A325825 -id:A325825 - cp's OEIS frontend

A325827 a(n) = A325825(2*n, sigma(n)).

1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 4, 1, 4, 3, 1, 1, 3, 1, 1, 1, 4, 1, 3, 5, 1, 1, 28, 1, 3, 1, 1, 3, 1, 1, 4, 1, 1, 1, 10, 14, 12, 1, 4, 3, 1, 1, 4, 5, 1, 3, 5, 1, 12, 1, 4, 1, 1, 1, 12, 5, 4, 1, 1, 1, 48, 1, 1, 3, 16, 1, 3, 5, 1, 5, 4, 1, 12, 1, 1, 1, 14, 1, 28, 1, 1, 3, 1, 1, 9, 16, 1, 5, 1, 1, 12, 5, 11, 3, 1, 34, 12, 1, 5, 3

Offset: 1

Antti Karttunen, May 22 2019

nonn

A091255 Square array computed from gcd(P(x),P(y)) where P(x) and P(y) are polynomials with coefficients in {0,1} given by the binary expansions of x and y, and the polynomial calculation is done over GF(2), with the result converted back to a binary number, and then expressed in decimal. Array is symmetric, and is read by falling antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 3, 2, 7, 2, 3, 2, 1, 2, 1, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1

Offset: 1

Antti Karttunen, Jan 03 2004

Array is read by antidiagonals, with (x,y) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...
Analogous to A003989.
"Coded in binary" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where a(k)=0 or 1).

			The top left 17 X 17 corner of the array:
      1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17
    +---------------------------------------------------------------
   1: 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   2: 1, 2, 1, 2, 1, 2, 1, 2, 1,  2,  1,  2,  1,  2,  1,  2,  1, ...
   3: 1, 1, 3, 1, 3, 3, 1, 1, 3,  3,  1,  3,  1,  1,  3,  1,  3, ...
   4: 1, 2, 1, 4, 1, 2, 1, 4, 1,  2,  1,  4,  1,  2,  1,  4,  1, ...
   5: 1, 1, 3, 1, 5, 3, 1, 1, 3,  5,  1,  3,  1,  1,  5,  1,  5, ...
   6: 1, 2, 3, 2, 3, 6, 1, 2, 3,  6,  1,  6,  1,  2,  3,  2,  3, ...
   7: 1, 1, 1, 1, 1, 1, 7, 1, 7,  1,  1,  1,  1,  7,  1,  1,  1, ...
   8: 1, 2, 1, 4, 1, 2, 1, 8, 1,  2,  1,  4,  1,  2,  1,  8,  1, ...
   9: 1, 1, 3, 1, 3, 3, 7, 1, 9,  3,  1,  3,  1,  7,  3,  1,  3, ...
  10: 1, 2, 3, 2, 5, 6, 1, 2, 3, 10,  1,  6,  1,  2,  5,  2,  5, ...
  11: 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 11,  1,  1,  1,  1,  1,  1, ...
  12: 1, 2, 3, 4, 3, 6, 1, 4, 3,  6,  1, 12,  1,  2,  3,  4,  3, ...
  13: 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1,  1, 13,  1,  1,  1,  1, ...
  14: 1, 2, 1, 2, 1, 2, 7, 2, 7,  2,  1,  2,  1, 14,  1,  2,  1, ...
  15: 1, 1, 3, 1, 5, 3, 1, 1, 3,  5,  1,  3,  1,  1, 15,  1, 15, ...
  16: 1, 2, 1, 4, 1, 2, 1, 8, 1,  2,  1,  4,  1,  2,  1, 16,  1, ...
  17: 1, 1, 3, 1, 5, 3, 1, 1, 3,  5,  1,  3,  1,  1,  15, 1, 17, ...
  ...
3, which is "11" in binary, encodes polynomial X + 1, while 7 ("111" in binary) encodes polynomial X^2 + X + 1, whereas 9 ("1001" in binary), encodes polynomial X^3 + 1. Now (X + 1)(X^2 + X + 1) = (X^3 + 1) when the polynomials are multiplied over GF(2), or equally, when multiplication of integers 3 and 7 is done as a carryless base-2 product (A048720(3,7) = 9). Thus it follows that A(3,9) = A(9,3) = 3 and A(7,9) = A(9,7) = 7.
Furthermore, 5 ("101" in binary) encodes polynomial X^2 + 1 which is equal to (X + 1)(X + 1) in GF(2)[X], thus A(5,9) = A(9,5) = 3, as the irreducible polynomial (X + 1) is the only common factor for polynomials X^2 + 1 and X^3 + 1.

Cf. A003987, A048720, A091256, A091257, A106449, A280500, A280501, A280503, A280505, A286153, A325634, A325635, A325825.

Cf. also A327856 (the upper left triangular section of this array), A327857.

A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2); \\ Antti Karttunen, Aug 12 2019

A(x,y) = A(y,x) = A(x, A003987(x,y)) = A(A003987(x,y), y), where A003987 gives the bitwise-XOR of its two arguments. - Antti Karttunen, Sep 28 2019

Data section extended up to a(105), examples added by Antti Karttunen, Sep 28 2019

A325820 Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 1, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 7, 12, 12, 7, 6, 0, 0, 7, 3, 15, 16, 15, 3, 7, 0, 0, 8, 5, 18, 11, 11, 18, 5, 8, 0, 0, 9, 4, 21, 24, 13, 24, 21, 4, 9, 0, 0, 10, 18, 24, 19, 21, 21, 19, 24, 18, 10, 0, 0, 11, 20, 27, 23, 26, 9, 26, 23, 27, 20, 11, 0, 0, 12, 19, 30, 36, 19, 15, 15, 19, 36, 30, 19, 12, 0

Offset: 0

Antti Karttunen, May 22 2019

			The array begins as:
  0,  0,  0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0, ...
  0,  1,  2,  3,  4,  5,  6,  7,  8,   9,  10,  11,  12, ...
  0,  2,  1,  6,  8,  7,  3,  5,  4,  18,  20,  19,  24, ...
  0,  3,  6,  9, 12, 15, 18, 21, 24,  27,  30,  33,  36, ...
  0,  4,  8, 12, 16, 11, 24, 19, 23,  36,  40,  44,  48, ...
  0,  5,  7, 15, 11, 13, 21, 26, 19,  45,  50,  52,  33, ...
  0,  6,  3, 18, 24, 21,  9, 15, 12,  54,  60,  57,  72, ...
  0,  7,  5, 21, 19, 26, 15, 13, 11,  63,  70,  68,  57, ...
  0,  8,  4, 24, 23, 19, 12, 11, 16,  72,  80,  76,  69, ...
  0,  9, 18, 27, 36, 45, 54, 63, 72,  81,  90,  99, 108, ...
  0, 10, 20, 30, 40, 50, 60, 70, 80,  90, 100,  83, 120, ...
  0, 11, 19, 33, 44, 52, 57, 68, 76,  99,  83,  91, 132, ...
  0, 12, 24, 36, 48, 33, 72, 57, 69, 108, 120, 132, 144, ...
  etc.
A(2,2) = 2*2 mod 3 = 1.

Cf. A169999 (the main diagonal).
Row/Column 0: A000004, Row/Column 1: A001477, Row/Column 2: A004488, Row/Column 3: A008585, Row/Column 4: A242399, Row/Column 9: A008591.
Cf. A007089, A207669, A207670, A263273, A325825.
Cf. A325821 (same table without the zero row and column).
Cf. A048720 (binary), A059692 (decimal), A004247 (full multiply).

up_to = 105;
A325820sq(b, c) = fromdigits(Vec(Pol(digits(b,3))*Pol(digits(c,3)))%3, 3);
A325820list(up_to) = { my(v = vector(up_to), i=0); for(a=0,oo, for(col=0,a, if(i++ > up_to, return(v)); v[i] = A325820sq(a-col,col))); (v); };
v325820 = A325820list(up_to);
A325820(n) = v325820[1+n];

A330740 a(n) = min(n, A004488(n)), where A004488(n) is base-3 sum n+n without carries.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 3, 5, 4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 9, 11, 10, 15, 17, 16, 12, 14, 13, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 29, 28, 33, 35, 34, 30, 32, 31, 45, 47, 46, 51, 53, 52, 48, 50, 49, 36, 38, 37, 42, 44, 43, 39, 41, 40, 81

Offset: 0

Antti Karttunen, Jan 11 2020

After the initial zero, sequence contains only terms of A132141, each appearing exactly twice.

Antti Karttunen, Table of n, a(n) for n = 0..59048

Cf. A004488, A132141.

Central diagonal of A325825 (after the initial zero).

A330740(n) = min(n,subst(Pol(apply(x->(3-x)%3, digits(n, 3)), 'x), 'x, 3));

a(n) = min(n, A004488(n)).

A325808 Numbers n such that sigma(n) can be obtained as the base-3 carryless product of 2n and some k.

Original entry on oeis.org

1, 6, 28, 120, 259, 496, 8128, 18990, 667296, 1858939, 2097414, 2383279, 4843717, 33550336, 150588313, 186695863, 188908297

Offset: 1

Antti Karttunen, May 22 2019

Numbers n that satisfy A000203(n) = A325820(2n, k) for some k.
Numbers n such that polynomial p divides polynomial q over GF(3), where p and q are obtained from the base-3 representations of 2n and sigma(n). (See the examples).
Conjecture: If we select only those n of these for which sigma(n) >= 2n, then we get a subsequence which contains only even terms: 6, 28, 120, 496, 8128, 18990, 667296, 2097414, 33550336, etc. If this is true, then there are no odd perfect numbers. See also conjectures in A325638 and A325639.

			2*120 has ternary representation (A007089) 22220_3, thus it encodes polynomial 2*x^4 + 2*x^3 + 2*x^2 + 2*x, while sigma(120) = 360 = 111100_3, encodes polynomial x^5 + x^4 + x^3 + x^2 which is a multiple of the former as it is equal to 2x(x^4 + x^3 + x^2 + x) when polynomial multiplication is done over GF(3). Thus 120 is included in this sequence.
2*259 = 201012_3 encodes polynomial 2*x^5 + x^3 + x + 2, while sigma(259) = 304 = 102021_3 encodes polynomial x^5 + 2*x^3 + 2*x + 1 = 2(2*x^5 + x^3 + x + 2), thus 259 is included.
2*18990 = 1221002200_3 encodes polynomial x^9 + 2*x^8 + 2*x^7 + x^6 + 2*x^3 + 2*x^2, while sigma(18990) = 49608 = 2112001100_3 encodes polynomial 2*x^9 + x^8 + x^7 + 2*x^6 + x^3 + x^2 = 2(x^9 + 2*x^8 + 2*x^7 + x^6 + 2*x^3), thus 18990 is included.
2*667296 = 2111210201100_3 encodes polynomial 2*x^12 + x^11 + x^10 + x^9 + 2*x^8 + x^7 + 2*x^5 + x^3 + x^2, while sigma(667296) = 2175264 = 11002111220100_3 encodes polynomial x^13 + x^12 + 2*x^9 + x^8 + x^7 + x^6 + 2*x^5 + 2*x^4 + x^2 = (2*x + 1)(2*x^12 + x^11 + x^10 + x^9 + 2*x^8 + x^7 + 2*x^5 + x^3 + x^2) [when polynomial multiplication is done over GF(3)], thus 667296 is included.

Cf. A000203, A325638, A325820, A325825, A325827.

Cf. A000396 (a subsequence).

isA325808(n) = { my(p=Pol(digits(n+n,3))*Mod(1, 3), q=Pol(digits(sigma(n),3))*Mod(1, 3)); !(q%p); };

cp's OEIS Frontend

A325827 a(n) = A325825(2*n, sigma(n)).

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A325820 Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

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A330740 a(n) = min(n, A004488(n)), where A004488(n) is base-3 sum n+n without carries.

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A325808 Numbers n such that sigma(n) can be obtained as the base-3 carryless product of 2n and some k.

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