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

1, 2, 3, 5, 7, 11, 13, 16, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 73, 78, 79, 81, 83, 89, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 127, 128, 130, 131, 137, 138, 139, 149, 151, 154, 157, 163, 165, 167, 170, 173, 174, 179

Offset: 1

Author

Douglas Latimer, Apr 12 2012

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

• Douglas Latimer, Table of n, a(n) for n = 1..1000

Crossrefs

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

Programs

Mathematica

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 *)

PARI

{plnt=1;k=1; print1(k, ", "); plnt++;
mxind=76 ; mxind++ ; for(k=2, 10^6,
M=factor(k);passes=1;
sz = matsize(M)[1];
for(k=1,sz, if(sz%2 != 1, passes=0;break()); if( M[k,2] % 3 != 1, passes=0));
if( passes == 1 , print1(k, ", "); plnt++) ; if(mxind ==  plnt, break() ))}