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

%I A066260 #17 Mar 21 2025 15:52:48
%S A066260 1,4,6,16,8,24,9,64,36,32,10,96,12,36,48,256,14,144,15,128,54,40,16,
%T A066260 384,64,48,216,144,18,192,20,1024,60,56,72,576,21,60,72,512,22,216,24,
%U A066260 160,288,64,25,1536,81,256,84,192,26,864,80,576,90,72,27,768,28,80,324
%N A066260 In the prime factorization of n replace the k-th prime with the k-th composite number, k > 0.
%H A066260 Harry J. Smith, <a href="/A066260/b066260.txt">Table of n, a(n) for n = 1..1000</a>
%H A066260 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F A066260 Completely multiplicative with a(p) = A002808(A049084(p)), p prime.
%e A066260 a(10) = a(2*5) = a(prime(1)*prime(3)) = a(prime(1))*a(prime(3)) = comp(1)*comp(3) = 4 * 8 = 32.
%p A066260 b:= proc(n) option remember; local k; if n=1 then 4 else
%p A066260       for k from 1+b(n-1) while isprime(k) do od; k fi
%p A066260     end:
%p A066260 a:= n-> mul(b(numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
%p A066260 seq(a(n), n=1..63);  # _Alois P. Heinz_, Mar 21 2025
%t A066260 nmax = 100;
%t A066260 compos = Select[Range[FindRoot[n == nmax + PrimePi[n] + 1,
%t A066260      {n, nmax, 2 nmax}][[1, 2]] // Floor], CompositeQ];
%t A066260 a[n_] := If[n == 1, 1, Product[{p, e} = pe; compos[[PrimePi[p]]]^e,
%t A066260      {pe, FactorInteger[n]}]];
%t A066260 Array[a, nmax] (* _Jean-François Alcover_, Nov 21 2021 *)
%o A066260 (PARI) Composite(n) = local(k); k=n + primepi(n) + 1; while (k != n + primepi(k) + 1, k = n + primepi(k) + 1); return(k)
%o A066260 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
%Y A066260 Cf. A002808, A049084, A066261, A066262.
%K A066260 nonn,mult
%O A066260 1,2
%A A066260 _Reinhard Zumkeller_, Dec 10 2001