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

1,2

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)

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.

Formula

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A235491 A374600 A256513 * A064505 A253288 A358971

Adjacent sequences: A358913 A358914 A358915 * A358917 A358918 A358919

Keyword

nonn

Author

David James Sycamore, Dec 05 2022

Extensions

More terms from Michael De Vlieger, Dec 07 2022

Status

approved