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

%I A382265 #11 Mar 22 2025 13:04:46
%S A382265 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,
%T A382265 16,24,18,1,36,12,48,16,20,14,40,6,21,32,22,9,96,15,24,4,64,36,48,10,
%U A382265 25,64,54,8,56,16,26,24,27,18,128,1,60,36,28,12,60,48,30,16,32,20,144
%N A382265 In the prime factorization of n replace the k-th prime with the k-th nonprime number.
%H A382265 Alois P. Heinz, <a href="/A382265/b382265.txt">Table of n, a(n) for n = 1..10000</a>
%F A382265 If n = Product prime(k)^e(k) then a(n) = Product nonprime(k)^e(k).
%p A382265 b:= proc(n) option remember; local k; if n=1 then 1 else
%p A382265       for k from 1+b(n-1) while isprime(k) do od; k fi
%p A382265     end:
%p A382265 a:= n-> mul(b(numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
%p A382265 seq(a(n), n=1..75);  # _Alois P. Heinz_, Mar 21 2025
%t A382265 nonPrime[n_] := FixedPoint[n + PrimePi@# &, n + PrimePi@ n]; (* _Robert G. Wilson v_ at A018252 *)
%t A382265 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 *)
%Y A382265 Cf. A000720, A003963, A018252, A066260.
%K A382265 nonn,mult
%O A382265 1,3
%A A382265 _Ilya Gutkovskiy_, Mar 19 2025