A235490 -id:A235490 - cp's OEIS frontend

A235034 Numbers whose prime divisors, when multiplied together without carry-bits (as encodings of GF(2)[X]-polynomials, with A048720), produce the original number; numbers for which A234741(n) = n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 37, 38, 40, 41, 43, 44, 46, 47, 48, 51, 52, 53, 56, 58, 59, 60, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 85, 86, 88, 89, 92, 94, 95, 96, 97, 101

Offset: 1

Antti Karttunen, Jan 02 2014

nonn

If n is present, then 2n is present also, as shifting binary representation left never produces any carries.

			All primes occur in this sequence as no multiplication -> no need to add any intermediate products -> no carry bits produced.
Composite numbers like 15 are also present, as 15 = 3*5, and when these factors (with binary representations '11' and '101') are multiplied as:
   101
  1010
  ----
  1111 = 15
we see that the intermediate products 1*5 and 2*5 can be added together without producing any carry-bits (as they have no 1-bits in the same columns/bit-positions), so A048720(3,5) = 3*5 and thus 15 is included in this sequence.

Gives the positions of zeros in A236378, i.e., n such that A234741(n) = n.
Intersection with A235035 gives A235032.
Other subsequences: A000040 (A091206 and also A091209), A045544 (A004729), A093641, A235040 (gives odd composites in this sequence), A235050, A235490.
Cf. A048720, A061858, A234741.

A235050 Squarefree numbers such that none of their prime factors share common 1-bits in the same bit-position of their binary representations.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 17, 19, 23, 26, 29, 31, 34, 37, 41, 43, 47, 53, 58, 59, 61, 67, 71, 73, 74, 79, 82, 83, 89, 97, 101, 103, 106, 107, 109, 113, 122, 127, 131, 137, 139, 146, 149, 151, 157, 163, 167, 173, 178, 179, 181, 191, 193, 194, 197, 199, 202, 211, 218, 223, 226, 227, 229, 233, 239, 241, 251, 257

Offset: 1

Antti Karttunen, Jan 22 2014

a(1)=1 is included on the grounds that it has no prime factors at all, thus no conflicting 1-bits.
After a(1)=1 all n such that A001414(n) = A072593(n), or equally, A001414(n) = A072594(n).
Union of noncomposites (A008578) and semiprimes of the form 2*A002144 (cf. also A235490).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Subsequences: A000040, A235490.
Subsequence of A005117.
Cf. also A001414, A072593, A072594, A235040.

A235488 Squarefree numbers which yield zero when their prime factors are xored together.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 22 2014

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.

			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.

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

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

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

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

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

cp's OEIS Frontend

A235034 Numbers whose prime divisors, when multiplied together without carry-bits (as encodings of GF(2)[X]-polynomials, with A048720), produce the original number; numbers for which A234741(n) = n.

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A235050 Squarefree numbers such that none of their prime factors share common 1-bits in the same bit-position of their binary representations.

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A235488 Squarefree numbers which yield zero when their prime factors are xored together.

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