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A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)). 57
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

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

LINKS

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

FORMULA

a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)) = a(A064216(n+1)). [Cf. the formula for A252463.]

Other identities. For all n >= 1, the following holds:

a(A000040(n)) = 1.

For all w >= 0, a(p_{i} * p_{j} * ... * p_{k}) = a(p_{i+w} * p_{j+w} * ... * p_{k+w}). [Follows directly from the definition.]

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}.

MATHEMATICA

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 *)

PROG

(PARI)

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)); };

for(n=1, 10000, write("b246277.txt", n, " ", A246277(n)));

(Scheme, 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)))))

(Python)

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 0 if n==1 else n/2 if n%2==0 else a(a064989(n))

print [a(n) for n in xrange(1, 101)] # Indranil Ghosh, Jun 15 2017

CROSSREFS

Cf. A078898 (has the same role with array A083221 as this sequence has with A246278).

Cf. A000040, A001222, A001358, 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.

Sequence in context: A325352 A305438 A078898 * A260739 A130747 A055440

Adjacent sequences:  A246274 A246275 A246276 * A246278 A246279 A246280

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 21 2014

STATUS

approved

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Last modified August 23 01:10 EDT 2019. Contains 326211 sequences. (Running on oeis4.)