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A285332 a(0) = 1, a(1) = 2, a(2n) = A019565(a(n)), a(2n+1) = A065642(a(n)). 16
1, 2, 3, 4, 6, 9, 5, 8, 15, 12, 14, 27, 10, 25, 7, 16, 210, 45, 35, 18, 105, 28, 462, 81, 21, 20, 154, 125, 30, 49, 11, 32, 10659, 420, 910, 75, 78, 175, 33, 24, 3094, 315, 385, 56, 780045, 924, 374, 243, 110, 63, 55, 40, 4389, 308, 170170, 625, 1155, 60, 286, 343, 42, 121, 13, 64, 54230826, 31977, 28405, 630, 1330665, 1820, 714 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Note the indexing: the domain starts from 0, while the range excludes zero.

This sequence can be represented as a binary tree. Each left hand child is produced as A019565(n), and each right hand child as A065642(n), when the parent node contains n >= 2:

                                     1

                                     |

                  ...................2...................

                 3                                       4

       6......../ \........9                   5......../ \........8

      / \                 / \                 / \                 / \

     /   \               /   \               /   \               /   \

    /     \             /     \             /     \             /     \

  15       12         14       27         10       25          7       16

210 45   35  18    105  28  462  81     21  20  154  125     30 49   11  32

etc.

Where will 38 appear in this tree? It is a reasonable assumption that by iterating A087207 starting from 38, as A087207(38) = 129, A087207(129) = 8194, A087207(8194) = 1501199875790187, ..., we will eventually hit a prime A000040(k), most likely with a largish index k. This prime occurs at the penultimate edge at right, as a(A000918(k)) = a((2^k)-2), and thus 38 occurs somewhere below it as a(m) = 38, m > k. All the numbers that share prime factors with 38, namely 76, 152, 304, 608, 722, ..., occur similarly late in this tree, as they form the rightward branch starting from 38. Alternatively, by iterating A285330 (each iteration moves one step towards the root) starting from 38, we might instead first hit some power of 3, or say, one of the terms of A033845 (the rightward branch starting from 6), in which case the first prime encountered would be a(2)=3 and 38 would appear on the left-hand side instead of the right-hand side subtree.

As long as it remains conjecture that A019565 has no cycles, it is certainly also an open question whether this is a permutation of the natural numbers: If A019565 has any cycles, then neither any of the terms in those cycles nor any A065642-trajectories starting from those terms (that is, numbers sharing same prime factors) may occur in this tree.

Sequence exhibits some outrageous swings, for example, a(703) = 224, but a(704) is 1427 decimal digits (4739 binary digits) long.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..703

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(0) = 1, a(1) = 2, a(2n) = A019565(a(n)), a(2n+1) = A065642(a(n)).

For n >= 0, a(2^n) = A109162(2+n). [The left edge of the tree.]

For n >= 0, a(A000225(n)) = A000079(n). [Powers of 2 occur at the right edge of the tree.]

For n >= 2, a(A000918(n)) = A000040(n). [And the next vertices inwards contain primes.]

For n >= 2, a(A036563(1+n)) = A001248(n). [Whose right children are their squares.]

For n >= 0, a(A055010(n)) = A000244(n). [Powers of 3 are at the rightmost edge of the left subtree.]

For n >= 2, a(A129868(n-1)) = A062457(n).

A048675(a(n)) = A285333(n).

PROG

(PARI)

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

A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014

A065642(n) = { my(r=A007947(n)); if(1==n, n, n = n+r; while(A007947(n) <> r, n = n+r); n); };

A285332(n) = { if(n<=1, n+1, if(!(n%2), A019565(A285332(n/2)), A065642(A285332((n-1)/2)))); };

for(n=0, 4095, write("b285332.txt", n, " ", A285332(n)));

(Scheme, with memoization-macro definec)

(definec (A285332 n) (cond ((<= n 1) (+ n 1)) ((even? n) (A019565 (A285332 (/ n 2)))) (else (A065642 (A285332 (/ (- n 1) 2))))))

(Python)

from operator import mul

from sympy import prime, primefactors

def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))

def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # This function from Chai Wah Wu

def a065642(n):

    if n==1: return 1

    r=a007947(n)

    n = n + r

    while a007947(n)!=r:

        n+=r

    return n

def a(n):

    if n<2: return n + 1

    if n%2==0: return a019565(a(n/2))

    else: return a065642(a((n - 1)/2))

print [a(n) for n in xrange(0, 51)] # Indranil Ghosh, Apr 18 2017

CROSSREFS

Inverse: A285331.

Cf. A007947, A019565, A033845, A048675, A065642, A087207, A109162, A285319, A285320, A285328, A285330, A285333.

Compare also to permutation A285112 and array A285321.

Sequence in context: A113199 A243572 A233559 * A185290 A141396 A159849

Adjacent sequences:  A285329 A285330 A285331 * A285333 A285334 A285335

KEYWORD

nonn,tabf

AUTHOR

Antti Karttunen, Apr 17 2017, with additional notes and formulas added Apr 19 2017.

STATUS

approved

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Last modified December 11 06:23 EST 2017. Contains 295868 sequences.