A385189 Intersection of A055932 and A002378.

2, 6, 12, 30, 72, 90, 210, 240, 420, 600, 1260, 6480, 15750, 50400, 147840, 194040, 291060, 510510, 2942940, 4324320, 5762400, 9147600, 19136250, 96049800, 153153000, 15178363200, 37822664880, 401392571580

Offset

1,1

Comments

These numbers are the products of two consecutive integers and also their squarefree part a primorial.

The last is 633555*633556 = 401392571580. See proof of finiteness in Clements link.

References

Ken Clements, Proof that the Equation A! x B! = C! Has Only One Solution for Integers 1 < A < B < C-1, submitted to INTEGERS, 2025.

Links

Table of n, a(n) for n=1..28.

Ken Clements, See: Appendix A of this proof

Formula

A007947(a(n)) is in A002110.

Example

a(1) = 2 = 1*2 = 2^1.
a(2) = 6 = 2*3 = 2^1 * 3^1.
a(3) = 12 = 3*4 = 2^2 * 3^1.
a(4) = 30 = 5*6 = 2^1 * 3^1 * 5^1.
a(5) = 72 = 8*9 = 2^3 * 3^2.
a(6) = 90 = 9*10 = 2^1 * 3^2 * 5^1.

Maple

q:= n-> (s-> nops(s)=numtheory[pi](max(s)))({ifactors(n)[2][.., 1][]}):
select(q, [i*(i+1)$i=1..640000])[];  # Alois P. Heinz, Jun 24 2025

Mathematica

Select[(#*(# + 1)) & /@ Range[633555], PrimePi[(f = FactorInteger[#1])[[-1, 1]]] == Length[f] &] (* Amiram Eldar, Jun 22 2025 *)

Prog

(Python)
from sympy import prime, primefactors
def is_pi_complete(n): # Check for complete set of
    factors = primefactors(n) # prime factors
    return factors[-1] == prime(len(factors))
def aupto(limit):
    result = []
    for i in range(1, limit+1):
        n = i * (i+1)
        if is_pi_complete(n):
            result.append(n)
    return result
print(aupto(100_000_000))
(PARI) lista(nn) = my(list=List()); for (n=1, nn, my(f=factor(n*(n+1))[, 1]~); if (f==primes(#f), listput(list, n*(n+1)))); Vec(list); \\ Michel Marcus, Jun 22 2025

Crossrefs

Cf. A002110, A007947, A055932, A002378.

Sequence in context: A384614 A032177 A095349 * A385956 A291518 A291445

Adjacent sequences: A385185 A385187 A385188 * A385190 A385191 A385192

Keyword

nonn,fini,full

Author

Ken Clements, Jun 20 2025

Status

approved