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

1,2

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.

Links

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

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).

Example

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

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 *)

Prog

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

Crossrefs

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

Main diagonal gives A352028.

Sequence in context: A055129 A133804 A185943 * A208337 A208335 A208597

Adjacent sequences: A351998 A351999 A352000 * A352002 A352003 A352004

Keyword

nonn,tabl

Author

Rémy Sigrist, Feb 27 2022

Status

approved