A355057 a(n) = product of prohibited prime factors of A090252(n).

1, 1, 2, 6, 15, 30, 70, 210, 462, 6006, 51051, 102102, 277134, 6374082, 10623470, 223092870, 588153930, 18232771830, 51893273670, 2127624220470, 5381637734130, 252936973504110, 6702829797858915, 13405659595717830, 41628100849860630, 2539314151841498430, 7397132529277408470

Offset: 1

Author

Michael De Vlieger, Jun 16 2022

Keywords

• nonn

Comments

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

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. - N. J. A. Sloane, Jun 17 2022

Examples

			a(1) = 1;
a(2) = 1 since s(1) = 1, and (2-1)/2 is not an integer;
a(3) = a(2) * K(s(2)) / K(s((3-1)/2)) = 1 * 2 / 1 = 2;
a(4) = a(3) * K(s(3)) = 2 * 3 = 6;
a(5) = a(4) * K(s(4)) / K(s((5-1)/2)) = 6 * 5 / 2 = 15;
a(6) = a(5) * K(s(5)) = 15 * 2 = 30;
a(7) = a(6) * K(s(6)) / K(s((7-1)/2)) = 30 * 7 / 3 = 70;
etc.
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..500
• Michael De Vlieger, Annotated chart of prime factors of a(n), 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.
• Michael De Vlieger, Plot of prime p | a(n), n = 1..2^12, plotting p | a(n) at (n, pi(p)) in black.

Crossrefs

See A354758 for another version.

A354765 is a binary encoding.

Cf. A007947, A090252, A354764.

Programs

Maple

# To get first M terms, from N. J. A. Sloane, Jun 18 2022
with(numtheory);
M:=20; ans:=[1,1,2];
for i from 4 to M do
S:={}; j1:=floor((i+1)/2); j2:=i-1;
  for j from j1 to j2 do S:={op(S), op(factorset(b252[j]))} od:
t2 := product(S[k], k = 1..nops(S));
ans:=[op(ans),t2];
od:
ans;

Mathematica

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]] ]

Python

from math import prod, lcm, gcd
from itertools import count, islice
from collections import deque
from sympy import primefactors
def A355057_gen(): # generator of terms
    aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True
    yield 1
    while True:
        for m in count(c):
            if m not in aset and gcd(m,b) == 1:
                yield prod(primefactors(b))
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = lcm(*aqueue)
                f = not f
                while c in aset:
                    c += 1
                break
A355057_list = list(islice(A355057_gen(),20)) # Chai Wah Wu, Jun 18 2022

Formula

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.