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A182120 Numbers for which the canonical prime factorization contains only exponents which are congruent to 2 modulo 3.

%I A182120 #20 Oct 21 2023 10:02:21
%S A182120 1,4,9,25,32,36,49,100,121,169,196,225,243,256,288,289,361,441,484,
%T A182120 529,676,800,841,900,961,972,1089,1156,1225,1369,1444,1521,1568,1681,
%U A182120 1764,1849,2048,2116,2209,2304,2601,2809,3025,3125,3249,3364,3481,3721,3844
%N A182120 Numbers for which the canonical prime factorization contains only exponents which are congruent to 2 modulo 3.
%C A182120 By convention 1 is included as the first term.
%H A182120 Amiram Eldar, <a href="/A182120/b182120.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Douglas Latimer)
%F A182120 Sum_{n>=1} 1/a(n) = zeta(3) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.56984817927051410948... . - _Amiram Eldar_, Oct 21 2023
%e A182120 100 is included, as its canonical prime factorization (2^2)*(5^2) contains only exponents which are congruent to 2 modulo 3.
%t A182120 Join[{1},Select[Range[5000],Union[Mod[Transpose[FactorInteger[#]][[2]],3]] == {2}&]] (* _Harvey P. Dale_, Aug 18 2014 *)
%o A182120 (PARI) {plnt=1; k=1; print1(k, ", "); plnt++;
%o A182120 mxind=76 ; mxind++ ; for(k=2, 2*10^6,
%o A182120 M=factor(k);passes=1;
%o A182120 sz = matsize(M)[1];
%o A182120 for(k=1,sz,  if( M[k,2] % 3 != 2, passes=0));
%o A182120 if( passes == 1 ,
%o A182120 print1(k, ", "); plnt++) ; if(mxind ==  plnt, break() ))}
%o A182120 (PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2]%3 != 2, return(0))); 1;} \\ _Amiram Eldar_, Oct 21 2023
%Y A182120 A062503 is a subsequence.
%Y A182120 Subsequence of A001694.
%Y A182120 Cf. A002117, A366762.
%K A182120 nonn,easy
%O A182120 1,2
%A A182120 _Douglas Latimer_, Apr 12 2012