A352780 Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, such that the row product is n and column k contains only (2^k)-th powers of squarefree numbers.

1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14

Offset: 1

Author

Antti Karttunen and Peter Munn, Apr 02 2022

Keywords

• nonn
• easy
• tabl

Comments

This is well-defined because positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2.

Each (infinite) row is the lexicographically earliest with product n and terms that are a (2^k)-th power for all k.

For all k, column k is column k+1 of A060176 conjugated by A225546.

Examples

			The top left corner of the array:
  n/k |   0   1   2   3   4   5   6
------+------------------------------
    1 |   1,  1,  1,  1,  1,  1,  1,
    2 |   2,  1,  1,  1,  1,  1,  1,
    3 |   3,  1,  1,  1,  1,  1,  1,
    4 |   1,  4,  1,  1,  1,  1,  1,
    5 |   5,  1,  1,  1,  1,  1,  1,
    6 |   6,  1,  1,  1,  1,  1,  1,
    7 |   7,  1,  1,  1,  1,  1,  1,
    8 |   2,  4,  1,  1,  1,  1,  1,
    9 |   1,  9,  1,  1,  1,  1,  1,
   10 |  10,  1,  1,  1,  1,  1,  1,
   11 |  11,  1,  1,  1,  1,  1,  1,
   12 |   3,  4,  1,  1,  1,  1,  1,
   13 |  13,  1,  1,  1,  1,  1,  1,
   14 |  14,  1,  1,  1,  1,  1,  1,
   15 |  15,  1,  1,  1,  1,  1,  1,
   16 |   1,  1, 16,  1,  1,  1,  1,
   17 |  17,  1,  1,  1,  1,  1,  1,
   18 |   2,  9,  1,  1,  1,  1,  1,
   19 |  19,  1,  1,  1,  1,  1,  1,
   20 |   5,  4,  1,  1,  1,  1,  1,
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..33153; the first 257 antidiagonals

Crossrefs

Sequences used in a formula defining this sequence: A000188, A007913, A060176, A225546.

Cf. A007913 (column 0), A335324 (column 1).

Range of values: {1} U A340682 (whole table), A005117 (column 0), A062503 (column 1), {1} U A113849 (column 2).

Row numbers of rows:

- with a 1 in column 0: A000290\{0};

- with a 1 in column 1: A252895;

- with a 1 in column 0, but not in column 1: A030140;

- where every 1 is followed by another 1: A337533;

- with 1's in all even columns: A366243;

- with 1's in all odd columns: A366242;

- where every term has an even number of distinct prime factors: A268390;

- where every term is a power of a prime: A268375;

- where the terms are pairwise coprime: A138302;

- where the last nonunit term is coprime to the earlier terms: A369938;

- where the last nonunit term is a power of 2: A335738.

Number of nonunit terms in row n is A331591(n); their positions are given (in reversed binary) by A267116(n); the first nonunit is in column A352080(n)-1 and the infinite run of 1's starts in column A299090(n).

Programs

PARI

up_to = 105;
A352780sq(n, k) = if(k==0, core(n), A352780sq(core(n, 1)[2], k-1)^2);
A352780list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, forstep(col=a-1,0,-1, i++; if(i > up_to, return(v)); v[i] = A352780sq(a-col,col))); (v); };
v352780 = A352780list(up_to);
A352780(n) = v352780[n];

Formula

A(n,0) = A007913(n); for k > 0, A(n,k) = A(A000188(n), k-1)^2.

A(n,k) = A225546(A060176(A225546(n), k+1)).

A331591(A(n,k)) <= 1.