A358971 a(1) = 1. Thereafter a(n) is least novel k != n such that rad(k) = rad(n), where rad is A007947.

1, 4, 9, 2, 25, 12, 49, 16, 3, 20, 121, 6, 169, 28, 45, 8, 289, 24, 361, 10, 63, 44, 529, 18, 5, 52, 81, 14, 841, 60, 961, 64, 99, 68, 175, 48, 1369, 76, 117, 50, 1681, 84, 1849, 22, 15, 92, 2209, 36, 7, 40, 153, 26, 2809, 72, 275, 98, 171, 116, 3481, 30, 3721

Offset: 1

Author

David James Sycamore, Dec 07 2022

Keywords

• nonn

Comments

In other words, for n > 1, a(n) is the least novel k other than n which has not occurred earlier whose squarefree kernel is equal to the squarefree kernel of n.

Conjectured to be a permutation of the positive integers with primes appearing in natural order. Primes are minima, 1 and primes squared are records.

From Michael De Vlieger, Dec 07 2022: (Start)

Some consequences of definition:

There are no fixed points outside of a(1) = 1.

Prime power p^e implies a(p^e) = p^(e+1) for odd e, else p^(e-1). Hence a(p) = p^2 comprise maxima, while a(p^2) = p comprise minima.

Let lpf(m) = least prime factor of m. Squarefree m implies a(m) = lpf(m)*m and a(lpf(m)*m) = m, as seen in scatterplot in rays with slope p and 1/p, respectively. Therefore squarefree numbers are sequestered along or below a(n/2) = n/2.

Let K = rad(n); a(n) and n (such that a(n) != n) belong to the same sequence K*R_K, where R_K is the list of K-regular numbers, 1 and those whose prime divisors are restricted to p | K. For example, if K = 6, then a(n) and n belong to 6*A003586, and if K = 10, then a(n) and n belong to 10*A003592.

Observation: For m in A286708, abs(a(m) - m) is relatively small. (End)

This sequence is a self-inverse permutation of the positive integers: for any squarefree number s > 1, let v_s be the list of numbers with radical s, then for any k > 0, a(v_s(2*k)) = v_s(2*k-1) and a(v_s(2*k-1)) = v_s(2*k). - Rémy Sigrist, Dec 08 2022

Examples

			a(2) = 4 because 4 is the least number (not equal to 2) which has the same squarefree kernel as 2.
a(4) = 2 because 2 is the least unused number (not equal to 4) having the same squarefree kernel as 4
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, highlighting primes in red, composite prime powers (in A246547) in gold, composite squarefree numbers (A120944) in green, numbers neither squarefree nor prime power (in A126706) in blue, with numbers in A286706 in large light blue. Gold and light blue numbers are in A001694. Maxima are a(p) = p^2, minima are a(p^2) = p.
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^10, using the same color coding as immediately above, labeled and showing quasi-rays with slopes p and 1/p.
• Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^16, showing n | a(n) in green, a(n) | n in red, and other terms in blue.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000040, A001248, A005117, A007947, A253288, A358916.

Programs

Mathematica

nn = 61; c[] = False; q[] = 1; f[n_] := f[n] = Times @@ FactorInteger[n][[All, 1]]; a[1] = 1; c[1] = True; u = 2; Do[Which[PrimeQ[n], k = n^2, PrimeQ@ Sqrt[n], k = Sqrt[n], True, k = f[n]; m = q[k]; While[Nand[! c[k m], k m != n, Divisible[k, f[m]]], m++]; While[Nor[c[q[k] k], Divisible[k, f[q[k]]]], q[k]++]; k *= m]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Dec 07 2022 *)

Formula

For squarefree n, a(a(n)) = n; a(p) = p^2 for p prime, and a(p^2) = p.

Extensions

More terms from Michael De Vlieger, Dec 07 2022