A117968 -id:A117968 - cp's OEIS frontend

A296073 Filter combining A296071(n) and A296072(n), related to the deficiencies of proper divisors of n.

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75, 76, 2, 77, 2, 78, 79, 80, 2, 81, 2, 82, 83, 84, 2, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 33

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

Construction: Pack the values of A296071(n) and A296072(n) to a single value with any injective N x N -> N packing function, like for example as f(n) = (1/2)*(2 + ((A296071(n)+A296072(n))^2) - A296071(n) - 3*A296072(n)) (the packing function here is the two-argument form of A000027). Then apply the restricted growth sequence transform to the sequence f(1), f(2), f(3), ... The transform assigns a unique increasing number for each newly encountered term of the sequence, and for any subsequent occurrences of the same term it gives the same number that term obtained for the first time.
For all i, j: a(i) = a(j) => A296074(i) = A296074(j).
Note that this is NOT restricted growth transform of A239968, which is A305800. Apart from 2's that occur at every prime, there are other duplicates also, first at a(125) = a(46) = 33.

			To see that a(46) and a(125) have the same value (33), consider the proper divisors of 46 = 1, 2, 23 and of 125 = 1, 5, 25. Their deficiencies are 1, 1, 22 and 1, 4, 19 respectively. When we look at their balanced ternary representations [as here all elements are positive, it can be obtained as A007089(A117967(n)) with 2's standing for -1's]:
   1 =    1
   1 =    1
  22 = 1211 (as 22 = 1*(3^3) + -1*(3^2) + 1*(3^1) + 1*(3^0))
and
   1 =    1
   4 =   11
  19 = 1201 (as 19 = 1*(3^3) + -1*(3^2) + 0*(3^1) + 1*(3^0)).
we see that in each column there is an equal number of 1's and an equal number of 2's. Moreover, this then implies also that the sums of those two sequences of deficiencies {1, 1, 22} and {1, 4, 19} are equal, as A296074(n) is a function of (can be computed from) a(n).

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A019565, A033879, A117967, A117968, A295882, A296071, A296072, A296074.
Cf. also A293226.
Differs from A305800 for the first time at n=125.

up_to = 65536;
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A289813(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From Rémy Sigrist
A289814(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From Rémy Sigrist
A295882(n) = { my(x = (2*n)-sigma(n)); if(x >= 0,A117967(x),A117968(-x)); };
A296071(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(A295882(d))))); m; };
A296072(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(A295882(d))))); m; };
Anotsubmitted3(n) = (1/2)*(2 + ((A296071(n)+A296072(n))^2) - A296071(n) - 3*A296072(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted3(n))),"b296073.txt");

Data section extended up to a(125) by Antti Karttunen, Jun 14 2018

A246208 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 2a(-(A117966(n))), otherwise a(n) = 1 + 2a(A117966(n)-1).

Original entry on oeis.org

0, 1, 2, 5, 11, 3, 10, 4, 22, 45, 91, 9, 19, 39, 183, 7, 21, 23, 90, 44, 182, 20, 6, 8, 38, 18, 78, 157, 315, 37, 75, 151, 631, 17, 77, 13, 27, 55, 155, 311, 623, 111, 1263, 35, 303, 47, 181, 43, 365, 41, 89, 367, 15, 79, 314, 156, 630, 76, 16, 36, 150, 74, 302, 180, 46, 88, 14, 366, 42, 40, 364, 12, 54, 26, 110, 34

Offset: 0

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where complementary pair A117968/A117967 (negative and positive part of inverse of balanced ternary enumeration of integers, respectively) is entangled with complementary pair A005843/A005408 (even and odd numbers respectively), with a(0) set to 0 and a(1) set to 1.

Thus this shares with A140264 the property that apart from a(0) = 0, even numbers occur only in positions given by A117968, and odd numbers only in positions given by A117967.

Antti Karttunen, Table of n, a(n) for n = 0..6561
Index entries for sequences that are permutations of the natural numbers

Inverse: A246207.
Related permutations: A140264, A054429, A246210, A246211.
Cf. A005408, A005843, A117966, A117967, A117968.

def a117966(n):
    if n==0: return 0
    if n%3==0: return 3*a117966(n//3)
    elif n%3==1: return 3*a117966((n - 1)//3) + 1
    else: return 3*a117966((n - 2)//3) - 1
def a(n):
    if n<2: return n
    x=a117966(n)
    if x<1: return 2*a(-x)
    else: return 1 + 2*a(x - 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 2*a(-(A117966(n))), otherwise a(n) = 1 + 2*a(A117966(n)-1).
As a composition of related permutations:
a(n) = A054429(A246210(n)).
a(n) = A246210(A246211(n)).

A323909 Balanced ternary representation of A004718, Per Nørgård's "infinity sequence".

Original entry on oeis.org

0, 1, 2, 5, 1, 0, 7, 3, 2, 5, 0, 1, 5, 2, 6, 4, 1, 0, 7, 3, 0, 1, 2, 5, 7, 3, 1, 0, 3, 7, 8, 17, 2, 5, 0, 1, 5, 2, 6, 4, 0, 1, 2, 5, 1, 0, 7, 3, 5, 2, 6, 4, 2, 5, 0, 1, 6, 4, 5, 2, 4, 6, 22, 15, 1, 0, 7, 3, 0, 1, 2, 5, 7, 3, 1, 0, 3, 7, 8, 17, 0, 1, 2, 5, 1, 0, 7, 3, 2, 5, 0, 1, 5, 2, 6, 4, 7, 3, 1, 0, 3, 7, 8, 17, 1, 0

Offset: 0

Antti Karttunen, Feb 10 2019

nonn

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

Antti Karttunen, Table of n, a(n) for n = 0..65536

Cf. A004718, A083866 (positions of zeros), A117966, A117967, A117968, A323907 (rgs-transform), A323908.

up_to = 65536;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ After code in A004718.
v004718 = A004718list(up_to);
A004718(n) = if(!n,n,v004718[n]);
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A323909(n) = { my(x = A004718(n)); if(x >= 0,A117967(x),A117968(-x)); };

If A004718(n) >= 0, then a(n) = A117967(A004718(n)), otherwise a(n) = A117968(-A004718(n)).

For all n >= 1, A117966(a(n)) = A004718(n).

A246210 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 1 + 2a(-(A117966(n))), otherwise a(n) = 2a(A117966(n)-1).

Original entry on oeis.org

0, 1, 3, 6, 12, 2, 13, 7, 25, 50, 100, 14, 28, 56, 200, 4, 26, 24, 101, 51, 201, 27, 5, 15, 57, 29, 113, 226, 452, 58, 116, 232, 904, 30, 114, 10, 20, 40, 228, 456, 912, 80, 1808, 60, 464, 48, 202, 52, 402, 54, 102, 400, 8, 112, 453, 227, 905, 115, 31, 59, 233, 117, 465, 203, 49, 103, 9, 401, 53, 55, 403, 11, 41, 21, 81, 61

Offset: 0

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where complementary pair A117967/A117968 (positive and negative part of inverse of balanced ternary enumeration of integers, respectively) is entangled with complementary pair A005843/A005408 (even and odd numbers respectively), with a(0) set to 0 and a(1) set to 1.

This implies that apart from a(1) = 1, even numbers occur only in positions given by A117967, and odd numbers only in positions given by A117968.

Antti Karttunen, Table of n, a(n) for n = 0..2187
Index entries for sequences that are permutations of the natural numbers

Inverse: A246209.
Similar or related permutations: A054429, A246208, A246211.
Cf. A005408, A005843, A117966, A117967, A117968.

def a117966(n):
    if n==0: return 0
    if n%3==0: return 3*a117966(n//3)
    elif n%3==1: return 3*a117966((n - 1)//3) + 1
    else: return 3*a117966((n - 2)//3) - 1
def a(n):
    if n<2: return n
    x=a117966(n)
    if x<1: return 1 + 2*a(-x)
    else: return 2*a(x - 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 1 + 2*a(-(A117966(n))), otherwise a(n) = 2*a(A117966(n)-1).
As a composition of related permutations:
a(n) = A054429(A246208(n)).
a(n) = A246208(A246211(n)).

A140265 Permutation of natural numbers: a(n) = A140263(n-1)+1.

Original entry on oeis.org

1, 2, 3, 6, 8, 4, 7, 5, 9, 18, 23, 16, 22, 17, 24, 12, 20, 10, 19, 11, 21, 15, 26, 13, 25, 14, 27, 54, 68, 52, 67, 53, 69, 48, 65, 46, 64, 47, 66, 51, 71, 49, 70, 50, 72, 36, 59, 34, 58, 35, 60, 30, 56, 28, 55, 29, 57, 33, 62, 31, 61, 32, 63, 45, 77, 43, 76, 44, 78, 39, 74, 37

Offset: 1

Antti Karttunen, May 19 2008

Inverse: A140266.

from sympy import ceiling
from sympy.ntheory.factor_ import digits
def a004488(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3)
def a117968(n):
    if n==1: return 2
    if n%3==0: return 3*a117968(n/3)
    elif n%3==1: return 3*a117968((n - 1)/3) + 2
    else: return 3*a117968((n + 1)/3) + 1
def a117967(n): return 0 if n==0 else a117968(-n) if n<0 else a004488(a117968(n))
def a001057(n): return -(-1)**n*ceiling(n/2)
def a(n): return a117967(a001057(n - 1)) + 1 # Indranil Ghosh, Jun 07 2017

(define (A140265 n) (+ 1 (A140263 (- n 1))))

A296071 a(n) = Product_{d|n, dA019565(A289813(A295882(d))); a product obtained from the 1's present in balanced ternary representation of the deficiencies of the proper divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 8, 6, 24, 2, 24, 2, 20, 36, 16, 2, 60, 2, 144, 30, 40, 2, 48, 12, 60, 30, 240, 2, 1080, 2, 32, 60, 56, 60, 120, 2, 28, 90, 576, 2, 3600, 2, 400, 900, 168, 2, 96, 10, 1008, 84, 1200, 2, 420, 120, 480, 42, 56, 2, 4320, 2, 84, 1500, 64, 180, 4200, 2, 784, 252, 90720, 2, 1200, 2, 140, 2520, 784, 100, 75600, 2, 1152, 210, 840, 2

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

Used as a part of filter A296073.

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

Cf. A033879, A019565, A289813, A295882, A296072, A296073.

Cf. also A293221.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A289813(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From Rémy Sigrist
A295882(n) = { my(x = (2*n)-sigma(n)); if(x >= 0,A117967(x),A117968(-x)); };
A296071(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(A295882(d))))); m; };

(define (A296071 n) (let loop ((m 1) (props (proper-divisors n))) (cond ((null? props) m) (else (loop (* m (A019565 (A289813 (A295882 (car props))))) (cdr props))))))
(define (proper-divisors n) (reverse (cdr (reverse (divisors n)))))
(define (divisors n) (let loop ((k n) (divs (list))) (cond ((zero? k) divs) ((zero? (modulo n k)) (loop (- k 1) (cons k divs))) (else (loop (- k 1) divs)))))

a(n) = Product_{d|n, dA019565(A289813(A295882(d))).

A296072 a(n) = Product_{d|n, dA019565(A289814(A295882(d))); a product obtained from the -1's present in balanced ternary representation of the deficiencies of the proper divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 12, 1, 2, 6, 1, 1, 12, 1, 1, 12, 3, 1, 12, 1, 1, 2, 15, 3, 216, 1, 5, 2, 6, 1, 6, 1, 2, 36, 5, 1, 180, 3, 10, 30, 1, 1, 1080, 1, 3, 10, 1, 1, 3240, 1, 1, 36, 1, 1, 20, 1, 450, 10, 30, 1, 45360, 1, 1, 30, 75, 3, 10, 1, 60, 360, 1, 1, 540, 15, 105, 2, 2, 1, 3240, 3, 50, 2, 35, 5, 2520, 1, 630, 60, 90, 1, 900

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

Used as a part of filter A296073.

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

Cf. A033879, A019565, A289814, A295882, A296071, A296073.

Cf. also A293222.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A289814(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From Rémy Sigrist
A295882(n) = { my(x = (2*n)-sigma(n)); if(x >= 0,A117967(x),A117968(-x)); };
A296072(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(A295882(d))))); m; };

(define (A296072 n) (let loop ((m 1) (props (proper-divisors n))) (cond ((null? props) m) (else (loop (* m (A019565 (A289814 (A295882 (car props))))) (cdr props))))))
(define (proper-divisors n) (reverse (cdr (reverse (divisors n)))))
(define (divisors n) (let loop ((k n) (divs (list))) (cond ((zero? k) divs) ((zero? (modulo n k)) (loop (- k 1) (cons k divs))) (else (loop (- k 1) divs)))))

a(n) = Product_{d|n, dA019565(A289814(A295882(d))).

A297154 Balanced ternary representation of A083254(n), 2*phi(n)-n.

Original entry on oeis.org

1, 0, 1, 0, 3, 7, 17, 0, 3, 7, 9, 8, 14, 7, 1, 0, 51, 21, 47, 8, 3, 7, 48, 19, 51, 7, 9, 8, 27, 67, 32, 0, 16, 7, 13, 24, 38, 7, 9, 19, 39, 63, 161, 8, 3, 7, 153, 68, 38, 20, 13, 8, 141, 63, 34, 19, 51, 7, 138, 56, 152, 7, 9, 0, 31, 55, 149, 8, 46, 71, 105, 57, 101, 7, 17, 8, 160, 60, 89, 68, 27, 7, 81, 72, 160, 7

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

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

Cf. A000010, A083254, A117966, A117967, A117968.

Cf. also A297153, A295882.

(define (A297154 n) (let ((x (A083254 n))) (if (>= x 0) (A117967 x) (A117968 (- x)))))

If A083254(n) >= 0, then a(n) = A117967(A083254(n)), otherwise a(n) = A117968(-A083254(n)).

For all n >= 1, A117966(a(n)) = A083254(n).

A140268 a(n) = negative integer -n presented in balanced ternary system.

Original entry on oeis.org

2, 21, 20, 22, 211, 210, 212, 201, 200, 202, 221, 220, 222, 2111, 2110, 2112, 2101, 2100, 2102, 2121, 2120, 2122, 2011, 2010, 2012, 2001, 2000, 2002, 2021, 2020, 2022, 2211, 2210, 2212, 2201, 2200, 2202, 2221, 2220, 2222, 21111, 21110, 21112

Offset: 1

Antti Karttunen, May 19 2008, prompted by Eric Angelini's posting on SeqFan mailing list on Sep 15 2005

Sequence A117968 in ternary. (See there for more references.)

			For example a(2) = 21, as -2 = -1*3 + 1*1.
Similarly, a(19) = 2102, as -19 = -1*27 + 1*9 + 0*3 + -1*1.

a(n) = A007089(A117968(n)). Cf. A140267.

from sympy.ntheory.factor_ import digits
def a117968(n):
    if n==1: return 2
    if n%3==0: return 3*a117968(n/3)
    elif n%3==1: return 3*a117968((n - 1)/3) + 2
    else: return 3*a117968((n + 1)/3) + 1
def a(n): return int("".join(map(str, digits(a117968(n), 3)[1:]))) # Indranil Ghosh, Jun 06 2017

A366793 Binary encoding of the ones in the balanced ternary representation of Per Nørgård's "infinity sequence".

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 0, 1, 2, 0, 0, 3, 1, 0, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 4, 0, 2, 0, 1, 2, 0, 0, 3, 0, 1, 0, 2, 1, 0, 1, 2, 2, 0, 0, 3, 0, 2, 0, 1, 0, 3, 2, 0, 3, 0, 3, 4, 1, 0, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 4, 0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 0, 1, 2, 0, 0, 3, 1, 2, 1, 0, 2, 1, 0, 4, 1, 0

Offset: 0

Antti Karttunen, Oct 24 2023

nonn

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

			A004718(254) = -7. In balanced ternary representation (see A117966) this is represented as -1*9 + 1*3 + -1*1. Taking the positive coefficients, and converting them to a binary string gives "10", which in base-2 (A007088) is equal to 2, therefore a(254) = 2.

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

Cf. A004718, A007088, A117966, A289813, A323909, A366794.

up_to = 65536;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ From the code in A004718.
v004718 = A004718list(up_to);
A004718(n) = if(!n,n,v004718[n]);
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A323909(n) = { my(x = A004718(n)); if(x >= 0,A117967(x),A117968(-x)); };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A366793(n) = A289813(A323909(n));

a(n) = A289813(A323909(n)).

cp's OEIS Frontend

A296073 Filter combining A296071(n) and A296072(n), related to the deficiencies of proper divisors of n.

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A246208 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 2*a(-(A117966(n))), otherwise a(n) = 1 + 2*a(A117966(n)-1).

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A323909 Balanced ternary representation of A004718, Per Nørgård's "infinity sequence".

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PARI

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A246210 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 1 + 2*a(-(A117966(n))), otherwise a(n) = 2*a(A117966(n)-1).

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A140265 Permutation of natural numbers: a(n) = A140263(n-1)+1.

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A296071 a(n) = Product_{d|n, dA019565(A289813(A295882(d))); a product obtained from the 1's present in balanced ternary representation of the deficiencies of the proper divisors of n.

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A296072 a(n) = Product_{d|n, dA019565(A289814(A295882(d))); a product obtained from the -1's present in balanced ternary representation of the deficiencies of the proper divisors of n.

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A297154 Balanced ternary representation of A083254(n), 2*phi(n)-n.

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A246208 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 2a(-(A117966(n))), otherwise a(n) = 1 + 2a(A117966(n)-1).

A246210 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 1 + 2a(-(A117966(n))), otherwise a(n) = 2a(A117966(n)-1).