%I #24 May 24 2020 10:20:33

%S 1,2,2,3,3,3,4,6,6,4,5,8,5,8,5,6,10,12,12,10,6,7,5,15,9,15,5,7,8,14,

%T 10,20,20,10,14,8,9,12,21,24,7,24,21,12,9,10,18,24,28,30,30,28,24,18,

%U 10,11,15,27,18,35,15,35,18,27,15,11,12,22,30,36,40,42,42,40,36,30,22,12,13,24,33,40,45,20,11,20,45,40,33,24,13

%N Square array A(n,k) = A225546(A225546(n) * A225546(k)), n >= 1, k >= 1, read by descending antidiagonals.

%C As a binary operation, this sequence defines a commutative monoid over the positive integers that is isomorphic to multiplication. The self-inverse permutation A225546(.) provides an isomorphism. This monoid therefore has unique factorization. Its primes are the even terms of A050376: 2, 4, 16, 256, ..., which in standard integer multiplication are the powers of 2 with powers of 2 as exponents.

%C In this monoid, in contrast, the powers of 2 run through the squarefree numbers, the k-th power of 2 being A019565(k). 4 is irreducible and its powers are the squares of the squarefree numbers, the k-th power of 4 being A019565(k)^2 (where "^2" denotes standard integer squaring); and so on with powers of 16, 256, ...

%C In many cases the product of two numbers is the same here as in standard integer multiplication. See the formula section for details.

%H Antti Karttunen, <a href="/A331590/b331590.txt">Antidiagonals n = 1..144, flattened</a>

%H Antti Karttunen, <a href="/A331590/a331590.txt">Data supplement: n, a(n) computed for n = 1..80200; (antidiagonals n = 1..400)</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Monoid.html">Monoid</a>

%F Alternative definition: A(n,1) = n; A(n,k) = A(A059897(n,k), A003961(A059895(n,k))).

%F Main derived identities: (Start)

%F A(n,k) = A(k,n).

%F A(1,n) = n.

%F A(n, A(m,k)) = A(A(n,m), k).

%F A(m,m) = A003961(m).

%F A(n^2, k^2) = A(n,k)^2.

%F A(A003961(n), A003961(k)) = A003961(A(n,k)).

%F A(A019565(n), A019565(k)) = A019565(n+k).

%F (End)

%F Characterization of conditions for A(n,k) = n * k: (Start)

%F The following 4 conditions are equivalent:

%F (1) A(n,k) = n * k;

%F (2) A(n,k) = A059897(n,k);

%F (3) A(n,k) = A059896(n,k);

%F (4) A059895(n,k) = 1.

%F If gcd(n,k) = 1, A(n,k) = n * k.

%F If gcd(n,k) = 1, A(A225546(n), A225546(k)) = A225546(n) * A225546(k).

%F The previous formula implies A(n,k) = n * k in the following cases:

%F (1) for n = A005117(m), k = j^2;

%F (2) more generally for n = A005117(m_1)^(2^i_1), k = A005117(m_2)^(2^i_2), with A004198(i_1, i_2) = 0.

%F (End)

%e From _Antti Karttunen_, Feb 02 2020: (Start)

%e The top left 16 X 16 corner of the array:

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...

%e 2, 3, 6, 8, 10, 5, 14, 12, 18, 15, 22, 24, 26, 21, 30, 32, ...

%e 3, 6, 5, 12, 15, 10, 21, 24, 27, 30, 33, 20, 39, 42, 7, 48, ...

%e 4, 8, 12, 9, 20, 24, 28, 18, 36, 40, 44, 27, 52, 56, 60, 64, ...

%e 5, 10, 15, 20, 7, 30, 35, 40, 45, 14, 55, 60, 65, 70, 21, 80, ...

%e 6, 5, 10, 24, 30, 15, 42, 20, 54, 7, 66, 40, 78, 35, 14, 96, ...

%e 7, 14, 21, 28, 35, 42, 11, 56, 63, 70, 77, 84, 91, 22, 105, 112, ...

%e 8, 12, 24, 18, 40, 20, 56, 27, 72, 60, 88, 54, 104, 84, 120, 128, ...

%e 9, 18, 27, 36, 45, 54, 63, 72, 25, 90, 99, 108, 117, 126, 135, 144, ...

%e 10, 15, 30, 40, 14, 7, 70, 60, 90, 21, 110, 120, 130, 105, 42, 160, ...

%e 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 13, 132, 143, 154, 165, 176, ...

%e 12, 24, 20, 27, 60, 40, 84, 54, 108, 120, 132, 45, 156, 168, 28, 192, ...

%e 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 17, 182, 195, 208, ...

%e 14, 21, 42, 56, 70, 35, 22, 84, 126, 105, 154, 168, 182, 33, 210, 224, ...

%e 15, 30, 7, 60, 21, 14, 105, 120, 135, 42, 165, 28, 195, 210, 35, 240, ...

%e 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 81, ...

%e (End)

%o (PARI)

%o up_to = 1275;

%o A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));

%o A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };

%o A331590sq(x,y) = if(1==x,y,if(1==y,x, my(fx=factor(x),fy=factor(y),u=max(#binary(vecmax(fx[, 2])),#binary(vecmax(fy[, 2]))),prodsx=vector(u,x,1),m=1); for(i=1,u,for(k=1,#fx~, if(bitand(fx[k,2],m),prodsx[i] *= fx[k,1])); for(k=1,#fy~, if(bitand(fy[k,2],m),prodsx[i] *= fy[k,1])); m<<=1); prod(i=1,u,A019565(A048675(prodsx[i]))^(1<<(i-1)))));

%o A331590list(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] = A331590sq(col,(a-(col-1))))); (v); };

%o v331590 = A331590list(up_to);

%o A331590(n) = v331590[n]; \\ _Antti Karttunen_, Feb 02 2020

%Y Isomorphic to A003991 with A225546 as isomorphism.

%Y Cf. A003961(main diagonal), A048675, A059895, A059896, A059897.

%Y Rows/columns, sorted in ascending order: 2: A000037, 3: A028983, 4: A252849.

%Y A019565 lists powers of 2 in order of increasing exponent.

%Y Powers of k, sorted in ascending order: k=2: A005117, k=3: A056911, k=4: A062503, k=5: A276378, k=6: intersection of A325698 and A005117, k=7: intersection of A007775 and A005117, k=8: A062838.

%Y Irreducibles are A001146 (even terms of A050376).

%K nonn,tabl

%O 1,2

%A _Peter Munn_, Jan 21 2020