1,1

A bijection from the positive integers to the nonsquares,A000037.

A003159(which has asymptotic density 2/3) lists index n such that a(n) = 2n. The sequence maps the terms ofA0031591:1 ontoA036554,defining a bijection between them.

Similarly, bijections are defined fromA007417toA325424,fromA325424toA145204\{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.

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 inA019565.-Antti Karttunen,Jun 08 2020

a(n) = n * m /A034386(m-1), where m =A053669(A007913(n)).

a(n) =A331590(2, n) =A225546(2 *A225546(n)).

a(A019565(n)) =A019565(n+1).

a(k * m^2) = a(k) * m^2.

a(A003961(n)) = 2 *A003961(n).

a(2 *A003961(n)) =A003961(a(n)).

a(A002110(n)) = prime(n+1).

A048675(a(n)) =A048675(n) + 1.

A008833(a(n)) =A008833(n).

A267116(a(n)) =A267116(n) OR 1, where OR denotes the bitwise operationA003986.

a(A003159(n)) =A036554(n) = 2 *A003159(n).

A334870(a(n)) = n. -Antti Karttunen,Jun 08 2020

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.

(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

Permutation ofA000037.

Row 2, and therefore column 2, ofA331590.Cf.A334748(row 3).

A007913,A034386,A053669,A225546are used in formulas defining the sequence.

The formula section details how the sequence maps the terms ofA002110,A003961,A019565;and how f(a(n)) relates to f(n) for f =A008833,A048675,A267116;making use ofA003986.

Subsequences:A016825(odd bisection),A036554,A329575.

Bijections are defined that relate toA003159,A007417,A036668,A145204,A325424,A332820,A332821,A332822.

Cf. also binary treesA334860,A334866andA334870(a left inverse).

Sequence in context:A211370A122479A351125*A285306A047403A286051

Adjacent sequences:A334744A334745A334746*A334748A334749A334750

nonn,easy

Peter Munn,May 09 2020

approved