A382265 In the prime factorization of n replace the k-th prime with the k-th nonprime number.

1, 1, 4, 1, 6, 4, 8, 1, 16, 6, 9, 4, 10, 8, 24, 1, 12, 16, 14, 6, 32, 9, 15, 4, 36, 10, 64, 8, 16, 24, 18, 1, 36, 12, 48, 16, 20, 14, 40, 6, 21, 32, 22, 9, 96, 15, 24, 4, 64, 36, 48, 10, 25, 64, 54, 8, 56, 16, 26, 24, 27, 18, 128, 1, 60, 36, 28, 12, 60, 48, 30, 16, 32, 20, 144

Offset: 1

Author

Ilya Gutkovskiy, Mar 19 2025

Keywords

• nonn
• mult

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A003963, A018252, A066260.

Programs

Maple

b:= proc(n) option remember; local k; if n=1 then 1 else
      for k from 1+b(n-1) while isprime(k) do od; k fi
    end:
a:= n-> mul(b(numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..75);  # Alois P. Heinz, Mar 21 2025

Mathematica

nonPrime[n_] := FixedPoint[n + PrimePi@# &, n + PrimePi@ n]; (* Robert G. Wilson v at A018252 *)
non[p_] := non[p] = nonPrime[PrimePi[p]]; f[p_, e_] := non[p]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 21 2025 *)

Formula

If n = Product prime(k)^e(k) then a(n) = Product nonprime(k)^e(k).