A066260 In the prime factorization of n replace the k-th prime with the k-th composite number, k > 0.

1, 4, 6, 16, 8, 24, 9, 64, 36, 32, 10, 96, 12, 36, 48, 256, 14, 144, 15, 128, 54, 40, 16, 384, 64, 48, 216, 144, 18, 192, 20, 1024, 60, 56, 72, 576, 21, 60, 72, 512, 22, 216, 24, 160, 288, 64, 25, 1536, 81, 256, 84, 192, 26, 864, 80, 576, 90, 72, 27, 768, 28, 80, 324

Offset: 1

Author

Reinhard Zumkeller, Dec 10 2001

Keywords

• nonn
• mult

Examples

			a(10) = a(2*5) = a(prime(1)*prime(3)) = a(prime(1))*a(prime(3)) = comp(1)*comp(3) = 4 * 8 = 32.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Index to divisibility sequences

Crossrefs

Cf. A002808, A049084, A066261, A066262.

Programs

Maple

b:= proc(n) option remember; local k; if n=1 then 4 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..63);  # Alois P. Heinz, Mar 21 2025

Mathematica

nmax = 100;
compos = Select[Range[FindRoot[n == nmax + PrimePi[n] + 1,
     {n, nmax, 2 nmax}][[1, 2]] // Floor], CompositeQ];
a[n_] := If[n == 1, 1, Product[{p, e} = pe; compos[[PrimePi[p]]]^e,
     {pe, FactorInteger[n]}]];
Array[a, nmax] (* Jean-François Alcover, Nov 21 2021 *)

PARI

Composite(n) = local(k); k=n + primepi(n) + 1; while (k != n + primepi(k) + 1, k = n + primepi(k) + 1); return(k)
for (n=1, 1000, f=factor(n); a=1; for (i=1, matsize(f)[1], a*=Composite(primepi(f[i, 1]))^f[i, 2]); write("b066260.txt", n, " ", a) ) \\ Harry J. Smith, Feb 07 2010

Formula

Completely multiplicative with a(p) = A002808(A049084(p)), p prime.