cp's OEIS Frontend

A385415 Products of three consecutive integers whose prime divisors are consecutive primes starting at 2.

%I A385415 #24 Jul 14 2025 21:45:29
%S A385415 6,24,60,120,210,720,3360,9240,117600,166320,970200,43243200,85765680
%N A385415 Products of three consecutive integers whose prime divisors are consecutive primes starting at 2.
%C A385415 The final term is 440*441*442 = 85765680.
%C A385415 Let tau(a(n)) be the number of divisors of a(n), sigma(a(n)) be the sum of divisors of a(n), S be the strictly increasing sequence of divisors of a(n), and S(i) be the i-th element of S. Since sigma(a(n)) is even and for every i in the interval [1, tau(a(n))-1], 2*S(i) >= S(i+1), a(n) is a Zumkeller number (see Proposition 17 in Rao/Peng JNT paper at A083207). - _Ivan N. Ianakiev_, Jul 09 2025
%e A385415 a(1) = 6 = 1*2*3 = 2^1 * 3^1.
%e A385415 a(2) = 24 = 2*3*4 = 2^3 * 3^1.
%e A385415 a(3) = 60 = 3*4*5 = 2^2 * 3^1 * 5^1.
%e A385415 a(4) = 120 = 4*5*6 = 2^3 * 3^1 * 5^1.
%e A385415 a(5) = 210 = 5*6*7 = 2^1 * 3^1 * 5^1 * 7^1.
%e A385415 a(6) = 720 = 8*9*10 = 2^4 * 3^2 * 5^1.
%e A385415 ...
%e A385415 a(13) = 85765680 = 440*441*442 = 2^4 * 3^2 * 5^1 * 7^2 * 11^1 * 13^1 * 17^1.
%t A385415 Select[(#*(# + 1)*(# + 2)) & /@ Range[500], PrimePi[(f = FactorInteger[#1])[[-1, 1]]] == Length[f] &] (* _Amiram Eldar_, Jun 28 2025 *)
%o A385415 (Python)
%o A385415 from sympy import prime, primefactors
%o A385415 def is_pi_complete(n): # Check for complete set of
%o A385415     factors = primefactors(n) # prime factors
%o A385415     return factors[-1] == prime(len(factors))
%o A385415 def aupto(limit):
%o A385415     result = []
%o A385415     for i in range(1, limit+1):
%o A385415         n = i * (i+1) * (i+2)
%o A385415         if is_pi_complete(n):
%o A385415             result.append(n)
%o A385415     return result
%o A385415 print(aupto(100_000))
%Y A385415 Intersection of A007531 and A055932.
%Y A385415 Cf. A385189, A083207 (supersequence).
%K A385415 nonn,fini,full
%O A385415 1,1
%A A385415 _Ken Clements_, Jun 28 2025