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
Examples
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.
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.
Programs
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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 *) -
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
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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
Formula
Common value for numbers of considered divisors is (A000005(a(n))-1)/2.
Comments