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A359747 Numbers k such that k*(k+1) has in its canonical prime factorization mutually distinct exponents.

%I A359747 #12 Jan 17 2023 09:23:42
%S A359747 1,3,4,7,8,16,24,27,31,48,63,71,72,107,108,124,127,199,242,243,256,
%T A359747 400,431,432,499,512,576,647,783,863,967,971,1024,1151,1152,1372,1567,
%U A359747 1600,1999,2187,2311,2400,2591,2592,2887,2916,3087,3136,3456,3887,3888,3968,4000
%N A359747 Numbers k such that k*(k+1) has in its canonical prime factorization mutually distinct exponents.
%C A359747 Equivalently, numbers k such that A002378(k) = k*(k+1) is a term of A130091.
%C A359747 Equivalently, numbers k such that the multisets of exponents in the prime factorizations of k and k+1 are disjoint and each have distinct elements.
%C A359747 Either k or k+1 is a powerful number (A001694). Except for k=8, are there terms k such that both k and k+1 are powerful (i.e., terms that are also in A060355)? None of the terms A060355(n) for n = 2..39 is in this sequence.
%H A359747 Amiram Eldar, <a href="/A359747/b359747.txt">Table of n, a(n) for n = 1..1000</a>
%e A359747 3 is a term since 3*4 = 12 = 2^2 * 3^1 has 2 distinct exponents in its prime factorization: 1 and 3.
%t A359747 q[n_] := UnsameQ @@ (FactorInteger[n*(n+1)][[;; , 2]]); Select[Range[4000], q]
%o A359747 (PARI) is(n) = { my(e = factor(n*(n+1))[, 2]); #Set(e) == #e; }
%Y A359747 Subsequence of A130091 and A342028.
%Y A359747 A359748 is a subsequence.
%Y A359747 Cf. A001694, A002378, A060355.
%K A359747 nonn
%O A359747 1,2
%A A359747 _Amiram Eldar_, Jan 13 2023