A292383 -id:A292383 - cp's OEIS frontend

A292943 a(n) = A292944(A243071(n)); Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+3 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

0, 0, 1, 0, 2, 2, 4, 0, 1, 4, 8, 4, 16, 8, 5, 0, 32, 2, 64, 8, 9, 16, 128, 8, 2, 32, 1, 16, 256, 10, 512, 0, 17, 64, 10, 4, 1024, 128, 33, 16, 2048, 18, 4096, 32, 9, 256, 8192, 16, 4, 4, 65, 64, 16384, 2, 18, 32, 129, 512, 32768, 20, 65536, 1024, 17, 0, 34, 34, 131072, 128, 257, 20, 262144, 8, 524288, 2048, 5, 256, 20, 66, 1048576, 32, 1, 4096

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences related to binary expansion of n

Cf. A005940, A163511, A243071, A292941, A292944, A292945.

Cf. also A292253, A292255, A292383.

(define (A292943 n) (A292944 (A243071 n)))
(define (A292943 n) (if (<= n 1) 0 (+ (if (= 3 (modulo n 6)) 1 0) (* 2 (A292943 (A252463 n))))))

a(n) = A292944(A243071(n)).
a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 3 (mod 6)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 6k+3, and 0 otherwise.
For n >= 0, a(A163511(n)) = A292944(n).
For n >= 1, A292941(n) + a(n) + A292945(n) = a(n) + A292253(n) + A292255(n) = A243071(n).

A292253 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2a(A252463(n)) + [n == 1 (mod 2)][J(3|n) == 1], where J is the Jacobi-symbol.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 8, 4, 4, 8, 17, 8, 35, 16, 8, 8, 70, 8, 140, 16, 16, 34, 281, 16, 9, 70, 8, 32, 562, 16, 1124, 16, 32, 140, 17, 16, 2249, 280, 68, 32, 4498, 32, 8996, 68, 16, 562, 17993, 32, 19, 18, 140, 140, 35986, 16, 32, 64, 280, 1124, 71973, 32, 143947, 2248, 32, 32, 64, 64, 287894, 280, 560, 34, 575789, 32, 1151579, 4498, 16, 560, 34, 136, 2303158, 64

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

Base-2 expansion of a(n) encodes the steps where numbers that are either of the form 12k+1 or of the form 12k+11 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n. An exception is the most significant bit of a(n) which corresponds with the final 1, but is shifted one bit-position towards right.

The AND - XOR formula(s) just restate the fact that J(3|n) = J(-1|n)*J(-3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A243071, A292254, A292255, A292263, A292941, A292943, A292383.

(define (A292253 n) (if (<= n 2) (- n 1) (+ (if (and (odd? n) (= 1 (jacobi-symbol 3 n))) 1 0) (* 2 (A292253 (A252463 n))))))

a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == 1], where J is the Jacobi-symbol, and [ ]'s are Iverson brackets, whose product gives 1 only if n is an odd number for which J(3|n) = +1, and 0 otherwise.
a(n) = A292263(n) AND (A292383(n) XOR A292941(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987). [See comments.]
For n >= 0, a(A163511(n)) = A292254(n).
For n >= 1, a(n) + A292255(n) + A292943(n) = A243071(n).

A292255 a(1) = 0, and for n > 1, a(n) = 2a(A252463(n)) + [n == 1 (mod 2)][J(3|n) == -1], where J is the Jacobi-symbol.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 0, 0, 2, 6, 0, 12, 6, 0, 0, 25, 0, 51, 4, 4, 12, 102, 0, 0, 24, 0, 12, 205, 0, 411, 0, 12, 50, 0, 0, 822, 102, 24, 8, 1645, 8, 3291, 24, 0, 204, 6582, 0, 0, 0, 48, 48, 13165, 0, 9, 24, 100, 410, 26330, 0, 52660, 822, 8, 0, 25, 24, 105321, 100, 204, 0, 210642, 0, 421284, 1644, 0, 204, 1, 48, 842569, 16, 0, 3290, 1685138, 16, 48, 6582

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

Base-2 expansion of a(n) encodes the steps where numbers that are either of the form 12k+5 or of the form 12k+7 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

The AND - XOR formulas just restate the fact that J(3|n) = J(-1|n)*J(-3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A243071, A292253, A292256, A292263, A292943.

Cf. also A292941, A292945.

(define (A292255 n) (if (<= n 1) 0 (+ (if (and (odd? n) (= -1 (jacobi-symbol 3 n))) 1 0) (* 2 (A292255 (A252463 n))))))

a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == -1], where J is the Jacobi-symbol, and [ ]'s are Iverson brackets, whose product gives 1 only if n is an odd number for which J(3|n) = -1, and 0 otherwise.
a(n) = A292263(n) AND (A292383(n) XOR A292945(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
a(n) = A292263(n) AND (A292385(n) XOR A292941(n)). [See comments.]
For n >= 0, a(A163511(n)) = A292256(n).
For n >= 1, a(n) + A292253(n) + A292943(n) = A243071(n).

A292382 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+2 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 8, 4, 4, 9, 16, 10, 32, 17, 10, 8, 64, 9, 128, 18, 18, 33, 256, 20, 8, 65, 8, 34, 512, 21, 1024, 16, 34, 129, 20, 18, 2048, 257, 66, 36, 4096, 37, 8192, 66, 20, 513, 16384, 40, 16, 17, 130, 130, 32768, 17, 36, 68, 258, 1025, 65536, 42, 131072, 2049, 36, 32, 68, 69, 262144, 258, 514, 41, 524288, 36, 1048576, 4097, 18, 514, 40

Offset: 1

Antti Karttunen, Sep 15 2017

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences related to binary expansion of n

Cf. A005940, A156552, A292272, A292372, A292380, A292381, A292383, A292384.

Table[FromDigits[Reverse@ NestWhileList[Function[k, Which[k == 1, 1, EvenQ@ k, k/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ k]], n, # > 1 &] /. k_ /; IntegerQ@ k :> If[Mod[k, 4] == 2, 1, 0], 2], {n, 77}] (* Michael De Vlieger, Sep 21 2017 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A292382(n) = if(1==n,0,(if(2==(n%4),1,0)+(2*A292382(A252463(n)))));

a(n) = my(m=factor(n),k=-2); sum(i=1,matsize(m)[1], 1 << (primepi(m[i,1]) + (k+=m[i,2]))); \\ Kevin Ryde, Dec 11 2020

from sympy.core.cache import cacheit
from sympy.ntheory.factor_ import digits
from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a292372(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join(['1' if i==2 else '0' for i in k]), 2)
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a252463(n): return 1 if n==1 else n//2 if n%2==0 else a064989(n)
@cacheit
def a292384(n): return 1 if n==1 else 4*a292384(a252463(n)) + n%4
def a(n): return a292372(a292384(n))
print([a(n) for n in range(1, 111)]) # Indranil Ghosh, Sep 21 2017

(define (A292382 n) (A292372 (A292384 n)))

a(n) = A292272(A156552(n)).
a(1) = 0; for n > 1, a(n) = 2*a(A252463(n)) + [n == 2 (mod 4)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 4k+2, and 0 otherwise.
a(n) = A292372(A292384(n)).
Other identities. For n >= 1:
a(n) AND A292380(n) = 0, where AND is a bitwise-AND (A004198).
a(n) + A292380(n) = A156552(n).
A000120(a(n)) + A000120(A292380(n)) = A001222(n).

A292941 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 6)].

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 9, 4, 4, 8, 18, 8, 37, 18, 8, 8, 74, 8, 149, 16, 16, 36, 298, 16, 9, 74, 8, 36, 596, 16, 1193, 16, 36, 148, 16, 16, 2387, 298, 72, 32, 4774, 32, 9549, 72, 16, 596, 19098, 32, 19, 18, 148, 148, 38196, 16, 33, 72, 296, 1192, 76392, 32, 152785, 2386, 32, 32, 72, 72, 305571, 296, 596, 32, 611142, 32, 1222285, 4774, 16, 596, 32

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+1 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n. An exception is the most significant bit of a(n) which corresponds with the final 1, but is shifted one bit-position towards right (less significant end).

The AND - XOR formulas just restate the fact that J(-3|n) = J(-1|n)*J(3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A243071, A292942, A292943, A292945.

Cf. also A292253, A292263, A292381, A292385.

(define (A292941 n) (if (<= n 2) (- n 1) (+ (if (= 1 (modulo n 6)) 1 0) (* 2 (A292941 (A252463 n))))))

a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 6)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 6k+1, and 0 otherwise.
Also, for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(-3|n) = 1], where J is the Jacobi-symbol.
a(n) = A292263(n) AND (A292253(n) XOR A292383(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
a(n) = A292263(n) AND (A292255(n) XOR A292385(n)). [See comments.]
For n >= 0, a(A163511(n)) = A292942(n).
For n >= 1, a(n) + A292943(n) + A292945(n) = A243071(n).

A292380 Base-2 expansion of a(n) encodes the steps where multiples of 4 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 0, 1, 0, 0, 0, 7, 0, 4, 0, 1, 0, 0, 0, 3, 4, 0, 6, 1, 0, 0, 0, 15, 0, 0, 0, 9, 0, 0, 0, 3, 0, 0, 0, 1, 2, 0, 0, 7, 8, 8, 0, 1, 0, 12, 0, 3, 0, 0, 0, 1, 0, 0, 2, 31, 0, 0, 0, 1, 0, 0, 0, 19, 0, 0, 8, 1, 0, 0, 0, 7, 14, 0, 0, 1, 0, 0, 0, 3, 0, 4, 0, 1, 0, 0, 0, 15, 0, 16, 2, 17, 0, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Sep 15 2017

			For n = 4, the starting value is a multiple of four, after which follows A252463(4) = 2, and A252463(2) = 1, the end point of iteration, and neither 2 nor 1 is a multiple of four, thus a(4) = 1*(2^0) + 0*(2^1) + 0*(2^2) = 1.
For n = 8, the starting value is a multiple of four, after which follows A252463(8) = 4 (also a multiple), continuing as before as 4 -> 2 -> 1, thus a(8) = 1*(2^0) + 1*(2^1) + 0*(2^2) + 0*(2^3) = 3.
For n = 9, the starting value is not a multiple of four, after which follows A252463(9) = 4 (which is), continuing as before as 4 -> 2 -> 1, thus a(9) = 0*(2^0) + 1*(2^1) + 0*(2^2) + 0*(2^3) = 2.

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

Cf. A005940, A048735, A156552, A292370, A292381, A292382, A292383, A292384.

Table[FromDigits[Reverse@ NestWhileList[Function[k, Which[k == 1, 1, EvenQ@ k, k/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ k]], n, # > 1 &] /. k_ /; IntegerQ@ k :> If[Mod[k, 4] == 0, 1, 0], 2], {n, 105}] (* Michael De Vlieger, Sep 21 2017 *)

a(n) = my(m=factor(n),k=-1,ret=0); for(i=1,matsize(m)[1], ret += bitneg(0,m[i,2]-1) << (primepi(m[i,1])+k); k+=m[i,2]); ret; \\ Kevin Ryde, Dec 11 2020

from sympy.core.cache import cacheit
from sympy.ntheory.factor_ import digits
from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a292370(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)
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a252463(n): return 1 if n==1 else n//2 if n%2==0 else a064989(n)
@cacheit
def a292384(n): return 1 if n==1 else 4*a292384(a252463(n)) + n%4
def a(n): return a292370(a292384(n))
print([a(n) for n in range(1, 111)]) # Indranil Ghosh, Sep 21 2017

(define (A292380 n) (A292370 (A292384 n)))

a(n) = A048735(A156552(n)).
a(n) = A292370(A292384(n)).
Other identities. For n >= 1:
a(n) AND A292382(n) = 0, where AND is a bitwise-AND (A004198).
a(n) + A292382(n) = A156552(n).
A000120(a(n)) + A000120(A292382(n)) = A001222(n).
A000035(a(n)) = A121262(n).

A292384 a(1) = 1; for n > 1, a(n) = 4*a(A252463(n)) + (n mod 4).

Original entry on oeis.org

1, 6, 27, 24, 109, 110, 439, 96, 97, 438, 1759, 440, 7037, 1758, 443, 384, 28149, 390, 112599, 1752, 1753, 7038, 450399, 1760, 389, 28150, 387, 7032, 1801597, 1774, 7206391, 1536, 7033, 112598, 1775, 1560, 28825565, 450398, 28155, 7008, 115302261, 7014, 461209047, 28152, 1761, 1801598, 1844836191, 7040, 1557, 1558, 112603, 112600

Offset: 1

Antti Karttunen, Sep 15 2017

a(n) encodes in its base-4 representation the succession of modulo 4 residues obtained when map x -> A252463(x), starting from x=n, is iterated down to the eventual 1.

Antti Karttunen, Table of n, a(n) for n = 1..1024

Cf. A010873, A252463, A292380, A292381, A292382, A292383.

Cf. also A292243.

from sympy.core.cache import cacheit
from sympy import factorint, prevprime, prod
def a064989(n):
    f = factorint(n)
    return 1 if n == 1 else prod(prevprime(i)**f[i] for i in f if i != 2)
def a252463(n): return 1 if n==1 else n//2 if n%2==0 else a064989(n)
@cacheit
def a(n): return 1 if n==1 else 4*a(a252463(n)) + n%4
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Sep 21 2017

a(1) = 1; for n > 1, a(n) = 4*a(A252463(n)) + A010873(n).

A292945 Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+5 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 5, 0, 10, 4, 0, 0, 21, 0, 42, 4, 4, 10, 85, 0, 0, 20, 0, 8, 171, 0, 342, 0, 8, 42, 1, 0, 684, 84, 20, 8, 1369, 8, 2738, 20, 0, 170, 5477, 0, 0, 0, 40, 40, 10955, 0, 8, 16, 84, 342, 21911, 0, 43822, 684, 8, 0, 17, 16, 87644, 84, 168, 2, 175289, 0, 350578, 1368, 0, 168, 3, 40, 701156, 16, 0, 2738, 1402313, 16, 40, 5476, 340, 40

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

The AND - XOR formulas are just a restatement of the fact that J(-3|n) = J(-1|n)*J(3|n), i.e., that Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A243071, A292941, A292943, A292946.

Cf. also A292253, A292255.

(define (A292945 n) (if (<= n 1) 0 (+ (if (= 5 (modulo n 6)) 1 0) (* 2 (A292945 (A252463 n))))))

a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 5 (mod 6)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 6k+5, and 0 otherwise.
Also, for n > 1, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(-3|n) = -1], where J is the Jacobi-symbol.
a(n) = A292263(n) AND (A292255(n) XOR A292383(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
a(n) = A292263(n) AND (A292253(n) XOR A292385(n)). [See comments.]
For n >= 0, a(A163511(n)) = A292946(n).
For n >= 1, A292941(n) + A292943(n) + a(n) = A243071(n).

A332896 a(1) = 0, and for n > 1, a(n) = 2*a(A332893(n)) + [n == 3 (mod 4)].

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 5, 0, 0, 4, 21, 4, 10, 10, 5, 0, 42, 0, 85, 8, 8, 42, 341, 8, 0, 20, 1, 20, 170, 10, 1365, 0, 40, 84, 11, 0, 682, 170, 21, 16, 2730, 16, 5461, 84, 8, 682, 21845, 16, 0, 0, 85, 40, 10922, 2, 43, 40, 168, 340, 87381, 20, 43690, 2730, 17, 0, 16, 80, 349525, 168, 680, 22, 1398101, 0, 174762, 1364, 1, 340, 32, 42, 5592405, 32, 0, 5460

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+3 are encountered when map x -> A332893(x) is iterated down to 1, starting from x=n. See the binary tree illustrated in A332815.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000975, A108546, A332815, A332893, A332895,

Cf. also A292383.

A332896(n) = if(1==n,n-1,2*A332896(A332893(n)) + (3==(n%4)));

a(1) = 0, and for n > 1, a(n) = 2*a(A332893(n)) + [n == 3 (mod 4)].
Other identities. For n >= 1:
a(2n) = 2*a(n).
a(A108546(n)) = A000975(n-1).

A292584 Compound filter: a(n) = P(A292583(n), A292585(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 2, 8, 5, 13, 2, 4, 8, 19, 5, 25, 13, 9, 2, 32, 4, 41, 8, 12, 19, 51, 5, 16, 25, 5, 13, 72, 9, 85, 2, 18, 32, 14, 4, 98, 41, 26, 8, 112, 12, 128, 19, 8, 51, 145, 5, 46, 16, 33, 25, 180, 5, 18, 13, 49, 72, 200, 9, 220, 85, 13, 2, 24, 18, 242, 32, 60, 14, 265, 4, 288, 98, 8, 41, 19, 26, 313, 8, 4, 112, 339, 12, 33, 128, 62, 19, 365, 8, 25, 51, 84, 145

Offset: 1

Antti Karttunen, Sep 20 2017

nonn

From Antti Karttunen, Sep 25 2017: (Start)
Some of the sequences this filter matches to:
For all i, j: a(i) = a(j) => A053866(i) = A053866(j).
For all i, j: a(i) = a(j) => A061395(i) = A061395(j). (also A006530, etc.)
For all i, j: a(i) = a(j) => A292378(i) = A292378(j).
The latter two implications follow simply because:
  A001222(A278222(A292383(n))) = A000120(A292383(n)) = A292377(n)
and, similarly, for n > 1,
  A001222(A278222(A292385(n))) = A000120(A292381(n)) = A292375(n),
and the sum of A292375(n) and A292377(n) is A061395(n) [index of the largest prime dividing n], while A292378 has been defined as 1 + their difference.
The case A053866 follows because of the component A292583, see comments under that entry. (End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A292383, A292385, A292583, A292585.

Cf. also A006530, A061395, A292378 (some of the matched sequences).

a(n) = (1/2)*(2 + ((A292583(n)+A292585(n))^2) - A292583(n) - 3*A292585(n)).

cp's OEIS Frontend

A292943 a(n) = A292944(A243071(n)); Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+3 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

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A292253 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == 1], where J is the Jacobi-symbol.

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A292255 a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == -1], where J is the Jacobi-symbol.

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A292382 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+2 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

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PARI

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A292941 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 6)].

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A292380 Base-2 expansion of a(n) encodes the steps where multiples of 4 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

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Mathematica

PARI

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A292384 a(1) = 1; for n > 1, a(n) = 4*a(A252463(n)) + (n mod 4).

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A292945 Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+5 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

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A292253 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2a(A252463(n)) + [n == 1 (mod 2)][J(3|n) == 1], where J is the Jacobi-symbol.

A292255 a(1) = 0, and for n > 1, a(n) = 2a(A252463(n)) + [n == 1 (mod 2)][J(3|n) == -1], where J is the Jacobi-symbol.