A329050 Square array A(n,k) = prime(n+1)^(2^k), read by descending antidiagonals (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...; Fermi-Dirac primes (A050376) in matrix form, sorted into rows by their prime divisor.

2, 4, 3, 16, 9, 5, 256, 81, 25, 7, 65536, 6561, 625, 49, 11, 4294967296, 43046721, 390625, 2401, 121, 13, 18446744073709551616, 1853020188851841, 152587890625, 5764801, 14641, 169, 17, 340282366920938463463374607431768211456, 3433683820292512484657849089281, 23283064365386962890625, 33232930569601, 214358881, 28561, 289, 19

Offset: 0

Author

Antti Karttunen and Peter Munn, Nov 02 2019

Keywords

• nonn
• tabl

Comments

This sequence is a permutation of A050376, so every positive integer is the product of a unique subset, S_factors, of its terms. If we restrict S_factors to be chosen from a subset, S_0, consisting of numbers from specified rows and/or columns of this array, there are notable sequences among those that may be generated. See the examples. Other notable sequences can be generated if we restrict the intersection of S_factors with specific rows/columns to have even cardinality. In any of the foregoing cases, the numbers in the resulting sequence form a group under the binary operation A059897(.,.).

Shares with array A246278 the property that columns grow downward by iterating A003961, and indeed, this array can be obtained from A246278 by selecting its columns 1, 2, 8, 128, ..., 2^((2^k)-1), for k >= 0.

A(n,k) is the image of the lattice point with coordinates X=n and Y=k under the inverse of the bijection f defined in the first comment of A306697. This geometric relationship can be used to construct an isomorphism from the polynomial ring GF(2)[x,y] to a ring over the positive integers, using methods similar to those for constructing A297845 and A306697. See A329329, the ring's multiplicative operator, for details.

Examples

			The top left 5 X 5 corner of the array:
  n\k |   0     1       2           3                   4
  ----+-------------------------------------------------------
   0  |   2,    4,     16,        256,              65536, ...
   1  |   3,    9,     81,       6561,           43046721, ...
   2  |   5,   25,    625,     390625,       152587890625, ...
   3  |   7,   49,   2401,    5764801,     33232930569601, ...
   4  |  11,  121,  14641,  214358881,  45949729863572161, ...
Column 0 continues as a list of primes, column 1 as a list of their squares, column 2 as a list of their 4th powers, and so on.
Every nonnegative power of 2 (A000079) is a product of a unique subset of numbers from row 0; every squarefree number (A005117) is a product of a unique subset of numbers from column 0. Likewise other rows and columns generate the sets of numbers from sequences:
Row 1:                 A000244 Powers of 3.
Column 1:              A062503 Squares of squarefree numbers.
Row 2:                 A000351 Powers of 5.
Column 2:              A113849 4th powers of squarefree numbers.
Union of rows 0 and 1:     A003586 3-smooth numbers.
Union of columns 0 and 1:  A046100 Biquadratefree numbers.
Union of row 0 / column 0: A122132 Oddly squarefree numbers.
Row 0 excluding column 0:  A000302 Powers of 4.
Column 0 excluding row 0:  A056911 Squarefree odd numbers.
All rows except 0:         A005408 Odd numbers.
All columns except 0:      A000290\{0} Positive squares.
All rows except 1:         A001651 Numbers not divisible by 3.
All columns except 1:      A252895 (have odd number of square divisors).
If, instead of restrictions on choosing individual factors of the product, we restrict the product to be of an even number of terms from each row of the array, we get A262675. The equivalent restriction applied to columns gives us A268390; applied only to column 0, we get A028260 (product of an even number of primes).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..77; the first 12 antidiagonals of array
• Wikipedia, Polynomial ring

Crossrefs

Transpose: A329049.

Permutation of A050376.

Rows 1-4: A001146, A011764, A176594, A165425 (after the two initial terms).

Columns 1-5: A000040, A001248, A030514, A179645, A030635.

Antidiagonal products: A191555.

Subtable of A182944, A242378, A246278, A329332.

A000290, A003961, A225546 are used to express relationship between terms of this sequence.

Related binary operations: A059897, A306697, A329329.

See also the table in the example section.

Programs

Mathematica

Table[Prime[#]^(2^k) &[m - k + 1], {m, 0, 7}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Dec 28 2019 *)

PARI

up_to = 105;
A329050sq(n,k) = (prime(1+n)^(2^k));
A329050list(up_to) = { my(v = vector(up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A329050sq(col, a-col))); (v); };
v329050 = A329050list(up_to);
A329050(n) = v329050[1+n];
for(n=0,up_to-1,print1(A329050(n),", ")); \\ Antti Karttunen, Nov 06 2019

Formula

A(0,k) = 2^(2^k), and for n > 0, A(n,k) = A003961(A(n-1,k)).

A(n,k) = A182944(n+1,2^k).

A(n,k) = A329332(2^n,2^k).

A(k,n) = A225546(A(n,k)).

A(n,k+1) = A000290(A(n,k)) = A(n,k)^2.

Extensions

Example annotated for clarity by Peter Munn, Feb 12 2020