A236378 -id:A236378 - cp's OEIS frontend

A234741 a(n) is the base-2 carryless product of the prime factors of n; Encoding of the product of the polynomials over GF(2) represented by the prime factors of n (with multiplicity).

1, 2, 3, 4, 5, 6, 7, 8, 5, 10, 11, 12, 13, 14, 15, 16, 17, 10, 19, 20, 9, 22, 23, 24, 17, 26, 15, 28, 29, 30, 31, 32, 29, 34, 27, 20, 37, 38, 23, 40, 41, 18, 43, 44, 17, 46, 47, 48, 21, 34, 51, 52, 53, 30, 39, 56, 53, 58, 59, 60, 61, 62, 27, 64, 57, 58, 67

Offset: 1

Antti Karttunen, Jan 22 2014

nonn

"Encoding" means the number whose binary representation is given by the coefficients of the polynomial, e.g., 13=1101[2] encodes X^3+X^2+1. The product is the usual multiplication of polynomials in GF(2)[X] (or binary multiplication without carry-bits, cf. A048720).

a(n) <= n. [As all terms of the table A061858 are nonnegative]

			a(9) = a(3*3) = 5, as when we multiply 3 ('11' in binary) with itself, and discard the carry-bits, using XOR (A003987) instead of normal addition, we get:
   11
  110
-----
  101
that is, 5, as '101' is its binary representation. In other words, a(9) = a(3*3) = A048720(3,3) = 5.
Alternatively, 9 = 3*3, and 3=11[2] encodes the polynomial X+1, and (X+1)*(X+1) = X^2+1 in GF(2)[X], which is encoded as 101[2] = 5, therefore a(9) = 5. - _M. F. Hasler_, Feb 16 2014

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to polynomials in ring GF(2)[X]

A235034 gives the k for which a(k)=k.
A236833(n) gives the number of times n occurs in this sequence.
A236841 gives the same sequence sorted and duplicates removed, A236834 gives the numbers that do not occur here, A236835 gives numbers that occur more than once.
A325562(n) gives the number of iterations needed before one of the fixed points (terms of A235034) is reached.
Cf. also A048720, A061858, A234742, A236378, A091202/A091203, A235041/A235042, A266195, A325561, A325562.

A234741(n)={n=factor(n);n[,1]=apply(t->Pol(binary(t)),n[,1]);sum(i=1,#n=Vec(factorback(n))%2,n[i]<<(#n-i))} \\ M. F. Hasler, Feb 18 2014

a(0)=0, a(1)=1, and for n > 1, a(n) = A048720(A020639(n),a(n/A020639(n))). [A048720 used as a bivariate function]
Equally, for n with its unique prime factorization n = p_1 * ... * p_k, with the p_i not necessarily distinct primes, a(n) = p_1 x ... x p_k, where x stands for carryless multiplication defined in A048720, which is isomorphic to multiplication in GF(2)[X].
a(2n) = 2*a(n).
More generally, if A061858(x,y) = 0, then a(x*y) = a(x)*a(y).
a(A235034(n)) = A235034(n).
A236378(n) = n - a(n).

Term a(0) = 0 removed and a new primary definition added by Antti Karttunen, May 10 2019

A061858 Differences between the ordinary multiplication table A004247 and the carryless multiplication table for GF(2)[X] polynomials A048720, i.e., the effect of the carry bits in binary multiplication.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 0, 0, 0, 0, 12, 0, 8, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 24, 0, 0, 0

Offset: 0

Antti Karttunen, May 11 2001

			From _Peter Munn_, Jan 28 2021: (Start)
The top left 12 X 12 corner of the table:
      |  0   1   2   3   4   5   6   7   8   9  10  11
------+------------------------------------------------
   0  |  0   0   0   0   0   0   0   0   0   0   0   0
   1  |  0   0   0   0   0   0   0   0   0   0   0   0
   2  |  0   0   0   0   0   0   0   0   0   0   0   0
   3  |  0   0   0   4   0   0   8  12   0   0   0   4
   4  |  0   0   0   0   0   0   0   0   0   0   0   0
   5  |  0   0   0   0   0   8   0   8   0   0  16  16
   6  |  0   0   0   8   0   0  16  24   0   0   0   8
   7  |  0   0   0  12   0   8  24  28   0   0  16  28
   8  |  0   0   0   0   0   0   0   0   0   0   0   0
   9  |  0   0   0   0   0   0   0   0   0  16   0  16
  10  |  0   0   0   0   0  16   0  16   0   0  32  32
  11  |  0   0   0   4   0  16   8  28   0  16  32  52
(End)

"Zoomed in" variant: A061859.
Rows/columns 3, 5 and 7 are given by A048728, A048729, A048730.
Main diagonal divided by 4: A213673.
Numbers that generate no carries when multiplied in binary by 11_2: A003714, by 101_2: A048716, by 1001_2: A115845, by 10001_2: A115847, by 100001_2: A114086.
Other sequences related to the presence/absence of a carry in binary multiplication: A116361, A235034, A235040, A236378, A266195, A289726.

a(n) = A004247(n) - A048720(n).

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.

Original entry on oeis.org

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.

A236379 How much n increases when it is remultiplied from GF(2)[X] to Z: a(n) = A234742(n) - n.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 0, 0, 12, 8, 0, 0, 0, 0, 12, 0, 64, 24, 0, 16, 28, 0, 16, 0, 0, 0, 36, 0, 4, 24, 0, 0, 60, 128, 56, 48, 0, 0, 60, 32, 0, 56, 32, 0, 144, 32, 0, 0, 28, 0, 192, 0, 4, 72, 0, 0, 60, 8, 0, 48, 0, 0, 84, 0, 376, 120, 0, 256, 52, 112, 112, 96, 0, 0, 276, 0, 100, 120, 96, 64, 88, 0, 148, 112, 644, 64

Offset: 0

Antti Karttunen, Jan 24 2014

nonn

All terms are divisible by 4.

Antti Karttunen, Table of n, a(n) for n = 0..8191

A235035 gives the positions of zeros.

Cf. also A234742, A236378, A236380, A091209, A236835, A236844.

Scheme
```
(define (A236379 n) (- (A234742 n) n))
```

a(n) = A234742(n) - n.

For all n, a(A091209(n)) > 0, and also a(A236844(n)) > 0 and a(A236835(n)) > 0.

A236380 Difference between value of n, when remultiplied "upward", from GF(2)[X] to N, and when remultiplied "downward", from N to GF(2)[X]: a(n) = A234742(n) - A234741(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 0, 0, 16, 8, 0, 0, 0, 0, 12, 0, 64, 32, 0, 16, 40, 0, 16, 0, 8, 0, 48, 0, 4, 24, 0, 0, 64, 128, 64, 64, 0, 0, 76, 32, 0, 80, 32, 0, 172, 32, 0, 0, 56, 16, 192, 0, 4, 96, 16, 0, 64, 8, 0, 48, 0, 0, 120, 0, 384, 128, 0, 256, 64, 128, 112, 128, 0, 0, 300, 0, 128, 152, 96, 64, 152, 0, 148, 160, 644, 64

Offset: 0

Antti Karttunen, Jan 24 2014

nonn

All terms are divisible by 4.

a(n) = 0 iff both A236378(n) and A236379(n) are zero, or in other words, iff A234741(n)=n and A234742(n)=n, which means that A235032 gives all such n, that a(n) = 0.

Antti Karttunen, Table of n, a(n) for n = 0..8191

A235032 gives the positions of zeros, A235033 the positions of nonzeros.

Cf. also A234741, A234742, A236378, A236379, A236851, A061858.

a(n) = A234742(n) - A234741(n).

a(n) = A236378(n) + A236379(n).

cp's OEIS Frontend

A234741 a(n) is the base-2 carryless product of the prime factors of n; Encoding of the product of the polynomials over GF(2) represented by the prime factors of n (with multiplicity).

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A061858 Differences between the ordinary multiplication table A004247 and the carryless multiplication table for GF(2)[X] polynomials A048720, i.e., the effect of the carry bits in binary multiplication.

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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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A236379 How much n increases when it is remultiplied from GF(2)[X] to Z: a(n) = A234742(n) - n.

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A236380 Difference between value of n, when remultiplied "upward", from GF(2)[X] to N, and when remultiplied "downward", from N to GF(2)[X]: a(n) = A234742(n) - A234741(n).

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