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

%I #20 Oct 21 2023 10:02:21

%S 1,4,9,25,32,36,49,100,121,169,196,225,243,256,288,289,361,441,484,

%T 529,676,800,841,900,961,972,1089,1156,1225,1369,1444,1521,1568,1681,

%U 1764,1849,2048,2116,2209,2304,2601,2809,3025,3125,3249,3364,3481,3721,3844

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

%C By convention 1 is included as the first term.

%H Amiram Eldar, <a href="/A182120/b182120.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Douglas Latimer)

%F 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 100 is included, as its canonical prime factorization (2^2)*(5^2) contains only exponents which are congruent to 2 modulo 3.

%t Join[{1},Select[Range[5000],Union[Mod[Transpose[FactorInteger[#]][[2]],3]] == {2}&]] (* _Harvey P. Dale_, Aug 18 2014 *)

%o (PARI) {plnt=1; k=1; print1(k, ", "); plnt++;

%o mxind=76 ; mxind++ ; for(k=2, 2*10^6,

%o M=factor(k);passes=1;

%o sz = matsize(M)[1];

%o for(k=1,sz, if( M[k,2] % 3 != 2, passes=0));

%o if( passes == 1 ,

%o print1(k, ", "); plnt++) ; if(mxind == plnt, break() ))}

%o (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 A062503 is a subsequence.

%Y Subsequence of A001694.

%Y Cf. A002117, A366762.

%K nonn,easy

%O 1,2

%A _Douglas Latimer_, Apr 12 2012