A211484 Numbers for which the canonical prime factorization contains only an even number of exponents, all of which are congruent to 1 modulo 3.

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 48, 51, 55, 57, 58, 62, 65, 69, 74, 77, 80, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 112, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 162, 166, 176, 177, 178, 183

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 involving this sequence.

Examples

			6 is included, as its canonical prime factorization (2^1)*(3^1) contains an even number of exponents, all of which are congruent to 1 modulo 3.
		

Links

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

Crossrefs

This sequence includes A030229 and hence A006881. The definition is similar to that for A211485.

Programs

Mathematica

pfQ[n_]:=Module[{f=Transpose[FactorInteger[n]][[2]]},EvenQ[Length[f]] && Union[ Mod[f,3]]=={1}]; Join[{1},Select[Range[200],pfQ]] (* Harvey P. Dale, Mar 24 2016 *)

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 != 0, passes=0;break()); if( M[k,2] % 3 != 1, passes=0));
if( passes == 1 , print1(k, ", "); plnt++) ; if(mxind ==  plnt, break() ))}

PARI

is(n,f=factor(n))=omega(f)%2==0 && factorback(f[,2]%3)==1 \\ Charles R Greathouse IV, Sep 07 2017