A352001 - OEIS

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).

%I #24 Mar 01 2022 16:25:59

%S 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,

%T 19,65,37,121,37,17,1,9,27,37,133,47,169,43,19,1,10,49,125,53,319,61,

%U 289,53,23,1,11,39,169,343,71,481,73,361,61,29,1

%N 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).

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

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

%H Michael De Vlieger, <a href="/A352001/b352001.txt">Table of n, a(n) for n = 1..11325</a> (antidiagonals n = 1..150 flattened)

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

%F A(n, 1) = n.

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

%F A(1, k) = 1.

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

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

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

%e Square array A(n, k) begins:

%e n\k| 1 2 3 4 5 6 7 8 9 10

%e ------------------------------------------------------------

%e 1| 1 1 1 1 1 1 1 1 1 1

%e 2| 2 3 5 7 11 13 17 19 23 29

%e 3| 3 7 13 19 29 37 43 53 61 71

%e 4| 4 9 25 49 121 169 289 361 529 841

%e 5| 5 13 23 37 47 61 73 89 103 113

%e 6| 6 21 65 133 319 481 731 1007 1403 2059

%e 7| 7 19 37 53 71 89 107 131 151 173

%e 8| 8 27 125 343 1331 2197 4913 6859 12167 24389

%e 9| 9 49 169 361 841 1369 1849 2809 3721 5041

%e 10| 10 39 115 259 517 793 1241 1691 2369 3277

%p A:= (n, k)-> mul(ithprime(k*numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):

%p seq(seq(A(d+1-k, k), k=1..d), d=1..12); # _Alois P. Heinz_, Feb 28 2022

%t 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 *)

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

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

%Y Main diagonal gives A352028.

%K nonn,tabl

%O 1,2

%A _Rémy Sigrist_, Feb 27 2022