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%I #8 Mar 30 2024 05:34:15
%S 1,12,48,72,192,288,432,768,1152,1260,1728,2592,3072,4608,5040,6912,
%T 10368,12288,12600,15552,18432,20160,27648,41472,45360,49152,50400,
%U 62208,73728,75600,80640,93312,110592,165888,181440,196608,201600,248832,264600,294912,302400
%N Numbers of least prime signature (A025487) whose prime factorization has equal number of even and odd exponents.
%H Amiram Eldar, <a href="/A371599/b371599.txt">Table of n, a(n) for n = 1..10000</a>
%e The prime signatures of the first 12 terms are:
%e n a(n) signature A162641(a(n)) = A162642(a(n))
%e -- ------- ------------ -----------------------------
%e 1 1 {} 0
%e 2 12 {2,1} 1
%e 3 48 {4,1} 1
%e 4 72 {3,2} 1
%e 5 192 {6,1} 1
%e 6 288 {5,2} 1
%e 7 432 {4,3} 1
%e 8 768 {8,1} 1
%e 9 1152 {7,2} 1
%e 10 1260 {2,2,1,1} 2
%e 11 1728 {6,3} 1
%e 12 2592 {5,4} 1
%t fun[p_, e_] := (-1)^e; q[n_] := Module[{f = FactorInteger[n]}, n == 1 || (f[[-1, 1]] == Prime[Length[f]] && Max@ Differences[f[[;; , 2]]] < 1 && Plus @@ fun @@@ f == 0)]; Select[Range[3*10^5], q]
%o (PARI) is(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); n == 1 || (prime(#p) == p[#p] && e == vecsort(e, , 4) && sum(i = 1, #e, (-1)^e[i]) == 0);}
%Y Intersection of A025487 and A187039.
%Y Cf. A162641, A162642, A371600.
%K nonn
%O 1,2
%A _Amiram Eldar_, Mar 29 2024
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