A331590 Square array A(n,k) = A225546(A225546(n) * A225546(k)), n >= 1, k >= 1, read by descending antidiagonals.

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

Offset: 1

Author

Peter Munn, Jan 21 2020

Keywords

• nonn
• tabl

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.

Examples

			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)
		

Links

• Antti Karttunen, Antidiagonals n = 1..144, flattened
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..80200; (antidiagonals n = 1..400)
• Eric Weisstein's World of Mathematics, Monoid

Crossrefs

Isomorphic to A003991 with A225546 as isomorphism.

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

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

A019565 lists powers of 2 in order of increasing exponent.

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.

Irreducibles are A001146 (even terms of A050376).

Programs

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

Formula

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

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.

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

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

(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.

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