A237417 Numbers that are the product of an odiousfree number and an evilfree number.

3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 21, 23, 24, 27, 29, 30, 33, 34, 35, 36, 39, 40, 42, 43, 45, 46, 48, 51, 53, 54, 55, 57, 58, 60, 63, 65, 66, 68, 70, 71, 72, 78, 80, 83, 84, 85, 86, 89, 90, 92, 93, 95, 96, 99, 101, 102, 105, 106, 108, 110, 111, 113, 114, 116, 117, 119, 120, 123, 126, 129

Offset: 1

Author

Irina Gerasimova, Feb 23 2014, following a suggestion from Juri-Stepan Gerasimov

Keywords

• nonn
• base

Comments

Odiousfree*evilfree numbers: numbers of the form odiousfree*evilfree.

Subsequence of this sequence (A237417): numbers that are not the products of two odious numbers or the products of two evil numbers: 3, 5, 6, 10, 12, 17, 20, 23, 24, 29, 33, 34, 39, 40, 43, 46, 48, 57, 58, 63, 65, 66, 68, 71, 78, 80, 83, 86, 89, 92, 95, 101, 105, 106, 111, 113, 114, 116, 119,...

Putting the 1 aside in A093688, these could be called odiousfree numbers, and are a subsequence of A001969. A093696 would be the evilfree numbers then, and are a subsequence of A000069.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A093688, A093696.

Programs

Maple

N:= 200: # to get all terms <= N
Ofree:= {$2..N}: Efree:= {$1..N/3}:
for n from 2 to N do
  t:= convert(convert(n,base,2),`+`) mod 2;
  if t = 0 then Efree:= Efree minus {seq(i,i=n..N/3,n)}
  else Ofree:= Ofree minus {seq(i,i=n..N,n)}
  fi
od:
sort(convert(select(`<=`,{seq(seq(s*t,s=Ofree),t=Efree)},N),list)); # Robert Israel, May 09 2019

Mathematica

odFreeQ[n_] := AllTrue[Rest @ Divisors[n], EvenQ[DigitCount[#, 2, 1]] &]; evFreeQ[n_] := AllTrue[Divisors[n], OddQ[DigitCount[#, 2, 1]] &]; m = 100; o = Select[Range[2, m], odFreeQ]; e = Select[Range[m], evFreeQ]; Union @ Select[Times @@@ Tuples[{o, e}], # <= m &] (* Amiram Eldar, Oct 16 2020 *)

PARI

isA093696(n)= fordiv(n, d, if(hammingweight(d)%2==0, return(0))); 1;
isA093688(n)= if (n==1, 0, sumdiv(n, d, hammingweight(d)%2)==1);
lista(nn) = {vn = vector(2*nn, i, i); vof = select(n->isA093696(n), vn); vef = select(n->isA093688(n), vn); vp = []; for (i = 1, #vof, for (j = 1, #vef, vp = Set(concat(vp, vof[i]*vef[j])););); for (i = 1, #vp, if (vp[i] <= nn, print1(vp[i], ", ")););} \\ Michel Marcus, Mar 05 2014

Formula

a(n) = A093688(k+1)*A093696(m).

Extensions

Definition corrected by Jon E. Schoenfield, Feb 26 2014