A246677 -id:A246677 - cp's OEIS frontend

A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 5, 11, 1, 12, 2, 13, 4, 14, 1, 15, 1, 16, 7, 17, 3, 18, 1, 19, 11, 20, 1, 21, 1, 22, 6, 23, 1, 24, 2, 25, 13, 26, 1, 27, 5, 28, 17, 29, 1, 30, 1, 31, 10, 32, 7, 33, 1, 34, 19, 35, 1, 36, 1, 37, 9, 38, 3, 39, 1, 40, 8, 41, 1, 42

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

If n >= 2, n occurs in column a(n) of A246278.

By convention, a(1) = 0 because 1 does not occur in A246278.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Terms of A348717 halved. A305897 is the restricted growth sequence transform.
Positions of terms 1 .. 8 in this sequence are given by the following sequences: A000040, A001248, A006094, A030078, A090076, A251720, A090090, A030514.
Cf. A078898 (has the same role with array A083221 as this sequence has with A246278).
Cf. A000040, A001222, A001358, A003961, A055396, A064989, A064216, A243055, A246272, A249810, A249820, A249735, A252463.
This sequence is also used in the definition of the following permutations: A246274, A246276, A246675, A246677, A246683, A249815, A249817 (A249818), A249823, A249825, A250244, A250245, A250247, A250249.
Also in the definition of arrays A249821, A251721, A251722.
Sum of prime indices of a(n) is A359358(n) + A001222(n) - 1, cf. A326844.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A005940, A241916, A253565, A356958, A358195.

a246277[n_Integer] := Module[{f, p, a064989, a},
  f[x_] := Transpose@FactorInteger[x];
  p[x_] := Which[
    x == 1, 1,
    x == 2, 1,
    True, NextPrime[x, -1]];
  a064989[x_] := Times @@ Power[p /@ First[f[x]], Last[f[x]]];
  a[1] = 0;
  a[x_] := If[EvenQ[x], x/2, NestWhile[a064989, x, OddQ]/2];
a/@Range[n]]; a246277[84] (* Michael De Vlieger, Dec 19 2014 *)

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)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2); \\ Antti Karttunen, Apr 30 2022

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

;; two different variants, the second one employing memoizing definec-macro)
(define (A246277 n) (if (= 1 n) 0 (let loop ((n n)) (if (even? n) (/ n 2) (loop (A064989 n))))))
(definec (A246277 n) (cond ((= 1 n) 0) ((even? n) (/ n 2)) (else (A246277 (A064989 n)))))

a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)) = a(A064216(n+1)). [Cf. the formula for A252463.]
Instead of the equation for a(2n+1) above, we may write a(A003961(n)) = a(n). - Peter Munn, May 21 2022
Other identities. For all n >= 1, the following holds:
For all w >= 0, a(p_{i} * p_{j} * ... * p_{k}) = a(p_{i+w} * p_{j+w} * ... * p_{k+w}).
For all n >= 2, A001222(a(n)) = A001222(n)-1. [a(n) has one less prime factor than n. Thus each semiprime (A001358) is mapped to some prime (A000040), etc.]
For all n >= 2, a(n) = A078898(A249817(n)).
For semiprimes n = p_i * p_j, j >= i, a(n) = A000040(1+A243055(n)) = p_{1+j-i}.
a(n) = floor(A348717(n)/2). - Antti Karttunen, Apr 30 2022
If n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=2..k} prime(x_i-x_1+1). The opposite version is A358195, prime indices A358172, even bisection A241916. - Gus Wiseman, Dec 29 2022

A246675 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A246277(n+1))-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 13, 10, 15, 64, 17, 128, 19, 18, 21, 256, 23, 12, 25, 14, 27, 512, 29, 1024, 31, 26, 33, 20, 35, 2048, 37, 42, 39, 4096, 41, 8192, 43, 22, 45, 16384, 47, 24, 49, 50, 51, 32768, 53, 36, 55, 66, 57, 65536, 59, 131072, 61, 38, 63, 52, 65, 262144, 67, 74, 69

Offset: 1

Antti Karttunen, Sep 01 2014

nonn

Consider the square array A246278, and also A246275 which is obtained from the former when one is subtracted from each term.
In A246278 the even numbers occur at the top row, and all the rows below that contain only odd numbers, those subsequent terms in each column having been obtained by shifting all primes present in the prime factorization of number immediately above to one larger indices with A003961.
To compute a(n): we do the same process in reverse, by shifting primes in the prime factorization of n+1 step by step to smaller primes, until after k >= 0 such shifts with A064989, the result is even, with the smallest prime present being 2.
We subtract one from this even number and shift the binary expansion of the resulting odd number k positions left (i.e. multiply it with 2^k), which will be the result of a(n).
In the essence, a(n) tells which number in the array A135764 is at the same position where n is in the array A246275. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e., a(2n+1) = 2n+1 for all n.
A055396(n+1) tells on which row of A246275 n is, which is equal to the row of A246278 on which n+1 is.
A246277(n+1) tells in which column of A246275 n is, which is equal to the column of A246278 in which n+1 is.

			Consider 54 = 55-1. To find 55's position in array A246278, we start shifting its prime factorization 55 = 5 * 11 = p_3 * p_5, step by step: p_2 * p_4 (= 3 * 7 = 21), until we get an even number: p_1 * p_3 = 2*5 = 10.
This tells us that 55 is on row 3 and column 5 (= 10/2) of array A246278, thus 54 occurs in the same position at array A246275. In array A135764 the same position contains number (2^(3-1)) * (10-1) = 4*9 = 36, thus a(54) = 36.

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences that are permutations of the natural numbers

Inverse: A246676.
More recursed variants: A246677, A246683.
Even bisection halved: A246679.
Other related permutations: A054582, A135764, A246274, A246275, A246276.
Cf. A000079, A003961, A005408, A055396, A064989, A246277, A246278.
a(n) differs from A156552(n+1) for the first time at n=13, where a(13) = 14, while A156552(14) = 17.

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)};
A246675(n) = { my(k=0); n++; while((n%2), n = A064989(n); k++); n--; while(k>0, n = 2*n; k--); n; };
for(n=1, 2048, write("b246675.txt", n, " ", A246675(n)));

(define (A246675 n) (* (A000079 (- (A055396 (+ 1 n)) 1)) (-1+ (* 2 (A246277 (+ 1 n))))))

a(n) = A000079(A055396(n+1)-1)  * ((2*A246277(n+1))-1).
As a composition of related permutations:
a(n) = A135764(A246276(n)).
a(n) = A054582(A246274(n)-1).
Other identities. For all n >= 0:
a(A005408(n)) = A005408(n). [Fixes the odd numbers.]

A269388 Permutation of natural numbers: a(1) = 0, after which, a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A269380(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 12, 19, 14, 33, 128, 23, 256, 65, 18, 35, 512, 21, 24, 31, 22, 129, 20, 27, 1024, 25, 34, 39, 2048, 29, 4096, 67, 30, 257, 8192, 47, 28, 513, 26, 131, 16384, 37, 48, 71, 38, 1025, 40, 43, 32768, 49, 66, 63, 36, 45, 65536, 259, 46, 41, 131072, 55, 96, 2049, 130, 51, 262144, 69, 44

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

Note the indexing: Domain starts from 1, range from 0.

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

Inverse: A269387.
Cf. A269380.
Related permutations: A260742, A269386, A269172.
Cf. also A252754, A269378.
Differs from A156552, A252754 and A246677(n-1) for the first time at n=19, which here a(19)=12, instead of 128.

a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A269380(n)).
As a composition of other permutations:
a(n) = A252754(A269172(n)).
a(n) = A269378(A260742(n)).

A246683 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2*a(A246277(n+1))) - 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 15, 32, 13, 10, 11, 64, 17, 128, 31, 18, 29, 256, 63, 12, 25, 14, 19, 512, 21, 1024, 127, 26, 33, 20, 255, 2048, 61, 58, 35, 4096, 57, 8192, 511, 30, 125, 16384, 23, 24, 49, 50, 27, 32768, 37, 36, 1023, 66, 41, 65536, 2047, 131072, 253, 62, 51, 52, 65, 262144, 39, 122, 509, 524288, 4095, 1048576, 121

Offset: 1

Antti Karttunen, Sep 06 2014

nonn

See the comments in A246675. This is otherwise similar permutation, except that after having reached an even number 2m when we have shifted the prime factorization of n+1 k steps towards smaller primes, here, in contrary to A246675, we don't shift the binary representation of the odd number 2m-1, but instead of an odd number (2*a(m))-1 the same number (k) of bit-positions leftward, i.e. multiply it with 2^k.

See also the comments at the inverse permutation A246684.

			Consider 44 = 45-1. To find 45's position in array A246278, we start shifting its prime factorization 45 = 3 * 3 * 5 = p_2 * p_2 * p_3, step by step, until we get an even number, which in this case happens immediately after the first step, as p_1 * p_1 * p_2 = 2*2*3 = 12. 12 is in the 6th column of A246278, thus we take [here a(6) is computed recursively in the same way:] (2*a(6))-1 = (2*8)-1 = 15, "1111" in binary, and shift it one bit left (that is, multiply by 2), to give 2*15 = 30, thus a(44) = 30.

Antti Karttunen, Table of n, a(n) for n = 1..2049
Index entries for sequences that are permutations of the natural numbers

Inverse: A246684.
Variants: A246675, A246677.
Cf. A000051, A000079, A005408, A055396, A246277, A246278,
A249813, A250244.
Differs from A249813 for the first time at n=20, where a(20) = 18, while A249813(20) = 14.

a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2*a(A246277(n+1))) - 1).
As a composition of other permutations:
a(n) = A249813(A250244(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back].
For all n >= 0, the following holds: a(A000051(n)) = A000051(n). [Numbers of the form 2^n + 1 are among the fixed points].

A246678 Permutation of natural numbers: a(1) = 1, a(2n+1) = 1 + 2*a(n), a(2n) = A242378(A001511(n), (1+A000265(n))) - 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 11, 24, 17, 26, 15, 10, 13, 20, 19, 34, 29, 44, 23, 48, 49, 32, 35, 124, 53, 80, 31, 12, 21, 74, 27, 54, 41, 62, 39, 76, 69, 38, 59, 174, 89, 134, 47, 120, 97, 50, 99, 64, 65, 98, 71, 342, 249, 104, 107, 624, 161, 242, 63, 16, 25, 56, 43, 244, 149, 224, 55, 90, 109, 68, 83

Offset: 1

Antti Karttunen, Sep 01 2014

nonn

See the comments in A246676. This is a similar permutation, except for odd numbers, which are here recursively permuted by the emerging permutation itself. The even bisection halved gives A246680, the odd bisection from a(3) onward with one subtracted and then halved gives this sequence back.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A246677.
Variants: A246676, A246684.
Even bisection halved: A246680.
Cf. A000265, A001511, A003961, A242378.

a(1) = 1, a(2n+1) = 1 + 2*a(n), a(2n) = A242378(A001511(n), (1+A000265(n))) - 1. [Where the bivariate function A242378(k,n) changes each prime p(i) in the prime factorization of n to p(i+k), i.e., it's the result of A003961 iterated k times starting from n].

A246679 Permutation of natural numbers, even bisection of A246675 halved: a(n) = A246675(2*n)/2.

Original entry on oeis.org

1, 2, 4, 3, 8, 16, 5, 32, 64, 9, 128, 6, 7, 256, 512, 13, 10, 1024, 21, 2048, 4096, 11, 8192, 12, 25, 16384, 18, 33, 32768, 65536, 19, 26, 131072, 37, 262144, 524288, 17, 20, 1048576, 15, 2097152, 42, 45, 4194304, 36, 57, 50, 8388608, 27, 16777216, 33554432, 29, 67108864, 134217728, 61, 268435456, 66, 43, 52, 24, 73, 14

Offset: 1

Antti Karttunen, Sep 02 2014

nonn

Equally: even bisection of A246677 halved.

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

Inverse: A246680.
Cf. A246675, A246677.
a(n) differs from A244153(n+1) for the first time at n=16, where a(16) = 13, while A244153(17) = 17.

(define (A246679 n) (/ (A246675 (* 2 n)) 2))

a(n) = A246675(2*n)/2.

a(n) = A246677(2*n)/2.

cp's OEIS Frontend

A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

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A246675 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A246277(n+1))-1).

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A269388 Permutation of natural numbers: a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A269380(n)).

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A246683 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2*a(A246277(n+1))) - 1).

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A246678 Permutation of natural numbers: a(1) = 1, a(2n+1) = 1 + 2*a(n), a(2n) = A242378(A001511(n), (1+A000265(n))) - 1.

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A246679 Permutation of natural numbers, even bisection of A246675 halved: a(n) = A246675(2*n)/2.

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A269388 Permutation of natural numbers: a(1) = 0, after which, a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A269380(n)).