A285331 Inverse for A285332: a(1) = 0, a(2) = 1, a(A019565(n)) = 2*a(n), a(A065642(n)) = 1 + 2*a(n).

0, 1, 2, 3, 6, 4, 14, 7, 5, 12, 30, 9, 62, 10, 8, 15, 126, 19, 254, 25, 24, 252, 510, 39, 13, 76, 11, 21, 1022, 28, 2046, 31, 38, 316, 18, 79, 4094

Offset

1,3

Comments

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

For the question whether this sequence and A285332 are permutations of natural numbers, see comments in A285332 and the conjecture stated in A019565.

As a practical problem, it seems next-to-impossible to compute even the value of a(38). Even though we know that 38 certainly is not in a finite cycle of A019565, because A048675(38) = 129, A048675(129) = 8194 and A048675(8194) = 4503599627370561 which factorizes as 3^2 * 37 * 71 * 190483425427 (thus is not squarefree and A285320(38) = 3), the value of a(38) is most likely so huge that it will not fit into the data section or even into a b-file. The same problem applies to all numbers that share prime factors with 38, namely 76, 152, 304, 608, 722, ...

Terms a(39) .. a(61) are [632, 51, 8190, 60, 16382, 505, 17, 72057594037927932, 32766, 159, 29, 103, 1016, 153, 65534, 319, 50, 43, 16376, 131014, 131070, 57, 262142].

The name is slightly misleading. The given definition of a(n) is not always very helpful to compute the terms (cf. example of n = 38), it is actually not clear whether the sequence is well defined. - M. F. Hasler, Mar 01 2018

Links

Table of n, a(n) for n=1..37.

Indranil Ghosh, Python Program to generate the sequence

Index entries for sequences that are permutations of the natural numbers

Formula

a(1) = 0, a(2) = 1, and for n > 2, if A008683(n) <> 0 [when n is squarefree], a(n) = 2*a(A048675(n)), otherwise a(n) = 1 + 2*a(A285328(n)).

For all n >= 0, a(A285332(n)) = n.

Example

a(1) = 0 and a(2) = 1 by definition.
a(3) = a(prime(2)) = a(A019565(2^1)) = 2*a(2) = 2.
a(4) = a(2^2) = a(A065642(2)) = 1 + 2*a(2) = 3.
a(5) = a(prime(3)) = a(A019565(2^2)) = 2*a(4) = 6.
a(9) = a(3^2) = a(A065642(3)) = 1 + 2*a(3) = 5.
a(10) = a(2*5) = a(prime(1)*prime(3)) = a(A019565(2^0+2^2)) = 2*a(1+4) = 12.
To compute a(38), write 38 = prime(1)*prime(8) = A019565(2^7+2^0), so a(38) = 2*a(129). To compute this, use 129 = prime(2)*prime(14) = A019565(2^13+2^1), so a(129) = 2*a(8194). But 8194 = prime(1)*prime(7)*prime(53) = A019565(2^0+2^6+2^52), so a(8194) = 2*a(4503599627370561)...

Prog

(PARI) A285331(n)={ if(n<=2, n-1, if(moebius(n)<>0, 2*A285331(A048675(n)), 1+2*A285331(A285328(n))))} \\ See A048675 & A285328 for respective PARI code. (We avoid content duplication leading to obsolete code.)
(Scheme, with memoization-macro definec)
(definec (A285331 n) (cond ((<= n 2) (- n 1)) ((not (zero? (A008683 n))) (* 2 (A285331 (A048675 n)))) (else (+ 1 (* 2 (A285331 (A285328 n)))))))

Crossrefs

Inverse: A285332.

Cf. A005117, A008683, A019565, A048675, A065642, A087207, A285319, A285320, A285328.

Compare also to permutation A285111.

Sequence in context: A268216 A346928 A245712 * A237125 A227296 A318846

Adjacent sequences: A285328 A285329 A285330 * A285332 A285333 A285334

Keyword

nonn,hard,more

Author

Antti Karttunen, Apr 17 2017, comments edited Apr 19 2017

Status

approved