A325827 -id:A325827 - cp's OEIS frontend

A325825 Square array giving the monic polynomial q satisfying q = gcd(P(x),P(y)) where P(x) and P(y) are polynomials in ring GF(3)[X] with coefficients in {0,1,2} given by the ternary expansions of x and y. The polynomial q is converted back to a ternary number, and then expressed in decimal.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 3, 5, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 3, 1, 1, 3, 1, 4, 3, 1, 1

Offset: 1

Antti Karttunen, May 22 2019

Array is symmetric, and is read by antidiagonals, with (x,y) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

If there is a polynomial q that satisfies q = gcd(P(x),P(y)), then also polynomial -q (which is obtained by changing all nonzero coefficients of q as 1 <--> 2, see A004488) satisfies the same relation, because there are two units (+1 and -1) in polynomial ring GF(3)[X]. Here we always choose the polynomial that is monic (i.e., with a leading coefficient +1), thus its base-3 encoding has "1" as its most significant digit, and the terms given here are all included in A132141.

			The array begins as:
   y
x      1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11,  12,  ...
   --+-----------------------------------------------------
   1 | 1,  1,  1,  1,  1,  1,  1,  1,  1,   1,  1,  1,  ...
   2 | 1,  1,  1,  1,  1,  1,  1,  1,  1,   1,  1,  1,  ...
   3 | 1,  1,  3,  1,  1,  3,  1,  1,  3,   1,  1,  3,  ...
   4 | 1,  1,  1,  4,  1,  1,  1,  4,  1,   1,  4,  4,  ...
   5 | 1,  1,  1,  1,  5,  1,  5,  1,  1,   1,  5,  1,  ...
   6 | 1,  1,  3,  1,  1,  3,  1,  1,  3,   1,  1,  3,  ...
   7 | 1,  1,  1,  1,  5,  1,  5,  1,  1,   1,  5,  1,  ...
   8 | 1,  1,  1,  4,  1,  1,  1,  4,  1,   1,  4,  4,  ...
   9 | 1,  1,  3,  1,  1,  3,  1,  1,  9,   1,  1,  3,  ...
  10 | 1,  1,  1,  1,  1,  1,  1,  1,  1,  10,  1,  1,  ...
  11 | 1,  1,  1,  4,  5,  1,  5,  4,  1,   1, 11,  4,  ...
  12 | 1,  1,  3,  4,  1,  3,  1,  4,  3,   1,  4, 12,  ...

Cf. A004488, A091255, A132141, A325820, A325821, A325827.

Central diagonal: A330740 (after its initial zero).

up_to = 105;
A004488(n) = subst(Pol(apply(x->(3-x)%3, digits(n, 3)), 'x), 'x, 3);
A325825sq(a,b) = { my(a=fromdigits(Vec(lift(gcd(Pol(digits(a,3))*Mod(1, 3),Pol(digits(b,3))*Mod(1, 3)))),3), b=A004488(a)); min(a,b); };
A325825list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A325825sq(col,(a-(col-1))))); (v); };
v325825 = A325825list(up_to);
A325825(n) = v325825[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

A325825 Square array giving the monic polynomial q satisfying q = gcd(P(x),P(y)) where P(x) and P(y) are polynomials in ring GF(3)[X] with coefficients in {0,1,2} given by the ternary expansions of x and y. The polynomial q is converted back to a ternary number, and then expressed in decimal.

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