A358916 a(1) = 1. Thereafter a(n) is the least novel k != n such that A007947(k)|n.

1, 4, 9, 2, 25, 3, 49, 16, 27, 5, 121, 6, 169, 7, 45, 8, 289, 12, 361, 10, 63, 11, 529, 18, 125, 13, 81, 14, 841, 15, 961, 64, 99, 17, 175, 24, 1369, 19, 117, 20, 1681, 21, 1849, 22, 75, 23, 2209, 32, 343, 40, 153, 26, 2809, 36, 275, 28, 171, 29, 3481, 30, 3721

Offset: 1

Author

David James Sycamore, Dec 05 2022

Keywords

• nonn

Comments

In other words, a(1) = 1, then for n > 1, a(n) is the least number k, not occurring earlier, whose squarefree kernel (rad(k)) is a divisor of n.

A permutation of the positive integers. - Robert Israel, Dec 11 2022

From Michael De Vlieger, Dec 06 2022, corrected by Robert Israel, Dec 11 2022: (Start)

Some consequences of definition:

Prime n = p implies a(p) = p^2, comprising maxima.

n = 2p implies a(2p) = p, n = 4p implies a(4p) = 2p.

n = 2^e with e >= 1 implies a(2^e) = 2^(e+1) if e is odd, 2^(e-1) if e is even.

n = p^e with e >= 1 and p an odd prime implies a(n) = p^(e+1).

Composite squarefree 2n implies a(2n) = n, comprising minima.

gcd(n, n +/- 1) = 1 implies gcd(a(n), a(n +/- 1)) = 1.

Let K = rad(n); a(n) is an element of R_K, 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) != n is in A003586, and if K = 10, then a(n) != n is in A003592. (End)

Examples

			a(5) = 25 because rad(25) = 5  and there is no smaller number not equal to 5 which has this property.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^12, highlighting and labeling primes in red, composite prime powers in gold and labeling in orange, composite squarefree in green, numbers that are products of composite prime powers in large light blue and labeling in blue, and all other numbers in dark blue. The first 36 terms are simply labeled in black.

Crossrefs

Cf. A000005, A000040, A001248, A005117, A007947, A032741, A358820, A358971.

Programs

Maple

N:= 100: # for a(1)..a(N)
R:= map(NumberTheory:-Radical, [$1..N^2]):
A[1]:= 1:
Agenda:= [$2..N^2]:
for n from 2 to N do
  if isprime(R[n]) then
    if R[n] = 2 and padic:-ordp(n,2)::even then A[n]:= n/2
    else A[n]:= R[n]*n
    fi;
    if A[n] <= N then Agenda:= subs(A[n]=NULL,Agenda) fi;
    next
  fi;
  found:= false;
  for j from 1 to nops(Agenda) do
    x:= Agenda[j];
    if x <> n  and n mod R[x] = 0 then
      A[n]:= x; Agenda:= subsop(j=NULL,Agenda); found:= true; break
    fi
  od;
  if not found then break fi;
od:
convert(A,list); # Robert Israel, Dec 11 2022

Mathematica

nn = 120; c[] = False; f[n] := f[n] = Times @@ FactorInteger[n][[All, 1]]; a[1] = 1; c[1] = True; u = 2; Do[Which[PrimeQ[n], k = n^2, PrimePowerQ[n], Set[{p, k}, {f[n], 1}]; While[Nand[! c[p^k], p^k != n], k++]; k = p^k, True, k = u; While[Nand[! c[k], k != n, Divisible[n, f[k]]], k++]]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Dec 06 2022 *)

Formula

For n = p^k, where p is prime and k >= 1, a(n) = p^(k+1). In particular, a(p) = p^2 (records).

Extensions

More terms from Michael De Vlieger, Dec 07 2022