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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).
3
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
Main diagonal gives A352028.
Sequence in context: A055129 A133804 A185943 * A208337 A208335 A208597
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist, Feb 27 2022
STATUS
approved