A334859 -id:A334859 - cp's OEIS frontend

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

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)

A334860 a(0) = 1, a(1) = 2, after which, a(2n) = A334747(a(n)), a(2n+1) = a(n)^2.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 8, 16, 5, 36, 18, 81, 12, 64, 32, 256, 10, 25, 72, 1296, 27, 324, 162, 6561, 24, 144, 128, 4096, 48, 1024, 512, 65536, 15, 100, 50, 625, 108, 5184, 2592, 1679616, 54, 729, 648, 104976, 243, 26244, 13122, 43046721, 20, 576, 288, 20736, 192, 16384, 8192, 16777216, 96, 2304, 2048, 1048576, 768, 262144, 131072, 4294967296, 30

Offset: 0

Antti Karttunen, Jun 08 2020

This irregular table can be represented as a binary tree. Each child to the left is obtained by applying A334747 to the parent, and each child to the right is obtained by squaring the parent:
                                      1
                                      |
                   ...................2...................
                  3                                       4
        6......../ \........9                   8......../ \........16
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    5       36         18       81         12       64         32      256
  10 25   72  1296   27  324 162  6561   24  144  128 4096   48 1024 512 65536
etc.
This is the mirror image of the tree in A334866.
Fermi-Dirac primes, A050376, occur at rightward growing branches that originate from primes situated at the left edge.
The tree illustrated in A163511 is expanded as x -> 2*x for the left child and x -> A003961(x) for the right child, while this tree is expanded as x -> A225546(2*A225546(x)) for the left child, and x -> A225546(A003961(A225546(x))) for the right child.

Index entries for sequences that are permutations of the natural numbers

Cf. A000290, A225546, A334204, A334747, A334859 (inverse), A334866 (mirror image).
Cf. A001146 (right edge of the tree), A019565 (left edge), A334110 (the right children of the left edge).
Composition of permutations A163511 and A225546.

A334747(n) = { my(c=core(n), m=n); forprime(p=2, , if(c % p, m*=p; break, m/=p)); m; }; \\ From A334747
A334860(n) = if(n<=1,1+n,if(!(n%2),A334747(A334860(n/2)),A334860((n-1)/2)^2));

a(0) = 1, a(1) = 2; and for n > 0, a(2n) = A334747(a(n)), a(2n+1) = a(n)^2.
a(n) = A225546(A163511(n)).
For n >= 0, a(2^n) = A019565(1+n), a(2^((2^n)-1)) = A000040(1+n).
A334109(a(n)) = A334204(n).
It seems that for n >= 1, A048675(a(n)) = A135529(n) = A048675(A163511(n)).

A334871 Number of steps needed to reach 1 when starting from n and iterating with A334870.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 3, 3, 5, 16, 4, 32, 9, 6, 3, 64, 4, 128, 6, 10, 17, 256, 5, 5, 33, 5, 10, 512, 7, 1024, 4, 18, 65, 12, 4, 2048, 129, 34, 7, 4096, 11, 8192, 18, 7, 257, 16384, 5, 9, 6, 66, 34, 32768, 6, 20, 11, 130, 513, 65536, 8, 131072, 1025, 11, 4, 36, 19, 262144, 66, 258, 13, 524288, 5, 1048576, 2049, 7, 130

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

Distance of n from the root (1) in binary trees like A334860 and A334866.
Each n > 0 occurs 2^(n-1) times.
a(n) is the size of the inner lining of the integer partition with Heinz number A225546(n), which is also the size of the largest hook of the same partition. (After Gus Wiseman's Apr 02 2019 comment in A252464).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A007913, A008833, A070939, A225546, A252464, A299090, A334859, A334860, A334865, A334866, A334869, A334870, A334872.

A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A334871(n) = { my(s=0); while(n>1,s++; n = A334870(n)); (s); };

\\ Much faster:
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A334871(n) = { my(s=0); while(n>1, if(issquare(n), s++; n = sqrtint(n), s += A048675(core(n)); n /= core(n))); (s); };

a(1) = 0; for n > 1, a(n) = 1 + a(A334870(n)).
a(n) = A252464(A225546(n)).
a(n) = A048675(A007913(n)) + a(A008833(n)).
For n > 1, a(n) = 1 + A048675(A007913(n)) + a(A000188(n)).
For n > 1, a(n) = A070939(A334859(n)) = A070939(A334865(n)).
For all n >= 1, a(n) >= A299090(n).
For all n >= 1, a(n) >= A334872(n).

A334872 Number of steps needed to reach either 1 or one of the "Fermi-Dirac primes" (A050376) when starting from n and iterating with A334870.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 1, 0, 4, 2, 1, 0, 3, 0, 1, 2, 8, 0, 3, 0, 1, 2, 1, 4, 2, 0, 1, 2, 5, 0, 3, 0, 16, 4, 1, 0, 2, 0, 1, 2, 32, 0, 3, 4, 9, 2, 1, 0, 6, 0, 1, 8, 2, 4, 3, 0, 64, 2, 5, 0, 3, 0, 1, 2, 128, 8, 3, 0, 4, 0, 1, 0, 10, 4, 1, 2, 17, 0, 5, 8, 256, 2, 1, 4, 3, 0, 1, 16, 2, 0, 3, 0, 33, 6

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

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

Cf. A050376 (positions of zeros after 1), A302777, A334859, A334865, A334870, A334871.

A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A334872(n) = { my(s=0); while(n>1 && !A302777(n), s++; n = A334870(n)); (s); };

\\ Much faster, A302777 like in above:
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A052126(n) = if(1==n,n,(n/vecmax(factor(n)[, 1])));
A334872(n) = { my(s=0); while(n>1 && !A302777(n), if(issquarefree(n), return(s+A048675(A052126(n)))); if(issquare(n), s++; n = sqrtint(n), s += A048675(core(n)); n /= core(n))); (s); };

If n = 1 or A302777(n) = 1, a(n) = 0, otherwise a(n) = 1 + a(A334870(n)).

For all n >= 1, a(n) <= A334871(n).

A334865 a(n) = A156552(A225546(n)).

Original entry on oeis.org

0, 1, 3, 2, 15, 7, 255, 5, 6, 31, 65535, 11, 4294967295, 511, 63, 4, 18446744073709551615, 13, 340282366920938463463374607431768211455, 47, 1023, 131071, 115792089237316195423570985008687907853269984665640564039457584007913129639935, 23, 30, 8589934591, 27, 767

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation of A334866. Composition of permutations A156552 and A225546, and also of A054429 and A334859.

Cf. A334871.

a(n) = A156552(A225546(n)).
a(n) = A054429(A334859(n)).
For n > 1, A070939(a(n)) = A334871(n).

cp's OEIS Frontend

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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A334860 a(0) = 1, a(1) = 2, after which, a(2n) = A334747(a(n)), a(2n+1) = a(n)^2.

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A334871 Number of steps needed to reach 1 when starting from n and iterating with A334870.

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Author

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Comments

Links

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Programs

PARI

PARI

Formula

A334872 Number of steps needed to reach either 1 or one of the "Fermi-Dirac primes" (A050376) when starting from n and iterating with A334870.

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Programs

PARI

PARI

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A334865 a(n) = A156552(A225546(n)).

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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)).