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

1,1

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

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A093688, A093696.

Sequence in context: A325428 A239064 A227455 * A165740 A241571 A080307

Adjacent sequences: A237414 A237415 A237416 * A237418 A237419 A237420

Keyword

nonn,base

Author

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

Extensions

Definition corrected by Jon E. Schoenfield, Feb 26 2014

Status

approved