cp's OEIS Frontend

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 A352001 #24 Mar 01 2022 16:25:59
%S A352001 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 A352001 19,65,37,121,37,17,1,9,27,37,133,47,169,43,19,1,10,49,125,53,319,61,
%U A352001 289,53,23,1,11,39,169,343,71,481,73,361,61,29,1
%N 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).
%C A352001 In other words, in prime factorization of n, replace prime(i) by prime(k*i).
%C A352001 For any k >= 1, n -> A(n, k) is completely multiplicative.
%H A352001 Michael De Vlieger, <a href="/A352001/b352001.txt">Table of n, a(n) for n = 1..11325</a> (antidiagonals n = 1..150 flattened)
%F A352001 A(A(n, k), k') = A(n, k*k').
%F A352001 A(n, 1) = n.
%F A352001 A(n, 2) = A297002(n).
%F A352001 A(1, k) = 1.
%F A352001 A(2, k) = prime(k) (the k-th prime number).
%F A352001 A(3, k) = prime(2*k) = A031215(k).
%F A352001 A(4, k) = A001248(k).
%e A352001 Square array A(n, k) begins:
%e A352001   n\k|   1   2    3    4     5     6     7     8      9     10
%e A352001   ------------------------------------------------------------
%e A352001     1|   1   1    1    1     1     1     1     1      1      1
%e A352001     2|   2   3    5    7    11    13    17    19     23     29
%e A352001     3|   3   7   13   19    29    37    43    53     61     71
%e A352001     4|   4   9   25   49   121   169   289   361    529    841
%e A352001     5|   5  13   23   37    47    61    73    89    103    113
%e A352001     6|   6  21   65  133   319   481   731  1007   1403   2059
%e A352001     7|   7  19   37   53    71    89   107   131    151    173
%e A352001     8|   8  27  125  343  1331  2197  4913  6859  12167  24389
%e A352001     9|   9  49  169  361   841  1369  1849  2809   3721   5041
%e A352001    10|  10  39  115  259   517   793  1241  1691   2369   3277
%p A352001 A:= (n, k)-> mul(ithprime(k*numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
%p A352001 seq(seq(A(d+1-k, k), k=1..d), d=1..12);  # _Alois P. Heinz_, Feb 28 2022
%t A352001 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 A352001 (PARI) A(n,k) = { my (f=factor(n)); prod (i=1, #f~, prime(k * primepi(f[i,1])) ^ f[i,2]) }
%Y A352001 Cf. A000040, A001248, A031215, A051128, A297002, A351995.
%Y A352001 Main diagonal gives A352028.
%K A352001 nonn,tabl
%O A352001 1,2
%A A352001 _Rémy Sigrist_, Feb 27 2022