cp's OEIS Frontend

A211485 Numbers for which the canonical prime factorization contains only an odd number of exponents, all of which are congruent to 1 modulo 3.

%I A211485 #14 Jul 28 2025 00:02:02
%S A211485 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 A211485 70,71,73,78,79,81,83,89,97,101,102,103,105,107,109,110,113,114,127,
%U A211485 128,130,131,137,138,139,149,151,154,157,163,165,167,170,173,174,179
%N A211485 Numbers for which the canonical prime factorization contains only an odd number of exponents, all of which are congruent to 1 modulo 3.
%C A211485 By convention 1 is included as the first term, in order to simplify the statement of certain results concerning this sequence.
%H A211485 Douglas Latimer, <a href="/A211485/b211485.txt">Table of n, a(n) for n = 1..1000</a>
%e A211485 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 A211485 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 A211485 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 A211485 (PARI) {plnt=1;k=1; print1(k, ", "); plnt++;
%o A211485 mxind=76 ; mxind++ ; for(k=2, 10^6,
%o A211485 M=factor(k);passes=1;
%o A211485 sz = matsize(M)[1];
%o A211485 for(k=1,sz, if(sz%2 != 1, passes=0;break()); if( M[k,2] % 3 != 1, passes=0));
%o A211485 if( passes == 1 , print1(k, ", "); plnt++) ; if(mxind ==  plnt, break() ))}
%Y A211485 The definition is similar to that for A211484. This sequence includes the prime numbers A000040, and includes A030059.
%K A211485 nonn,easy
%O A211485 1,2
%A A211485 _Douglas Latimer_, Apr 12 2012