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A359749 Numbers k such that k and k+1 do not share a common exponent in their prime factorizations.

%I A359749 #11 Jan 17 2023 09:23:37
%S A359749 1,3,4,7,8,9,15,16,24,25,26,27,31,32,35,36,48,63,64,71,72,81,100,107,
%T A359749 108,120,121,124,125,127,128,143,144,168,169,195,196,199,200,215,216,
%U A359749 224,225,242,243,255,256,287,289,323,342,361,391,392,399,400,431,432,440
%N A359749 Numbers k such that k and k+1 do not share a common exponent in their prime factorizations.
%C A359749 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.
%C A359749 A002496(k)-1, A078324(k)-1, A078325(k)-1, and A049533(k)^2 are terms for all k >= 1.
%H A359749 Amiram Eldar, <a href="/A359749/b359749.txt">Table of n, a(n) for n = 1..10000</a>
%e A359749 3 is a term since 3 has the exponent 1 in its prime factorization, and 3 + 1 = 4 = 2^2 has a different exponent in its prime factorization, 2.
%t A359749 q[n_] := UnsameQ @@ Join @@ (Union[FactorInteger[#][[;; , 2]]]& /@ (n + {0, 1})); Join[{1}, Select[Range[400], q]]
%o A359749 (PARI) lista(nmax) = {my(e1 = [], e2); for(n = 2, nmax, e2 = Set(factor(n)[,2]); if(setintersect(e1, e2) == [], print1(n-1, ", ")); e1 = e2); }
%Y A359749 Subsequences: A000079, A000225 \ {0}, A075408, A359747.
%Y A359749 Cf. A001694, A002496, A049533, A060355, A078324, A078325.
%K A359749 nonn
%O A359749 1,2
%A A359749 _Amiram Eldar_, Jan 13 2023