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

Author

Ken Clements, Jun 20 2025

Keywords

• nonn
• fini
• full

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.

Examples

			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.
		

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

• Ken Clements, See: Appendix A of this proof

Crossrefs

Cf. A002110, A007947, A055932, A002378.

Programs

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 *)

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

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))

Formula

A007947(a(n)) is in A002110.