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A385189 Intersection of A055932 and A002378.

%I A385189 #34 Jun 27 2025 23:34:45
%S A385189 2,6,12,30,72,90,210,240,420,600,1260,6480,15750,50400,147840,194040,
%T A385189 291060,510510,2942940,4324320,5762400,9147600,19136250,96049800,
%U A385189 153153000,15178363200,37822664880,401392571580
%N A385189 Intersection of A055932 and A002378.
%C A385189 These numbers are the products of two consecutive integers and also their squarefree part a primorial.
%C A385189 The last is 633555*633556 = 401392571580. See proof of finiteness in Clements link.
%D A385189 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.
%H A385189 Ken Clements, <a href="https://github.com/kenatiod/Proofs/blob/main/ABC_Factorial_Proof.pdf">See: Appendix A of this proof</a>
%F A385189 A007947(a(n)) is in A002110.
%e A385189 a(1) = 2 = 1*2 = 2^1.
%e A385189 a(2) = 6 = 2*3 = 2^1 * 3^1.
%e A385189 a(3) = 12 = 3*4 = 2^2 * 3^1.
%e A385189 a(4) = 30 = 5*6 = 2^1 * 3^1 * 5^1.
%e A385189 a(5) = 72 = 8*9 = 2^3 * 3^2.
%e A385189 a(6) = 90 = 9*10 = 2^1 * 3^2 * 5^1.
%p A385189 q:= n-> (s-> nops(s)=numtheory[pi](max(s)))({ifactors(n)[2][.., 1][]}):
%p A385189 select(q, [i*(i+1)$i=1..640000])[];  # _Alois P. Heinz_, Jun 24 2025
%t A385189 Select[(#*(# + 1)) & /@ Range[633555], PrimePi[(f = FactorInteger[#1])[[-1, 1]]] == Length[f] &] (* _Amiram Eldar_, Jun 22 2025 *)
%o A385189 (Python)
%o A385189 from sympy import prime, primefactors
%o A385189 def is_pi_complete(n): # Check for complete set of
%o A385189     factors = primefactors(n) # prime factors
%o A385189     return factors[-1] == prime(len(factors))
%o A385189 def aupto(limit):
%o A385189     result = []
%o A385189     for i in range(1, limit+1):
%o A385189         n = i * (i+1)
%o A385189         if is_pi_complete(n):
%o A385189             result.append(n)
%o A385189     return result
%o A385189 print(aupto(100_000_000))
%o A385189 (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
%Y A385189 Cf. A002110, A007947, A055932, A002378.
%K A385189 nonn,fini,full
%O A385189 1,1
%A A385189 _Ken Clements_, Jun 20 2025