A356803 - OEIS

%I #11 Sep 08 2022 01:34:57

%S 1,1,2,6,15,105,385,2310,6006,102102,277134,6374082,16804398,

%T 520936338,3038795305,66853496710,190275336790,7420738134810,

%U 17897074325130,769574195980590,1903683537425670,100895227483560510,258818192240437830,15787909726666707630,36475515575402393490

%N a(n) = product of prohibited prime factors of A354790(n).

%C Let s(n) = A354790(n), a squarefree number by definition. Prime p | s(n) implies p does not divide s(n+j), 1 <= j <= n. Therefore a(n) is the product of primes p that cannot divide s(n). a(n) = product of distinct primes that divide a(j) for floor((n+1)/2) <= j <= n-1. (After _N. J. A. Sloane_ in A355057.)

%C Analogous to A355057.

%H Michael De Vlieger, <a href="/A356803/b356803.txt">Table of n, a(n) for n = 1..585</a>

%H Michael De Vlieger, <a href="/A356803/a356803.png">Annotated plot of prime p | m at (n, pi(p))</a> for m = a(n) in blue and A354790(n) in red and n = 1..80.

%H Michael De Vlieger, <a href="/A356803/a356803_1.png">Plot of prime p | m at (n, pi(p))</a> for m = a(n) in cyan and A354790(n) in red and n = 1..1024.

%F a(n) = a(n-1) * s(n-1) / s((n-1)/2), where the last operation is only carried out iff (n-1)/2 is an integer.

%e a(1) = 1;

%e a(2) = 1 since s(1) = 1, and (2-1)/2 is not an integer;

%e a(3) = a(2) * s(2) / s((3-1)/2) = 1 * 2 / 1 = 2;

%e a(4) = a(3) * s(3) = 2 * 3 = 6;

%e a(5) = a(4) * s(4) / s((5-1)/2) = 6 * 5 / 2 = 15;

%e a(6) = a(5) * s(5) = 15 * 7 = 105;

%e a(7) = a(6) * s(6) / s((7-1)/2) = 105 * 11 / 3 = 385; etc.

%t Block[{s = Import["https://oeis.org/A354790/b354790.txt", "Data"][[1 ;; 26, -1]], ww, m = 1, t, w = 3, k = 3}, Reap[Do[m *= Times @@ FactorInteger[s[[If[# == 0, 1, #] &[i - 1]]]][[All, 1]]; If[IntegerQ[#] && # > 0, m /= Times @@ FactorInteger[s[[#]]][[All, 1]]] &[(i - 1)/2]; Sow[m], {i, Length[s] - 1}]][[-1, -1]] ]

%Y Cf. A354790, A355057.

%K nonn

%O 1,3

%A _Michael De Vlieger_, Sep 06 2022