A227891 Numbers for which the number of odious proper divisors (A000069) equals the number of evil proper divisors (A001969).

1, 9, 25, 289, 441, 529, 625, 841, 1849, 2809, 3249, 5041, 6889, 7225, 7569, 7921, 10201, 12769, 15129, 15625, 19321, 21025, 22201, 26569, 31329, 38809, 46225, 48841, 53361, 55225, 66049, 69169, 72361, 76729, 78961, 83521, 85849, 93025, 96721, 100489, 103041

Offset

1,2

Comments

All terms are odd squares (see Shevelev links).

Links

Donovan Johnson, Table of n, a(n) for n = 1..1000

Vladimir Shevelev, A set of sequences of perfect squares.

Vladimir Shevelev, A question of Donovan Johnson.

Formula

Common value for numbers of considered divisors is (A000005(a(n))-1)/2.

Example

1 has no proper divisors, so it is in the sequence.
9 has two proper divisors 1 (odious) and 3 (evil). Thus 9 is in the sequence.

Mathematica

isQ[n_] := Sum[Switch[Mod[Total[IntegerDigits[d, 2]], 2], 0, 1, 1, -1], {d, Most[Divisors[n]]}] == 0; Select[(2*Range[200]-1)^2, isQ] (* Jean-François Alcover, Dec 04 2015 *)

Prog

(PARI) is(n)=sumdiv(n, d, (-1)^hammingweight(d))==(-1)^hammingweight(n)
select(is, vector(10^4, i, (2*i-1)^2)) \\ Charles R Greathouse IV, Oct 26 2013
(PARI) c=0; forstep(i=1, 8135, 2, n=i^2; nd=numdiv(n); d=divisors(n); ce=0; co=0; for(j=1, nd-1, if(hammingweight(d[j])%2==0, ce++, co++)); if(ce==co, c++; write("b227891.txt", c " " n))) \\ Donovan Johnson, Oct 30 2013

Crossrefs

Cf. A000005, A227872, A227873, A227889, A000069, A001969.

Sequence in context: A014760 A013397 A293720 * A193065 A278911 A330828

Adjacent sequences: A227888 A227889 A227890 * A227892 A227893 A227894

Keyword

base,nonn

Author

Vladimir Shevelev and Peter J. C. Moses, Oct 26 2013

Status

approved