A348475 - OEIS

A348475

a(n) = Product_{k=1..A003056(n)} prime(T(n,k)), where T(n,k) = k-th term in row n of A235791.

%I #9 Oct 22 2021 11:36:17

%S 2,3,10,14,33,78,170,190,483,1218,2046,4070,5330,8385,33558,37842,

%T 47082,127490,169510,269445,825630,1250886,1404858,1969926,4270910,

%U 6988190,13239105,27506490,29387490,61019322,74958702,77319606,191617790,254235170,715103895,1549364190

%N a(n) = Product_{k=1..A003056(n)} prime(T(n,k)), where T(n,k) = k-th term in row n of A235791.

%C Prime product compactification of A235791.

%C All terms are squarefree.

%H Michael De Vlieger, <a href="/A348475/b348475.txt">Table of n, a(n) for n = 1..10000</a>

%e Row 3 of A235791 contains {3,1}, thus, a(3) = prime(3)*prime(1) = 5*2 = 10.

%e Omar E. Pol's diagram of rows of A235791:

%e . y row a(n) of A067255* a(n)

%e Row _| --------------------------

%e 1 _|1| -> 1 = 2

%e 2 _|2 _| -> .1 = 3

%e 3 _|3 |1| -> 1.1 = 10

%e 4 _|4 _|1| -> 1..1 = 14

%e 5 _|5 |2 _| -> .1..1 = 33

%e 6 _|6 _|2|1| -> 11...1 = 78

%e 7 _|7 |3 |1| -> 1.1...1 = 170

%e 8 _|8 _|3 _|1| -> 1.1....1 = 190

%e 9 _|9 |4 |2 _| -> .1.1....1 = 483

%e 10 _|10 _|4 |2|1| -> 11.1.....1 = 1218

%e 11 _|11 |5 _|2|1| -> 11..1.....1 = 2046

%e 12 _|12 _|5 |3 |1| -> 1.1.1......1 = 4070

%e 13 _|13 |6 |3 _|1| -> 1.1..1......1 = 5330

%e 14 _|14 _|6 _|3|2 _| -> .11..1.......1 = 8385

%e 15 _|15 |7 |4 |2|1| -> 11.1..1.......1 = 33558

%e 16 _|16 _|7 |4 |2|1| -> 11.1..1........1 = 37842

%e ...

%e * we replace 0 with "." for clarity.

%t Table[Times @@ Array[Prime@ Ceiling[(n + 1)/# - (# + 1)/2] &, Floor[(Sqrt[8 n + 1] - 1)/2]], {n, 35}]

%t (* Use the b-file to generate k rows of A235791 *)

%t With[{k = 120}, MapIndexed[Reverse[PrimePi /@ FactorInteger[#][[All, 1]]] &, Import["https://oeis.org/A348475/b348475.txt", "Data"][[1 ;; k, -1]]]] (* _Michael De Vlieger_, Oct 21 2021 *)

%Y Cf. A000040, A003056, A067255, A235791.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Oct 19 2021