login
a(n) = product of prohibited prime factors of A090252(n).
6

%I #22 Jul 05 2022 22:08:37

%S 1,1,2,6,15,30,70,210,462,6006,51051,102102,277134,6374082,10623470,

%T 223092870,588153930,18232771830,51893273670,2127624220470,

%U 5381637734130,252936973504110,6702829797858915,13405659595717830,41628100849860630,2539314151841498430,7397132529277408470

%N a(n) = product of prohibited prime factors of A090252(n).

%C Let s(n) = A090252(n) and let K(n) = A007947(n) = squarefree kernel of n.

%C Prime p | s(n) implies p does not divide s(n+j), 1 <= j <= n.

%C Therefore a(n) is the product of primes p that cannot divide s(n).

%C a(n) = product of distinct primes that divide a(j) for floor((n+1)/2) <= j <= n-1. - _N. J. A. Sloane_, Jun 17 2022

%H N. J. A. Sloane, <a href="/A355057/b355057.txt">Table of n, a(n) for n = 1..500</a>

%H Michael De Vlieger, <a href="/A355057/a355057.png">Annotated chart of prime factors of a(n)</a>, n = 1..64, plotting prime p | a(n) at (n, pi(p)) in black, and prime q | s(n) at (n, pi(q)) in red.

%H Michael De Vlieger, <a href="/A355057/a355057_1.png">Plot of prime p | a(n)</a>, n = 1..2^12, plotting p | a(n) at (n, pi(p)) in black.

%F a(n) = a(n-1) * K(s(n-1)) / K(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) * K(s(2)) / K(s((3-1)/2)) = 1 * 2 / 1 = 2;

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

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

%e a(6) = a(5) * K(s(5)) = 15 * 2 = 30;

%e a(7) = a(6) * K(s(6)) / K(s((7-1)/2)) = 30 * 7 / 3 = 70;

%e etc.

%p # To get first M terms, from _N. J. A. Sloane_, Jun 18 2022

%p with(numtheory);

%p M:=20; ans:=[1,1,2];

%p for i from 4 to M do

%p S:={}; j1:=floor((i+1)/2); j2:=i-1;

%p for j from j1 to j2 do S:={op(S), op(factorset(b252[j]))} od:

%p t2 := product(S[k], k = 1..nops(S));

%p ans:=[op(ans),t2];

%p od:

%p ans;

%t Block[{s = Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; 120, -1]], m = 1}, 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]] ]

%o (Python)

%o from math import prod, lcm, gcd

%o from itertools import count, islice

%o from collections import deque

%o from sympy import primefactors

%o def A355057_gen(): # generator of terms

%o aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True

%o yield 1

%o while True:

%o for m in count(c):

%o if m not in aset and gcd(m,b) == 1:

%o yield prod(primefactors(b))

%o aset.add(m)

%o aqueue.append(m)

%o if f: aqueue.popleft()

%o b = lcm(*aqueue)

%o f = not f

%o while c in aset:

%o c += 1

%o break

%o A355057_list = list(islice(A355057_gen(),20)) # _Chai Wah Wu_, Jun 18 2022

%Y See A354758 for another version.

%Y A354765 is a binary encoding.

%Y Cf. A007947, A090252, A354764.

%K nonn

%O 1,3

%A _Michael De Vlieger_, Jun 16 2022