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A386951 Intersection of A025487 and A007531.

%I A386951 #7 Aug 15 2025 00:04:22
%S A386951 6,24,60,120,210,720,3360,9240,166320,970200,43243200
%N A386951 Intersection of A025487 and A007531.
%C A386951 These numbers are the products of three consecutive integers that are also Hardy-Ramanujan integers; that is, of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n. This sequence is finite with last term a(11) = 350*351*352 = 43243200.
%e A386951 a(1) = 6 = 1*2*3 = 2^1 * 3^1.
%e A386951 a(2) = 24 = 2*3*4 = 2^3 * 3^1.
%e A386951 a(3) = 60 = 3*4*5 = 2^2 * 3^1 * 5^1.
%e A386951 a(4) = 120 = 4*5*6 = 2^3 * 3^1 * 5^1.
%e A386951 a(5) = 210 = 5*6*7 = 2^1 * 3^1 * 5^1 * 7^1.
%e A386951 a(6) = 720 = 8*9*10 = 2^4 * 3^2 * 5^1.
%t A386951 Select[FactorialPower[Range[0, 1000], 3], (Max@ Differences[(f = FactorInteger[#])[[;; , 2]]] < 1 && f[[-1, 1]] == Prime[Length[f]]) &] (* _Amiram Eldar_, Aug 10 2025 *)
%o A386951 (Python)
%o A386951 from sympy import prime, factorint
%o A386951 def is_Hardy_Ramanujan(n):
%o A386951     factors = factorint(n)
%o A386951     p_idx = len(factors)
%o A386951     if list(factors.keys())[-1] != prime(p_idx):
%o A386951         return False
%o A386951     expos = list(factors.values())
%o A386951     e = expos[0]
%o A386951     for i in range(1, p_idx):
%o A386951         if expos[i] > e:
%o A386951             return False
%o A386951         e = expos[i]
%o A386951     return True
%o A386951 print([ n*(n+1)*(n+2) for n in range(1, 1000) if is_Hardy_Ramanujan(n*(n+1)*(n+2))])
%Y A386951 Cf. A055932, A025487, A007531, A385189, A385956.
%K A386951 nonn,fini,full
%O A386951 1,1
%A A386951 _Ken Clements_, Aug 10 2025