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A328400 Smallest number with the same set of distinct prime exponents as n.

%I A328400 #18 Oct 18 2019 11:28:41
%S A328400 1,2,2,4,2,2,2,8,4,2,2,12,2,2,2,16,2,12,2,12,2,2,2,24,4,2,8,12,2,2,2,
%T A328400 32,2,2,2,4,2,2,2,24,2,2,2,12,12,2,2,48,4,12,2,12,2,24,2,24,2,2,2,12,
%U A328400 2,2,12,64,2,2,2,12,2,2,2,72,2,2,12,12,2,2,2,48,16,2,2,12,2,2,2,24,2,12,2,12,2,2,2,96,2,12,12,4,2,2,2,24,2
%N A328400 Smallest number with the same set of distinct prime exponents as n.
%C A328400 A variant of A046523 which gives the smallest number with the same prime signature as n. However, in this sequence, if any prime exponent occurs multiple times in n, the extra occurrences are removed and the signature is that of one of the numbers where only distinct values of prime exponents occur (A130091).
%H A328400 Antti Karttunen, <a href="/A328400/b328400.txt">Table of n, a(n) for n = 1..16384</a>
%H A328400 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A328400 a(n) = A181821(A007947(A181819(n))).
%F A328400 For all n, a(n) = a(A046523(n)).
%e A328400 90 = 2^1 * 3^2 * 5^1 has prime signature (1,1,2). The smallest number with prime signature (1,2) is 12 = 2^2 * 3, thus a(90) = 12.
%t A328400 Array[Times @@ MapIndexed[Prime[#2[[1]]]^#1 &, Reverse[Flatten[Cases[FactorInteger[#], {p_, k_} :> Table[PrimePi[p], {k}]]]]] &[Times @@ FactorInteger[#][[All, 1]]] &@ If[# == 1, 1, Times @@ Prime@ FactorInteger[#][[All, -1]]] &, 105] (* _Michael De Vlieger_, Oct 17 2019, after _Gus Wiseman_ at A181821 *)
%o A328400 (PARI)
%o A328400 A007947(n) = factorback(factorint(n)[, 1]);
%o A328400 A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
%o A328400 A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); };
%o A328400 A328400(n) = A181821(A007947(A181819(n)));
%Y A328400 Cf. A007947, A046523, A181819, A181821, A328401 (rgs-transform).
%Y A328400 Cf. A005117 (gives indices of terms <= 2), A062503 (after its initial 1, gives indices of 4's in this sequence).
%K A328400 nonn
%O A328400 1,2
%A A328400 _Antti Karttunen_, Oct 15 2019