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Squarefree numbers which yield zero when their prime factors are xored together.
4

%I #20 Apr 01 2017 10:15:19

%S 70,646,1798,2145,3526,5865,6006,9177,11305,13110,16422,20553,20806,

%T 21489,23529,28905,28985,30305,31465,37961,38086,38454,42441,44022,

%U 44998,45353,45942,46345,53985,54230,55913,60630,60697,61705,62049,64790,78406,80934,81158

%N Squarefree numbers which yield zero when their prime factors are xored together.

%C All n for which A008683(n) <> 0 and A072594(n) = 0.

%C It seems that an analogous case as A072595 for GF(2)[X]-polynomials is just the squares of GF(2)[X]-polynomials (A000695), thus in that ring, the sequence analogous to this one would be empty.

%C This sequence happens also to encode in the prime factorization of n a certain subset of the Nim game positions that are second-player win.

%H Antti Karttunen and Charles R Greathouse IV, <a href="/A235488/b235488.txt">Table of n, a(n) for n = 1..10000</a> (first 75 terms from Karttunen)

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Nim">Nim</a>

%e 70 is included, as 70 = 2*5*7, whose binary representations are '10', '101' and '111', which when all are xored (cf. A003987) together, cancel all 1-bits, thus yielding zero.

%e 212585 is included, as 212585 = 5*17*41*61, and when we xor their base-2 representations together:

%e 101

%e 10001

%e 101001

%e 111101

%e --------

%e 000000

%e we get only zeros, because in each column (bit-position), there is an even number of 1-bits.

%t Select[Range[82000],SquareFreeQ[#]&&BitXor@@FactorInteger[#][[All,1]]==0&] (* _Harvey P. Dale_, Apr 01 2017 *)

%o (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define A235488 (MATCHING-POS 1 1 (lambda (n) (and (not (zero? (A008683 n))) (zero? (A072594 n))))))

%o (PARI) is(n)=if(n<9, return(0)); my(f=factor(n)); vecmax(f[,2])==1 && fold(bitxor, f[,1])==0 \\ _Charles R Greathouse IV_, Aug 06 2016

%Y Intersection of A005117 and A072595 (equally: of A005117 and A072596).

%Y Cf. A003987, A235490, A079599, A048833, A050314.

%K nonn,base

%O 1,1

%A _Antti Karttunen_, Jan 22 2014