A297171 -id:A297171 - cp's OEIS frontend

A297161 Restricted growth sequence transform of A297171, which is Möbius transform of A243071.

1, 2, 3, 2, 4, 5, 6, 5, 5, 7, 8, 9, 10, 11, 3, 12, 13, 5, 14, 15, 16, 17, 18, 19, 12, 20, 12, 21, 22, 12, 23, 24, 25, 26, 9, 12, 27, 28, 29, 30, 31, 32, 33, 34, 19, 35, 36, 37, 24, 12, 38, 39, 40, 12, 41, 42, 43, 44, 45, 4, 46, 47, 30, 48, 49, 50, 51, 52, 53, 7, 54, 24, 55, 56, 12, 57, 58, 59, 60, 61, 24, 62, 63, 64, 65, 66, 67, 68, 69, 19

Offset: 1

Antti Karttunen, Dec 27 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A243071, A297171.

Cf. also A297113, A297157, A297162.

up_to = 8192;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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])); }
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)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
A297171(n) = sumdiv(n,d,moebius(n/d)*A243071(d));
write_to_bfile(1,rgs_transform(vector(up_to,n,A297171(n))),"b297161.txt");

A364571 a(n) = A297171(A163511(n)), where A297171 is the Möbius transform of the inverse permutation of A163511.

Original entry on oeis.org

0, 1, 1, 3, 2, 2, 2, 7, 4, 4, 2, 4, 5, 3, 6, 15, 8, 8, 4, 8, 4, 4, 4, 8, 10, 10, 4, 5, 13, 11, 14, 31, 16, 16, 8, 16, 8, 8, 8, 16, 8, 8, 4, 8, 8, 8, 8, 16, 20, 20, 10, 20, 7, 9, 6, 9, 26, 26, 12, 21, 29, 27, 30, 63, 32, 32, 16, 32, 16, 16, 16, 32, 16, 16, 8, 16, 16, 16, 16, 32, 16, 16, 8, 16, 8, 8, 8, 16, 16, 16

Offset: 0

Antti Karttunen, Aug 05 2023

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A163511, A243071, A297171.

Cf. also A364567.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A163511(n) = if(!n,1,A005940(1+A054429(n)));
A243071(n) = if(n<=2, n-1, my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p*p2*(2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); ((3<<#binary(res\2))-res-1)); \\ (Combining programs given in A156552 and A054429) - Antti Karttunen, Aug 05 2023
A297171(n) = sumdiv(n,d,moebius(n/d)*A243071(d));
A364571(n) = A297171(A163511(n));

a(n) = A297171(A163511(n)).

A243071 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A064989(2n+1)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 31, 12, 63, 30, 13, 8, 127, 10, 255, 28, 29, 62, 511, 24, 11, 126, 9, 60, 1023, 26, 2047, 16, 61, 254, 27, 20, 4095, 510, 125, 56, 8191, 58, 16383, 124, 25, 1022, 32767, 48, 23, 22, 253, 252, 65535, 18, 59, 120, 509, 2046, 131071

Offset: 1

Antti Karttunen, Jun 20 2014

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

See also the comments at A163511, which is the inverse permutation to this one.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A163511.
Cf. A000040, A000225, A007814, A054429, A064989, A064216, A122111, A209229, A245611 (= (a(2n-1)-1)/2, for n > 1), A245612, A292383, A292385, A297171 (Möbius transform).
Cf. A007283 (known positions where a(n)=n), A364256 [= gcd(n,a(n))], A364288 [= n-a(n)], A364289 [where a(n)>=n], A364290 [where a(n)A364291 [where a(n)<=n], A364497 [where n|a(n)].
Cf. A156552 (variant with inverted binary code), A253566, A332215, A332811, A334859 (other variants).

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)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n))))); \\ Antti Karttunen, Jul 18 2020

A243071(n) = if(n<=2, n-1, my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p*p2*(2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); ((3<<#binary(res\2))-res-1)); \\ (Combining programs given in A156552 and A054429) - Antti Karttunen, Jul 28 2023

from functools import reduce
from sympy import factorint, prevprime
from operator import mul
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 a(n): return n - 1 if n<3 else 2*a(n//2) if n%2==0 else 1 + 2*a(a064989(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

;; With memoizing definec-macro from Antti Karttunen's IntSeq-library.
(definec (A243071 n) (cond ((<= n 2) (- n 1)) ((even? n) (* 2 (A243071 (/ n 2)))) (else (+ 1 (* 2 (A243071 (A064989 n)))))))

a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A064989(2n+1)).
For n >= 1, a(A000040(n)) = A000225(n).
For n >= 1, a(2n+1) = 1 + 2*a(A064216(n+1)).
From Antti Karttunen, Jul 18 2020: (Start)
a(n) = A245611(A048673(n)).
a(n) = A253566(A122111(n)).
a(n) = A334859(A225546(n)).
For n >= 2, a(n) = A054429(A156552(n)).
a(n) = A292383(n) + A292385(n) = A292383(n) OR A292385(n).
For n > 1, A007814(a(n)) = A007814(n) - A209229(n). [This map  preserves the 2-adic valuation of n, except when n is a power of two, in which cases it is decremented by one.]
(End)

A297112 Möbius transform of A156552.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 8, 4, 4, 4, 16, 4, 32, 8, 4, 8, 64, 4, 128, 8, 8, 16, 256, 8, 8, 32, 8, 16, 512, 4, 1024, 16, 16, 64, 8, 8, 2048, 128, 32, 16, 4096, 8, 8192, 32, 8, 256, 16384, 16, 16, 8, 64, 64, 32768, 8, 16, 32, 128, 512, 65536, 8, 131072, 1024, 16, 32, 32, 16, 262144, 128, 256, 8, 524288, 16, 1048576, 2048, 8

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A008683, A064989, A156552, A297113.

Cf. also A297106, A297156, A297171, A297172.

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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A297112(n) = sumdiv(n,d,moebius(n/d)*A156552(d));

;; With memoization-macro definec.
(definec (A297112 n) (cond ((<= n 2) (- n 1)) ((odd? n) (* 2 (A297112 (A064989 n)))) ((= 2 (modulo n 4)) (A297112 (/ n 2))) (else (* 2 (A297112 (/ n 2)))))) ;; Antti Karttunen, Dec 27 2017

a(1) = 0, a(2) = 1, after which, a(2n+1) = 2*a(A064989(2n+1)), a(4n) = 2*a(2n), a(4n+2) = a(2n+1).
a(n) = Sum_{d|n} A008683(n/d)*A156552(d).
For n > 1, a(n) = A000079(A297113(n)-1).

A297156 Möbius transform of A243354.

Original entry on oeis.org

0, 1, 3, 1, 7, 2, 15, 3, 1, 6, 31, 6, 63, 14, 2, 5, 127, 2, 255, 14, 10, 30, 511, 10, 1, 62, 7, 30, 1023, 4, 2047, 11, 26, 126, 2, 2, 4095, 254, 58, 26, 8191, 12, 16383, 62, 14, 510, 32767, 22, 1, 2, 122, 126, 65535, 6, 18, 58, 250, 1022, 131071, 4, 262143, 2046, 30, 21, 50, 28, 524287, 254, 506, 4, 1048575, 6

Offset: 1

Antti Karttunen, Dec 28 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A006068, A156552, A243354, A297157 (rgs-transform of this sequence).

Cf. also A297112, A297171, A297172.

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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ Essentially Joerg Arndt's Jul 19 2012 code.
A243354(n) = A006068(A156552(n));
A297156(n) = sumdiv(n,d,moebius(n/d)*A243354(d));

A297172 Möbius transform of A253564.

Original entry on oeis.org

0, 1, 3, 1, 7, 1, 15, 2, 3, 3, 31, 3, 63, 7, 3, 4, 127, 2, 255, 7, 9, 15, 511, 6, 7, 31, 6, 15, 1023, 3, 2047, 8, 21, 63, 7, 4, 4095, 127, 45, 14, 8191, 7, 16383, 31, 9, 255, 32767, 12, 15, 4, 93, 63, 65535, 4, 21, 30, 189, 511, 131071, 5, 262143, 1023, 21, 16, 49, 15, 524287, 127, 381, 5, 1048575, 8, 2097151, 2047, 6

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A008683, A064989, A122111, A156552, A253564, A297162 (rgs-transform of this sequence).

Cf. also A297112, A297156, A297171.

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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A253564(n) = A156552(A122111(n));
A297172(n) = sumdiv(n,d,moebius(n/d)*A253564(d));

a(n) = Sum_{d|n} A008683(n/d)*A253564(d).

cp's OEIS Frontend

A297161 Restricted growth sequence transform of A297171, which is Möbius transform of A243071.

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A364571 a(n) = A297171(A163511(n)), where A297171 is the Möbius transform of the inverse permutation of A163511.

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A243071 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A064989(2n+1)).

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A297112 Möbius transform of A156552.

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A297156 Möbius transform of A243354.

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A297172 Möbius transform of A253564.

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A243071 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A064989(2n+1)).