A325820 -id:A325820 - cp's OEIS frontend

A008585 a(n) = 3*n.

0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177

Offset: 0

N. J. A. Sloane

If n != 1 and n^2+2 is prime then n is a member of this sequence. - Cino Hilliard, Mar 19 2007
Multiples of 3. Positive members of this sequence are the third transversal numbers (or 3-transversal numbers): Numbers of the 3rd column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 3rd column in the square array A057145. - Omar E. Pol, May 02 2008
Numbers n for which polynomial 27*x^6-2^n is factorizable. - Artur Jasinski, Nov 01 2008
1/7 in base-2 notation = 0.001001001... = 1/2^3 + 1/2^6 + 1/2^9 + ... - Gary W. Adamson, Jan 24 2009
A165330(a(n)) = 153 for n > 0; subsequence of A031179. - Reinhard Zumkeller, Sep 17 2009
A011655(a(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
A215879(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2012
Moser conjectured, and Newman proved, that the terms of this sequence are more likely to have an even number of 1s in binary than an odd number. The excess is an undulating multiple of n^(log 3/log 4). See also Coquet, who refines this result. - Charles R Greathouse IV, Jul 17 2013
Integer areas of medial triangles of integer-sided triangles.
Also integer subset of A188158(n)/4.
A medial triangle MNO is formed by joining the midpoints of the sides of a triangle ABC. The area of a medial triangle is A/4 where A is the area of the initial triangle ABC. - Michel Lagneau, Oct 28 2013
From Derek Orr, Nov 22 2014: (Start)
Let b(0) = 0, and b(n) = the number of distinct terms in the set of pairwise sums {b(0), ... b(n-1)} + {b(0), ... b(n-1)}. Then b(n+1) = a(n), for n > 0.
Example: b(1) = the number of distinct sums of {0} + {0}. The only possible sum is {0} so b(1) = 1. b(2) = the number of distinct sums of {0,1} + {0,1}. The possible sums are {0,1,2} so b(2) = 3. b(3) = the number of distinct sums of {0,1,3} + {0,1,3}. The possible sums are {0, 1, 2, 3, 4, 6} so b(3) = 6. This continues and one can see that b(n+1) = a(n). (End)
Number of partitions of 6n into exactly 2 parts. - Colin Barker, Mar 23 2015
Partial sums are in A045943. - Guenther Schrack, May 18 2017
Number of edges in a maximal planar graph with n+2 vertices, n > 0 (see A008486 comments). - Jonathan Sondow, Mar 03 2018
Also numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 3. - Stefano Spezia, Jul 08 2025

			G.f.: 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + ...

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
J. Coquet, A summation formula related to the binary digits, Inventiones Mathematicae 73 (1983), pp. 107-115.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
John Graham-Cumming, The hollow triangular numbers are divisible by three (2013)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 315
Tanya Khovanova, Recursive Sequences
D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Wikipedia, Maximal planar graphs
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Row / column 3 of A004247 and of A325820.
Cf. A016957, A057145, A139600, A139606, A001651 (complement), A032031 (partial products), A190944 (binary), A061819 (base 4).
Cf. A031179, A008486, A008591, A017197, A017233, A045943.

a008585 = (* 3)
a008585_list = iterate (+ 3) 0  -- Reinhard Zumkeller, Feb 19 2013

[3*n: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Range[0, 500, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)

makelist(3*n,n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n)=3*n \\ Charles R Greathouse IV, Jun 28 2013

G.f.: 3*x/(1-x)^2. - R. J. Mathar, Oct 23 2008
a(n) = A008486(n), n > 0. - R. J. Mathar, Oct 28 2008
G.f.: A(x) - 1, where A(x) is the g.f. of A008486. - Gennady Eremin, Feb 20 2021
a(n) = Sum_{k=0..inf} A030308(n,k)*A007283(k). - Philippe Deléham, Oct 17 2011
E.g.f.: 3*x*exp(x). - Ilya Gutkovskiy, May 18 2016
From Guenther Schrack, May 18 2017: (Start)
a(3*k) = a(a(k)) = A008591(n).
a(3*k+1) = a(a(k) + 1) = a(A016777(n)) = A017197(n).
a(3*k+2) = a(a(k) + 2) = a(A016789(n)) = A017233(n). (End)

Partially edited by Joerg Arndt, Mar 11 2010

A048720 Multiplication table {0..i} X {0..j} of binary polynomials (polynomials over GF(2)) interpreted as binary vectors, then written in base 10; or, binary multiplication without carries.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 5, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 12, 15, 16, 15, 12, 7, 0, 0, 8, 14, 10, 20, 20, 10, 14, 8, 0, 0, 9, 16, 9, 24, 17, 24, 9, 16, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 20, 27, 32, 27, 20, 27, 32, 27, 20, 11, 0

Offset: 0

Antti Karttunen, Apr 26 1999

Essentially same as A091257 but computed starting from offset 0 instead of 1.
Each polynomial in GF(2)[X] is encoded as the number whose binary representation is given by the coefficients of the polynomial, e.g., 13 = 2^3 + 2^2 + 2^0 = 1101_2 encodes 1*X^3 + 1*X^2 + 0*X^1 + 1*X^0 = X^3 + X^2 + X^0. - Antti Karttunen and Peter Munn, Jan 22 2021
To listen to this sequence, I find instrument 99 (crystal) works well with the other parameters defaulted. - Peter Munn, Nov 01 2022

			Top left corner of array:
  0  0  0  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 13 14 15 ...
  0  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30 ...
  0  3  6  5 12 15 10  9 24 27 30 29 20 23 18 17 ...
  ...
From _Antti Karttunen_ and _Peter Munn_, Jan 23 2021: (Start)
Multiplying 10 (= 1010_2) and 11 (= 1011_2), in binary results in:
     1011
  *  1010
  -------
   c1011
  1011
  -------
  1101110  (110 in decimal),
and we see that there is a carry-bit (marked c) affecting the result.
In carryless binary multiplication, the second part of the process (in which the intermediate results are summed) looks like this:
    1011
  1011
  -------
  1001110  (78 in decimal).
(End)

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Antti Karttunen, Scheme-program for computing this sequence.
N. J. A. Sloane, Transforms: Maple implementation of binary eXclusive OR (XORnos).
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A051776 (Nim-product), A091257 (subtable).
Carryless multiplication in other bases: A325820 (3), A059692 (10).
Ordinary {0..i} * {0..j} multiplication table: A004247 and its differences from this: A061858 (which lists further sequences related to presence/absence of carry in binary multiplication).
Carryless product of the prime factors of n: A234741.
Binary irreducible polynomials ("X-primes"): A014580, factorization table: A256170, table of "X-powers": A048723, powers of 3: A001317, rearranged subtable with distinct terms (comparable to A054582): A277820.
See A014580 for further sequences related to the difference between factorization into GF(2)[X] irreducibles and ordinary prime factorization of the integer encoding.
Row/column 3: A048724 (even bisection of A003188), 5: A048725, 6: A048726, 7: A048727; main diagonal: A000695.
Associated additive operation: A003987.
Equivalent sequences, as compared with standard integer multiplication: A048631 (factorials), A091242 (composites), A091255 (gcd), A091256 (lcm), A280500 (division).
See A091202 (and its variants) and A278233 for maps from/to ordinary multiplication.
See A115871, A115872 and A277320 for tables related to cross-domain congruences.

trinv := n -> floor((1+sqrt(1+8*n))/2); # Gives integral inverses of the triangular numbers
# Binary multiplication of nn and mm, but without carries (use XOR instead of ADD):
Xmult := proc(nn,mm) local n,m,s; n := nn; m := mm; s := 0; while (n > 0) do if(1 = (n mod 2)) then s := XORnos(s,m); fi; n := floor(n/2); # Shift n right one bit. m := m*2; # Shift m left one bit. od; RETURN(s); end;

trinv[n_] := Floor[(1 + Sqrt[1 + 8*n])/2];
Xmult[nn_, mm_] := Module[{n = nn, m = mm, s = 0}, While[n > 0, If[1 == Mod[n, 2], s = BitXor[s, m]]; n = Floor[n/2]; m = m*2]; Return[s]];
a[n_] := Xmult[(trinv[n] - 1)*((1/2)*trinv[n] + 1) - n, n - (trinv[n]*(trinv[n] - 1))/2];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 16 2015, updated Mar 06 2016 after Maple *)

up_to = 104;
A048720sq(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A048720list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A048720sq(col, a-col))); (v); };
v048720 = A048720list(up_to);
A048720(n) = v048720[1+n]; \\ Antti Karttunen, Feb 15 2021

a(n) = Xmult( (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n), (n-((trinv(n)*(trinv(n)-1))/2)) );
T(2b, c)=T(c, 2b)=T(b, 2c)=2T(b, c); T(2b+1, c)=T(c, 2b+1)=2T(b, c) XOR c - Henry Bottomley, Mar 16 2001
For n >= 0, A003188(2n) = T(n, 3); A003188(2n+1) = T(n, 3) XOR 1, where XOR is the bitwise exclusive-or operator, A003987. - Peter Munn, Feb 11 2021

A004488 Tersum n + n.

Original entry on oeis.org

0, 2, 1, 6, 8, 7, 3, 5, 4, 18, 20, 19, 24, 26, 25, 21, 23, 22, 9, 11, 10, 15, 17, 16, 12, 14, 13, 54, 56, 55, 60, 62, 61, 57, 59, 58, 72, 74, 73, 78, 80, 79, 75, 77, 76, 63, 65, 64, 69, 71, 70, 66, 68, 67, 27, 29, 28, 33, 35, 34, 30, 32, 31, 45, 47, 46, 51

Offset: 0

N. J. A. Sloane

Could also be described as "Write n in base 3, then replace each digit with its base-3 negative" as with A048647 for base 4. - Henry Bottomley, Apr 19 2000
a(a(n)) = n, a self-inverse permutation of the nonnegative integers. - Reinhard Zumkeller, Dec 19 2003
First 3^n terms of the sequence form a permutation s(n) of 0..3^n-1, n>=1; the number of inversions of s(n) is A016142(n-1). - Gheorghe Coserea, Apr 23 2018

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 6561 terms from Alois P. Heinz)
Index entries for sequences related to carryless arithmetic
Index entries for sequences that are permutations of the natural numbers

Column k=0 of A253586, A253587.
Column k=3 of A248813.
Row / column 2 of A325820.
Main diagonal of A004489.
Cf. A048647, A055115, A055116, A055120, A059249, A117966, A117967, A117968, A225901, A242399, A244042, A263273, A289813, A289814, A289815, A289816, A289831, A289838, A300222, A321464.

a004488 0 = 0
a004488 n = if d == 0 then 3 * a004488 n' else 3 * a004488 n' + 3 - d
            where (n', d) = divMod n 3
-- Reinhard Zumkeller, Mar 12 2014

a:= proc(n) local t, r, i;
      t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+3^i *irem(2*irem(t, 3, 't'), 3)
      od; r
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 07 2011

a[n_] := FromDigits[Mod[3-IntegerDigits[n, 3], 3], 3]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Mar 03 2014 *)

a(n) = my(b=3); fromdigits(apply(d->(b-d)%b, digits(n, b)), b);
vector(67, i, a(i-1))  \\ Gheorghe Coserea, Apr 23 2018

from sympy.ntheory.factor_ import digits
def a(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3) # Indranil Ghosh, Jun 06 2017

Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g., 5 + 8 = "21" + "22" = "10" = 1.
a(n) = Sum(3-d(i)-3*0^d(i): n=Sum(d(i)*3^d(i): 0<=d(i)<3)). - Reinhard Zumkeller, Dec 19 2003
a(3*n) = 3*a(n), a(3*n+1) = 3*a(n)+2, a(3*n+2) = 3*a(n)+1. - Robert Israel, May 09 2014

A004247 Multiplication table read by antidiagonals: T(i,j) = ij (i>=0, j>=0). Alternatively, multiplication triangle read by rows: P(i,j) = j(i-j) (i>=0, 0<=j<=i).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 12, 15, 16, 15, 12, 7, 0, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, 0, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30

Offset: 0

David W. Wilson

Table of x*y, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
Or, triangle read by rows, in which row n gives the numbers 0, n*1, (n-1)*2, (n-2)*3, ..., 2*(n-1), 1*n, 0.
Letting T(n,k) be the (k+1)st entry in the (n+1)st row (same numbering used for Pascal's triangle), T(n,k) is the dimension of the space of all k-dimensional subspaces of a (fixed) n-dimensional real vector space. - Paul Boddington, Oct 21 2003
From Dennis P. Walsh, Nov 10 2009: (Start)
Triangle P(n,k), 0<=k<=n, equals n^2 x the variance of a binary data set with k zeros and (n-k) ones. [For the case when n=0, let the variance of the empty set be defined as 0.]
P(n,k) is also the number of ways to form an opposite-sex dance couple from k women and (n-k) men. (End)
P(n,k) is the number of negative products of two numbers from a set of n real numbers, k of which are negative. - Logan Pipes, Jul 08 2021

			As the triangle P, sequence begins:
  0;
  0,0;
  0,1,0;
  0,2,2,0;
  0,3,4,3,0;
  0,4,6,6,4,0,;
  0,5,8,9,8,5,0;
  ...
From _Dennis P. Walsh_, Nov 10 2009: (Start)
P(5,2)=T(2,3)=6 since the variance of the data set <0,0,1,1,1> equals 6/25.
P(5,2)=6 since, with 2 women, say Alice and Betty, and with 3 men, say Charles, Dennis, and Ed, the dance couple is one of the following: {Alice, Charles}, {Alice, Dennis}, {Alice, Ed}, {Betty, Charles}, {Betty, Dennis} and {Betty, Ed}. (End)

T. D. Noe, Rows n = 0..50 of triangle, flattened
Dennis Walsh, Variance bounds on binary data sets

See A003991 for another version with many more comments.

Cf. A002262, A025581, A003056, A004197, A003984, A048720, A325820, A000292 (row sums of triangle), A002620.

seq(seq(k*(n-k),k=0..n),n=0..13); # Dennis P. Walsh, Nov 10 2009

Table[(x - y) y, {x, 0, 13}, {y, 0, x}] // Flatten (* Robert G. Wilson v, Oct 06 2007 *)

T(i,j)=i*j \\ Charles R Greathouse IV, Jun 23 2017

a(n) = A002262(n) * A025581(n). - Antti Karttunen
From Ridouane Oudra, Dec 14 2019: (Start)
a(n) = A004197(n)*A003984(n).
a(n) = (3/4 + n)*t^2 - (1/4)*t^4 - (1/2)*t - n^2 - n, where t = floor(sqrt(2*n+1)+1/2). (End)
P(n,k) = (P(n-1,k-1) + P(n-1,k) + n) / 2. - Robert FERREOL, Jan 16 2020
P(n,floor(n/2)) = A002620(n). - Logan Pipes, Jul 08 2021
From Stefano Spezia, Aug 19 2024: (Start)
G.f. as array: x*y/((1 - x)^2*(1 - y)^2).
E.g.f. as array: exp(x+y)*x*y. (End)

Edited by N. J. A. Sloane, Sep 30 2007

A169999 Carryless product n X n in base 3.

Original entry on oeis.org

0, 1, 1, 9, 16, 13, 9, 13, 16, 81, 100, 91, 144, 142, 157, 117, 112, 133, 81, 91, 100, 117, 133, 112, 144, 157, 142, 729, 784, 757, 900, 961, 931, 819, 877, 853, 1296, 1369, 1333, 1278, 1249, 1237, 1413, 1381, 1456, 1053, 1117, 1099, 1008, 997, 1030, 1197, 1183, 1141

Offset: 0

N. J. A. Sloane, Aug 29 2010

Antti Karttunen, Table of n, a(n) for n = 0..19682
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version
Index entries for sequences related to carryless arithmetic

Main diagonal of A325820.

For bases 2 through 10 see A000695, A169999, A170985-A170990 and A059729.

A325820sq(b, c) = fromdigits(Vec(Pol(digits(b,3))*Pol(digits(c,3)))%3, 3);
A169999(n) = A325820sq(n,n); \\ Antti Karttunen, Jun 17 2019

A242399 Write n and 3n in ternary representation and add all trits modulo 3.

Original entry on oeis.org

0, 4, 8, 12, 16, 11, 24, 19, 23, 36, 40, 44, 48, 52, 47, 33, 28, 32, 72, 76, 80, 57, 61, 56, 69, 64, 68, 108, 112, 116, 120, 124, 119, 132, 127, 131, 144, 148, 152, 156, 160, 155, 141, 136, 140, 99, 103, 107, 84, 88, 83, 96, 91, 95, 216, 220, 224, 228, 232

Offset: 0

Reinhard Zumkeller, May 13 2014

			n = 25, 3*n = 75:
.  A007089(25) =  221
.  A007089(75) = 2210
.   add trits    ----
.    modulo 3    2101 = A007089(64), hence a(25) = 64.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system
Index entries for sequences related to carryless arithmetic

Cf. A004488, A007089, A008586, A030341, A048724, A242400, A242407.

Row / column 4 of A325820.

a242399 n = foldr (\t v -> 3 * v + t) 0 $
                  map (flip mod 3) $ zipWith (+) ([0] ++ ts) (ts ++ [0])
            where ts = a030341_row n

a(n) <= 4*n; a(m) = 4*m iff m is a term of A242407.

a(n) = A008586(n) - A242400(n).

A059692 Table of carryless products i * j, i>=0, j>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 0, 2, 2, 0, 6, 0, 0, 7, 2, 5, 6, 5, 2, 7, 0, 0, 8, 4, 8, 0, 0, 8, 4, 8, 0, 0, 9, 6, 1, 4, 5, 4, 1, 6, 9, 0, 0, 10, 8, 4, 8, 0, 0, 8, 4, 8, 10, 0, 0, 11, 20, 7, 2, 5, 6, 5, 2, 7, 20, 11, 0

Offset: 0

Henry Bottomley, Feb 19 2001

			Table begins:
  0, 0, 0, 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, 13, 14, 15 ...
  0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 20 ...
  0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 30, 33, 36, 39, 32, 35 ...
  0, 4, 8, 2, 6, 0, 4, 8, 2, 6, 40, 44, 48, 42, 46, 40 ...
  ...
T(12, 97) = 954 since we have 12 X 97 = carryless sum of 900, (180 mod 100=)80, 70 and (14 mod 10=)4 = 954.

Stefano Spezia, First 140 antidiagonals of the table, flattened
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A055120 for carryless 9 X n, A008592 for carryless 10 X n.

Cf. A048720 (binary), A325820 (ternary).

len[num_]:=Length[IntegerDigits[num]]; digit[num_,d_]:=Part[IntegerDigits[num],d]; T[i_, j_] := FromDigits[Reverse[CoefficientList[PolynomialMod[Sum[digit[i,c]*x^(len[i]-c), {c, len[i]}]*Sum[digit[j,r]*x^(len[j]-r), {r, len[j]}], 10], x]]]; Flatten[Table[T[i - j, j], {i, 0, 12}, {j, 0, i}]] (* Stefano Spezia, Sep 26 2022 *)

T(n,k) = fromdigits(lift(Vec( Mod(Pol(digits(n)),10) * Pol(digits(k))))); \\ Kevin Ryde, Sep 27 2022

Minor edits by N. J. A. Sloane, Aug 24 2010

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.

Original entry on oeis.org

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];

A325821 Multiplication table for carryless product i X j in base 3 for i >= 1 and j >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 6, 6, 4, 5, 8, 9, 8, 5, 6, 7, 12, 12, 7, 6, 7, 3, 15, 16, 15, 3, 7, 8, 5, 18, 11, 11, 18, 5, 8, 9, 4, 21, 24, 13, 24, 21, 4, 9, 10, 18, 24, 19, 21, 21, 19, 24, 18, 10, 11, 20, 27, 23, 26, 9, 26, 23, 27, 20, 11, 12, 19, 30, 36, 19, 15, 15, 19, 36, 30, 19, 12, 13, 24, 33, 40, 45, 12, 13, 12, 45, 40, 33, 24, 13, 14, 26, 36, 44, 50, 54, 11, 11, 54, 50, 44, 36, 26, 14

Offset: 1

Antti Karttunen, May 22 2019

This is table A325820 without the zero row and column. See there for more comments.

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

Cf. A003991, A091257, A325820.

up_to = 105;
A325820sq(b, c) = fromdigits(Vec(Pol(digits(b,3))*Pol(digits(c,3)))%3, 3);
A325821list(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] = A325820sq(col,(a-(col-1))))); (v); };
v325821 = A325821list(up_to);
A325821(n) = v325821[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

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