A352001 Square array A(n, k), n, k >= 1, read by antidiagonals upwards; A(n, k) = Product_{ i >= 1 } prime(k*i)^e_i where n = Product_{ i >= 1 } prime(i)^e_i (where prime(i) denotes the i-th prime number).

1, 2, 1, 3, 3, 1, 4, 7, 5, 1, 5, 9, 13, 7, 1, 6, 13, 25, 19, 11, 1, 7, 21, 23, 49, 29, 13, 1, 8, 19, 65, 37, 121, 37, 17, 1, 9, 27, 37, 133, 47, 169, 43, 19, 1, 10, 49, 125, 53, 319, 61, 289, 53, 23, 1, 11, 39, 169, 343, 71, 481, 73, 361, 61, 29, 1

Offset: 1

Author

Rémy Sigrist, Feb 27 2022

Keywords

• nonn
• tabl

Comments

In other words, in prime factorization of n, replace prime(i) by prime(k*i).

For any k >= 1, n -> A(n, k) is completely multiplicative.

Examples

			Square array A(n, k) begins:
  n\k|   1   2    3    4     5     6     7     8      9     10
  ------------------------------------------------------------
    1|   1   1    1    1     1     1     1     1      1      1
    2|   2   3    5    7    11    13    17    19     23     29
    3|   3   7   13   19    29    37    43    53     61     71
    4|   4   9   25   49   121   169   289   361    529    841
    5|   5  13   23   37    47    61    73    89    103    113
    6|   6  21   65  133   319   481   731  1007   1403   2059
    7|   7  19   37   53    71    89   107   131    151    173
    8|   8  27  125  343  1331  2197  4913  6859  12167  24389
    9|   9  49  169  361   841  1369  1849  2809   3721   5041
   10|  10  39  115  259   517   793  1241  1691   2369   3277
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (antidiagonals n = 1..150 flattened)

Crossrefs

Cf. A000040, A001248, A031215, A051128, A297002, A351995.

Main diagonal gives A352028.

Programs

Maple

A:= (n, k)-> mul(ithprime(k*numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
seq(seq(A(d+1-k, k), k=1..d), d=1..12);  # Alois P. Heinz, Feb 28 2022

Mathematica

Table[If[# == 1, 1, Times @@ Map[Prime[#3*PrimePi[#1]]^#2 & @@ Flatten[{#1, k}] &, FactorInteger[#]]] &[n - k + 1], {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Feb 28 2022 *)

PARI

A(n,k) = { my (f=factor(n)); prod (i=1, #f~, prime(k * primepi(f[i,1])) ^ f[i,2]) }

Formula

A(A(n, k), k') = A(n, k*k').

A(n, 1) = n.

A(n, 2) = A297002(n).

A(1, k) = 1.

A(2, k) = prime(k) (the k-th prime number).

A(3, k) = prime(2*k) = A031215(k).

A(4, k) = A001248(k).