OFFSET
1,2
COMMENTS
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.
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, ...
In many cases the product of two numbers is the same here as in standard integer multiplication. See the formula section for details.
LINKS
Antti Karttunen, Antidiagonals n = 1..144, flattened
Eric Weisstein's World of Mathematics, Monoid
FORMULA
Main derived identities: (Start)
A(n,k) = A(k,n).
A(1,n) = n.
A(n, A(m,k)) = A(A(n,m), k).
A(m,m) = A003961(m).
A(n^2, k^2) = A(n,k)^2.
(End)
Characterization of conditions for A(n,k) = n * k: (Start)
The following 4 conditions are equivalent:
(1) A(n,k) = n * k;
(2) A(n,k) = A059897(n,k);
(3) A(n,k) = A059896(n,k);
(4) A059895(n,k) = 1.
If gcd(n,k) = 1, A(n,k) = n * k.
The previous formula implies A(n,k) = n * k in the following cases:
(1) for n = A005117(m), k = j^2;
(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.
(End)
EXAMPLE
From Antti Karttunen, Feb 02 2020: (Start)
The top left 16 X 16 corner of the array:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
2, 3, 6, 8, 10, 5, 14, 12, 18, 15, 22, 24, 26, 21, 30, 32, ...
3, 6, 5, 12, 15, 10, 21, 24, 27, 30, 33, 20, 39, 42, 7, 48, ...
4, 8, 12, 9, 20, 24, 28, 18, 36, 40, 44, 27, 52, 56, 60, 64, ...
5, 10, 15, 20, 7, 30, 35, 40, 45, 14, 55, 60, 65, 70, 21, 80, ...
6, 5, 10, 24, 30, 15, 42, 20, 54, 7, 66, 40, 78, 35, 14, 96, ...
7, 14, 21, 28, 35, 42, 11, 56, 63, 70, 77, 84, 91, 22, 105, 112, ...
8, 12, 24, 18, 40, 20, 56, 27, 72, 60, 88, 54, 104, 84, 120, 128, ...
9, 18, 27, 36, 45, 54, 63, 72, 25, 90, 99, 108, 117, 126, 135, 144, ...
10, 15, 30, 40, 14, 7, 70, 60, 90, 21, 110, 120, 130, 105, 42, 160, ...
11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 13, 132, 143, 154, 165, 176, ...
12, 24, 20, 27, 60, 40, 84, 54, 108, 120, 132, 45, 156, 168, 28, 192, ...
13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 17, 182, 195, 208, ...
14, 21, 42, 56, 70, 35, 22, 84, 126, 105, 154, 168, 182, 33, 210, 224, ...
15, 30, 7, 60, 21, 14, 105, 120, 135, 42, 165, 28, 195, 210, 35, 240, ...
16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 81, ...
(End)
PROG
(PARI)
up_to = 1275;
A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
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)))));
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); };
v331590 = A331590list(up_to);
A331590(n) = v331590[n]; \\ Antti Karttunen, Feb 02 2020
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Munn, Jan 21 2020
STATUS
approved