f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; Array[If[# == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater]]] &@ f[FromDigits[IntegerDigits[#, 3] /. k_ /; k < 3 :> If[k == 0, 1, 0], 2], 1, 1] &, 96] (* Michael De Vlieger, Sep 11 2017 *)

a(n) = A278222(A291770(n)).

A292244 Base-2 expansion of a(n) encodes the steps where multiples of 3 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 0, 2, 5, 0, 0, 1, 0, 0, 1, 12, 6, 7, 14, 0, 1, 0, 4, 1, 8, 10, 3, 0, 0, 21, 24, 0, 1, 28, 2, 3, 2, 0, 1, 0, 0, 5, 2, 2, 1, 22, 24, 17, 0, 12, 33, 32, 14, 35, 42, 28, 45, 24, 0, 1, 16, 2, 11, 48, 0, 59, 0, 8, 3, 0, 2, 5, 0, 16, 1, 4, 20, 3, 6, 6, 7, 8, 0, 1, 56, 0, 3, 0, 42, 5, 0, 48, 5, 0, 0, 1, 14, 2, 65, 64, 56, 49, 44, 4, 49, 64, 6, 57, 0

Offset: 1

Antti Karttunen, Sep 15 2017

			For n = 3, the starting value is a multiple of three, after which follows A253889(3) = 1, the end point of iteration, which is not a multiple of three, thus a(3) = 1*(2^0) = 1.
For n = 8, the starting value is not a multiple of three, after which follows A253889(8) = 3, which is, thus a(8) = 0*(2^0) + 1*(2^1) = 2.
For n = 9, the starting value is a multiple of three, after which follows A253889(9) = 8 (which is not), while A253889(8) = 3 (which is), thus a(9) = 1*(2^0) + 0*(2^1) + 1*(2^2) = 5.

Cf. A048673, A064216, A253889, A291770, A292243, A292247.

Cf. also A292245, A292246, and A292381, A292383, A292385, and A292590, A292591 for similarly constructed sequences, and also A292250.

f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1];g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n;Table[FromDigits[#, 2] &@ Map[Boole[Divisible[#, 3]] &,  Reverse@ NestWhileList[Floor@ g[Floor[f[#]/2]] &, n, # > 1 &]], {n, 109}] (* Michael De Vlieger, Sep 16 2017 *)

(define (A292244 n) (A291770 (A292243 n)))

a(n) = A291770(A292243(n)).
Other identities. For all n >= 1:
a(A048673(n)) = A292247(n).
a(n) + A292245(n) = A064216(n).
a(n) AND A292245(n) = a(n) AND A292246(n) = 0, where AND is a bitwise-AND (A004198).

A292250 Binary encoding of 0-digits in ternary representation of A048673(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 3, 0, 2, 2, 1, 4, 6, 2, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 5, 6, 1, 4, 1, 4, 0, 8, 0, 14, 1, 4, 12, 0, 7, 0, 4, 2, 1, 0, 5, 8, 2, 0, 4, 2, 4, 0, 4, 6, 5, 0, 5, 8, 3, 12, 2, 4, 2, 8, 0, 4, 1, 8, 3, 2, 1, 16, 16, 2, 3, 28, 0, 0, 1, 8, 0, 26, 8, 0, 9, 0, 15, 0, 1, 10, 14, 4, 0, 4, 7, 0, 4, 12, 6, 16, 5, 6, 8, 0, 2, 10, 9, 4

Offset: 1

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A048673, A291759, A291770, A292251, A292252.

Cf. also A292240, A292260.

Map[FromDigits[IntegerDigits[#, 3] /. k_ /; k < 3 :> If[k == 0, 1, 0], 2] &, Table[(Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n, {n, 116}]] (* Michael De Vlieger, Sep 12 2017 *)

a(n) = A291770(A048673(n)).

A292260 Binary encoding of 0-digits in ternary representation of A245612(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 2, 0, 0, 3, 2, 1, 0, 16, 8, 10, 4, 1, 2, 4, 0, 12, 0, 1, 4, 6, 0, 0, 0, 41, 34, 0, 16, 33, 22, 11, 8, 2, 0, 10, 4, 21, 10, 12, 0, 5, 26, 23, 0, 4, 4, 3, 8, 5, 14, 2, 0, 5, 2, 3, 0, 146, 80, 132, 68, 43, 2, 180, 32, 0, 68, 81, 44, 12, 16, 2, 16, 33, 6, 48, 0, 9, 22, 33, 8, 54, 40, 8, 20, 11, 26, 2, 0, 126, 8, 9, 52, 0, 48, 52, 0

Offset: 0

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A245612, A291763, A291770, A292261, A292262.

Cf. also A292240, A292260.

a(n) = A291770(A245612(n)).

A292240 Binary encoding of 0-digits in ternary representation of A254103(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 2, 0, 0, 0, 0, 2, 0, 0, 0, 1, 4, 3, 2, 1, 0, 3, 2, 1, 0, 7, 6, 0, 0, 3, 2, 4, 0, 0, 0, 1, 8, 0, 0, 6, 4, 4, 4, 2, 0, 8, 8, 6, 4, 4, 4, 5, 0, 0, 0, 1, 12, 3, 2, 0, 0, 8, 8, 9, 4, 11, 10, 0, 0, 3, 2, 4, 0, 0, 0, 2, 16, 3, 2, 1, 0, 15, 14, 0, 8, 11, 10, 1, 8, 7, 6, 5, 0, 19, 18, 1, 16, 15, 14, 13, 8, 11

Offset: 0

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A254103, A291760, A291770, A292241, A292242.

Cf. also A292250, A292260.

a(n) = A291770(A254103(n)).

A292370 A binary encoding of the zeros in base-4 representation of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 7, 6, 6, 6, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 3, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 2, 2, 2, 1, 0, 0, 0, 1

Offset: 0

Antti Karttunen, Sep 15 2017

			   n      a(n)     base-4(n)  binary(a(n))
                  A007090(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        0          1           0
   2        0          2           0
   3        0          3           0
   4        1         10           1
   5        0         11           0
   6        0         12           0
   7        0         13           0
   8        1         20           1
   9        0         21           0
  10        0         22           0
  11        0         23           0
  12        1         30           1
  13        0         31           0
  14        0         32           0
  15        0         33           0
  16        3        100          11
  17        2        101          10

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A007088, A007090, A160380, A292371, A292372, A292373.

Cf. A291770 (analogous sequence for base-3).

Table[FromDigits[IntegerDigits[n, 4] /. k_ /; IntegerQ@ k :> If[k == 0, 1, 0], 2], {n, 0, 120}] (* Michael De Vlieger, Sep 21 2017 *)

from sympy.ntheory.factor_ import digits
def a(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join('1' if i==0 else '0' for i in k), 2)
print([a(n) for n in range(111)]) # Indranil Ghosh, Sep 21 2017

(define (A292370 n) (if (zero? n) n (let loop ((n n) (b 1) (s 0)) (if (< n 4) s (let ((d (modulo n 4))) (if (zero? d) (loop (/ n 4) (+ b b) (+ s b)) (loop (/ (- n d) 4) (+ b b) s)))))))

For all n >= 0, A000120(a(n)) = A160380(n).

A343230 A binary encoding of the digits "0" in balanced ternary representation of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 2, 3, 2, 0, 1, 0, 0, 1, 0, 2, 3, 2, 0, 1, 0, 4, 5, 4, 6, 7, 6, 4, 5, 4, 0, 1, 0, 2, 3, 2, 0, 1, 0, 0, 1, 0, 2, 3, 2, 0, 1, 0, 4, 5, 4, 6, 7, 6, 4, 5, 4, 0, 1, 0, 2, 3, 2, 0, 1, 0, 8, 9, 8, 10, 11, 10, 8, 9, 8, 12, 13, 12, 14, 15, 14, 12

Offset: 0

Rémy Sigrist, Apr 08 2021

The ones in the binary representation of a(n) correspond to the nonleading digits "0" in the balanced ternary representation of n.

We can extend this sequence to negative indices: a(-n) = a(n) for any n >= 0.

			The first terms, alongside the balanced ternary representation of n (with "T" instead of digits "-1") and the binary representation of a(n), are:
  n   a(n)  ter(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     0      1T          0
   3     1      10          1
   4     0      11          0
   5     0     1TT          0
   6     1     1T0          1
   7     0     1T1          0
   8     2     10T         10
   9     3     100         11
  10     2     101         10
  11     0     11T          0
  12     1     110          1
  13     0     111          0
  14     0    1TTT          0
  15     1    1TT0          1

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Wikipedia, Balanced ternary

Cf. A059095, A140267, A291770, A343228, A343229, A343231, A147991 (indices of 0's).

a(n) = { my (v=0, b=1, t); while (n, t=centerlift(Mod(n, 3)); if (t==0, v+=b); n=(n-t)\3; b*=2); v }

A305297 Restricted growth sequence transform of A292260, formed from 0-digits in ternary representation of A245612(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 4, 1, 1, 3, 4, 2, 1, 5, 6, 7, 8, 2, 4, 8, 1, 9, 1, 2, 8, 10, 1, 1, 1, 11, 12, 1, 5, 13, 14, 15, 6, 4, 1, 7, 8, 16, 7, 9, 1, 17, 18, 19, 1, 8, 8, 3, 6, 17, 20, 4, 1, 17, 4, 3, 1, 21, 22, 23, 24, 25, 4, 26, 27, 1, 24, 28, 29, 9, 5, 4, 5, 13, 10, 30, 1, 31, 14, 13, 6, 32, 33, 6, 34, 15, 18, 4, 1, 35, 6, 31, 36, 1, 30, 36, 1, 37

Offset: 0

Antti Karttunen, May 31 2018

nonn

For all i, j: a(i) = a(j) => A292261(i) = A292261(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A291770, A292260, A245612, A292261.
Cf. also A305296, A305298.
Cf. also A304750.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A254049(n) = A048673((2*n)-1);
A245612(n) = if(n<2,1+n,if(!(n%2),(3*A245612(n/2))-1,A254049(A245612((n-1)/2))));
A291770(n) = { my(s=0, b=1, d); while(n>2, if(!(n%3), s += b); b <<= 1; n \= 3); (s); };
A292260(n) = A291770(A245612(n));
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v305297 = rgs_transform(vector(65538,n,A292260(n-1)));
A305297(n) = v305297[1+n];

A304750 Restricted growth sequence transform of A292240(n), formed from 0-digits in ternary representation of A254103(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 4, 1, 1, 1, 1, 4, 1, 1, 1, 2, 5, 3, 4, 2, 1, 3, 4, 2, 1, 6, 7, 1, 1, 3, 4, 5, 1, 1, 1, 2, 8, 1, 1, 7, 5, 5, 5, 4, 1, 8, 8, 7, 5, 5, 5, 9, 1, 1, 1, 2, 10, 3, 4, 1, 1, 8, 8, 11, 5, 12, 13, 1, 1, 3, 4, 5, 1, 1, 1, 4, 14, 3, 4, 2, 1, 15, 16, 1, 8, 12, 13, 2, 8, 6, 7, 9, 1, 17, 18, 2, 14, 15, 16, 19, 8, 12

Offset: 0

Antti Karttunen, May 30 2018

nonn

For all i, j: a(i) = a(j) => A292241(i) = A292241(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A254103, A291770, A292240.

Cf. also A292241, A304740, A304746.

A254103(n) = if(!n,n,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2));
A291770(n) = { my(s=0, b=1, d); while(n>2, if(!(n%3), s += b); b <<= 1; n \= 3); (s); };
A292240(n) = A291770(A254103(n));
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v304750 = rgs_transform(vector(65538,n,A292240(n-1)));
A304750(n) = v304750[1+n];

cp's OEIS Frontend

A291771 Filter based on runlengths of 0-digits in base-3 expansion of n: a(n) = A278222(A291770(n)).

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A292244 Base-2 expansion of a(n) encodes the steps where multiples of 3 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

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A292250 Binary encoding of 0-digits in ternary representation of A048673(n).

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A292260 Binary encoding of 0-digits in ternary representation of A245612(n).

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A292240 Binary encoding of 0-digits in ternary representation of A254103(n).

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A292370 A binary encoding of the zeros in base-4 representation of n.

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A343230 A binary encoding of the digits "0" in balanced ternary representation of n.

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A305297 Restricted growth sequence transform of A292260, formed from 0-digits in ternary representation of A245612(n).

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A304750 Restricted growth sequence transform of A292240(n), formed from 0-digits in ternary representation of A254103(n).

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