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%I #37 Mar 03 2018 17:35:29
%S 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,
%T 76,11,21,1022,28,2046,31,38,316,18,79,4094
%N Inverse for A285332: a(1) = 0, a(2) = 1, a(A019565(n)) = 2*a(n), a(A065642(n)) = 1 + 2*a(n).
%C Note the indexing: the domain starts from 1, while the range includes also zero.
%C For the question whether this sequence and A285332 are permutations of natural numbers, see comments in A285332 and the conjecture stated in A019565.
%C 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, ...
%C 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].
%C 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
%H Indranil Ghosh, <a href="/A285331/a285331.txt">Python Program to generate the sequence</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F 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)).
%F For all n >= 0, a(A285332(n)) = n.
%e a(1) = 0 and a(2) = 1 by definition.
%e a(3) = a(prime(2)) = a(A019565(2^1)) = 2*a(2) = 2.
%e a(4) = a(2^2) = a(A065642(2)) = 1 + 2*a(2) = 3.
%e a(5) = a(prime(3)) = a(A019565(2^2)) = 2*a(4) = 6.
%e a(9) = a(3^2) = a(A065642(3)) = 1 + 2*a(3) = 5.
%e a(10) = a(2*5) = a(prime(1)*prime(3)) = a(A019565(2^0+2^2)) = 2*a(1+4) = 12.
%e 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)...
%o (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.)
%o (Scheme, with memoization-macro definec)
%o (definec (A285331 n) (cond ((<= n 2) (- n 1)) ((not (zero? (A008683 n))) (* 2 (A285331 (A048675 n)))) (else (+ 1 (* 2 (A285331 (A285328 n)))))))
%Y Inverse: A285332.
%Y Cf. A005117, A008683, A019565, A048675, A065642, A087207, A285319, A285320, A285328.
%Y Compare also to permutation A285111.
%K nonn,hard,more
%O 1,3
%A _Antti Karttunen_, Apr 17 2017, comments edited Apr 19 2017
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