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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 (list; graph; refs; listen; history; text; internal format)
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 second-player 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 1-bits, thus yielding zero.

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

     101

   10001

  101001

  111101

--------

  000000

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

MATHEMATICA

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

PROG

(Scheme, with Antti Karttunen's IntSeq-library)

(define A235488 (MATCHING-POS 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

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Last modified October 1 19:46 EDT 2022. Contains 357171 sequences. (Running on oeis4.)