|
%I #29 Jun 19 2020 04:24:47
%S 2,3,6,8,10,5,14,12,18,15,22,24,26,21,30,32,34,27,38,40,42,33,46,20,
%T 50,39,54,56,58,7,62,48,66,51,70,72,74,57,78,60,82,35,86,88,90,69,94,
%U 96,98,75,102,104,106,45,110,84,114,87,118,120,122,93,126,128,130,55
%N Let p be the smallest prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller primes.
%C A bijection from the positive integers to the nonsquares, A000037.
%C A003159 (which has asymptotic density 2/3) lists index n such that a(n) = 2n. The sequence maps the terms of A003159 1:1 onto A036554, defining a bijection between them.
%C Similarly, bijections are defined from A007417 to A325424, from A325424 to A145204\{0}, and from the first in each of the following pairs to the nonsquare integers in the second: (A145204\{0}, A036668), (A036668, A007417), (A036554, A003159), (A332820, A332821), (A332821, A332822), (A332822, A332820). Note that many of these are between sets where membership depends on whether a number's squarefree part divides by 2 and/or 3.
%C Starting from 1, and iterating the sequence as a(1) = 2, a(2) = 3, a(3) = 6, a(6) = 5, a(5) = 10, etc., runs through the squarefree numbers in the order they appear in A019565. - _Antti Karttunen_, Jun 08 2020
%F a(n) = n * m / A034386(m-1), where m = A053669(A007913(n)).
%F a(n) = A331590(2, n) = A225546(2 * A225546(n)).
%F a(A019565(n)) = A019565(n+1).
%F a(k * m^2) = a(k) * m^2.
%F a(A003961(n)) = 2 * A003961(n).
%F a(2 * A003961(n)) = A003961(a(n)).
%F a(A002110(n)) = prime(n+1).
%F A048675(a(n)) = A048675(n) + 1.
%F A008833(a(n)) = A008833(n).
%F A267116(a(n)) = A267116(n) OR 1, where OR denotes the bitwise operation A003986.
%F a(A003159(n)) = A036554(n) = 2 * A003159(n).
%F A334870(a(n)) = n. - _Antti Karttunen_, Jun 08 2020
%e 168 = 42*4 has squarefree part 42 (and square part 4). The smallest prime absent from 42 = 2*3*7 is 5 and the product of all smaller primes is 2*3 = 6. So a(168) = 168*5/6 = 140.
%o (PARI) a(n) = {my(c=core(n), m=n); forprime(p=2, , if(c % p, m*=p; break, m/=p)); m;} \\ _Michel Marcus_, May 22 2020
%Y Permutation of A000037.
%Y Row 2, and therefore column 2, of A331590. Cf. A334748 (row 3).
%Y A007913, A034386, A053669, A225546 are used in formulas defining the sequence.
%Y The formula section details how the sequence maps the terms of A002110, A003961, A019565; and how f(a(n)) relates to f(n) for f = A008833, A048675, A267116; making use of A003986.
%Y Subsequences: A016825 (odd bisection), A036554, A329575.
%Y Bijections are defined that relate to A003159, A007417, A036668, A145204, A325424, A332820, A332821, A332822.
%Y Cf. also binary trees A334860, A334866 and A334870 (a left inverse).
%K nonn,easy
%O 1,1
%A _Peter Munn_, May 09 2020
|
