A211485 - OEIS

%I #11 Jul 03 2019 15:25:22

%S 1,2,3,5,7,11,13,16,17,19,23,29,30,31,37,41,42,43,47,53,59,61,66,67,

%T 70,71,73,78,79,81,83,89,97,101,102,103,105,107,109,110,113,114,127,

%U 128,130,131,137,138,139,149,151,154,157,163,165,167,170,173,174,179

%N Numbers n for which the canonical prime factorization contains only an odd number of exponents, all of which are congruent to 1 modulo 3.

%C By convention 1 is included as the first term, in order to simplify the statement of certain results concerning this sequence.

%H Douglas Latimer, <a href="/A211485/b211485.txt">Table of n, a(n) for n = 1..1000</a>

%e 3 is included, as its canonical prime factorization 3^1 contains only an odd number of exponents, all of which are congruent to 1 modulo 3.

%e 81 is in the sequence, because its canonical prime factorization is 3^4, and that one exponent, 4, is congruent to 1 modulo 3.

%t oneQ[n_]:=Module[{f=FactorInteger[n][[All,2]]},OddQ[Length[f]]&&Union[ Mod[ f,3]]=={1}]; Select[Range[200],oneQ] (* _Harvey P. Dale_, Jul 03 2019 *)

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

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

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

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

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

%o if( passes == 1 , print1(k, ", "); plnt++) ; if(mxind ==  plnt, break() ))}

%Y The definition is similar to that for A211484. This sequence includes the prime numbers A000040, and includes A030059.

%K nonn,easy

%O 1,2

%A _Douglas Latimer_, Apr 12 2012