This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The array begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---+------------------------------------------------------------
0| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1| 1 3 2 15 12 9 11 10 6 8 7 5 14 13 4 --> A051917(n)
2| 2 1 3 5 4 14 13 15 11 12 9 10 7 6 8
3| 3 2 1 10 8 7 6 5 13 4 14 15 9 11 12
4| 4 12 8 1 13 15 7 3 14 11 10 2 5 9 6
5| 5 15 10 14 1 6 12 9 8 3 13 7 11 4 2
6| 6 13 11 4 9 1 10 12 5 7 3 8 2 15 14
7| 7 14 9 11 5 8 1 6 3 15 4 13 12 2 10
8| 8 4 12 2 6 5 9 1 7 13 15 3 10 14 11
9| 9 7 14 13 10 12 2 11 1 5 8 6 4 3 15
10| 10 5 15 7 2 11 4 14 12 1 6 9 13 8 3
11| 11 6 13 8 14 2 15 4 10 9 1 12 3 5 7
12| 12 8 4 3 11 10 14 2 9 6 5 1 15 7 13
13| 13 11 6 12 7 3 5 8 15 14 2 4 1 10 9
14| 14 9 7 6 15 4 3 13 2 10 12 11 8 1 5
15| 15 10 5 9 3 13 8 7 4 2 11 14 6 12 1
| | | | |
| | | | A004474(n)
| | | A004477(n)
| | A004480(n)
| A006015(n)
A004468(n)
Nim-product for 0..3: 0 1 2 3 ------- 0|0 0 0 0 1|0 1 2 3 2|0 2 3 1 3|0 3 1 2 factors of 3 are 1 and 3.
function a = A348291( max_n ) a(1) = 0; multable = zeros(max_n , max_n); for n = 1:max_n for m = 1:max_n multable(m,n) = nimprod(m,n); end end for n = 1:max_n [i,j] = find(multable(1:n,1:n) == n); a(n+1) = length(find(unique([i j]) < n)); end end % highest power of 2 that divides a given number function h2 = hpo2( in ) h2 = bitand(in,bitxor(in,2^32-1)+1); end % base 2 logarithm of the highest power of 2 dividing a given number function lhp2 = lhpo2( in ) lhp2 = 0; m = hpo2(in); q = 0; while m > 1 m = m/2; lhp2 = lhp2+1; end end % nim-product of two numbers function prod = nimprod(x,y) if (x < 2 || y < 2) prod = x * y; else h = hpo2(x); if (x > h) prod = bitxor(nimprod(h, y),nimprod(bitxor(x,h), y)); else if (hpo2(y) < y) prod = nimprod(y, x); else xp = lhpo2(x); yp = lhpo2(y); comp = bitand(xp,yp); if (comp == 0) prod = x * y; else h = hpo2(comp); prod = nimprod(nimprod(bitshift(x,-h),bitshift(y,-h)),bitshift(3,(h - 1))); end end end end end
For n=3, giving 2^n=8 and 2^(2^n)=256: let x be the nimber *255. Then the powers of x (under nim-multiplication) are *1, *255, *156, *61, *205, *200, *38, *71, *179. Under nim-addition, the subset of these powers *1 + *61 + *200 + *71 + *179 sum to *0. That is, 1+x^3+x^5+x^7+x^8 = 0. No sum of the powers up to and including x^7 is zero. So the polynomial 1+x^3+x^5+x^7+x^8 over GF(2) is the minimal polynomial of *255. Therefore the sequence entry for n=3 is the integer obtained by reinterpreting this polynomial as one over the integers and evaluating it at 2, i.e. 1+2^3+2^5+2^7+2^8 = 425.
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