

A235488


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


4



70, 646, 1798, 2145, 3526, 5865, 6006, 9177, 11305, 13110, 16422, 20553, 20806, 21489, 23529, 28905, 28985, 30305, 31465, 37961, 38086, 38454, 42441, 44022, 44998, 45353, 45942, 46345, 53985, 54230, 55913, 60630, 60697, 61705, 62049, 64790, 78406, 80934, 81158
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OFFSET

1,1


COMMENTS

All n for which A008683(n) <> 0 and A072594(n) = 0.
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.
This sequence happens also to encode in the prime factorization of n a certain subset of the Nim game positions that are secondplayer win.


LINKS

Antti Karttunen and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 75 terms from Karttunen)
Wikipedia, Nim


EXAMPLE

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 1bits, thus yielding zero.
212585 is included, as 212585 = 5*17*41*61, and when we xor their base2 representations together:
101
10001
101001
111101

000000
we get only zeros, because in each column (bitposition), there is an even number of 1bits.


MATHEMATICA

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


PROG

(Scheme, with Antti Karttunen's IntSeqlibrary)
(define A235488 (MATCHINGPOS 1 1 (lambda (n) (and (not (zero? (A008683 n))) (zero? (A072594 n))))))
(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


CROSSREFS

Intersection of A005117 and A072595 (equally: of A005117 and A072596).
Cf. A003987, A235490, A079599, A048833, A050314.
Sequence in context: A183715 A104475 A169712 * A199829 A271495 A232587
Adjacent sequences: A235485 A235486 A235487 * A235489 A235490 A235491


KEYWORD

nonn,base


AUTHOR

Antti Karttunen, Jan 22 2014


STATUS

approved



