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A385956 Intersection of A025487 and A002378.

%I A385956 #14 Aug 15 2025 00:05:22
%S A385956 2,6,12,30,72,210,240,420,1260,6480,50400,147840,510510,4324320
%N A385956 Intersection of A025487 and A002378.
%C A385956 These numbers are the products of two 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(14) = 2079*2080 = 4324320.
%e A385956 a(1) = 2 = 1*2 = 2^1.
%e A385956 a(2) = 6 = 2*3 = 2^1 * 3^1.
%e A385956 a(3) = 12 = 3*4 = 2^2 * 3^1.
%e A385956 a(4) = 30 = 5*6 = 2^1 * 3^1 * 5^1.
%e A385956 a(5) = 72 = 8*9 = 2^3 * 3^2.
%e A385956 a(6) = 210 = 14*15 = 2^1 * 3^1 * 5^1 * 7^1.
%t A385956 Select[FactorialPower[Range[0, 3000], 2], (Max@Differences[(f = FactorInteger[#])[[;; , 2]]] < 1 && f[[-1, 1]] == Prime[Length[f]]) &] (* _Amiram Eldar_, Aug 10 2025 *)
%o A385956 (Python)
%o A385956 from sympy import prime, factorint
%o A385956 def is_Hardy_Ramanujan(n):
%o A385956     factors = factorint(n)
%o A385956     p_idx = len(factors)
%o A385956     if list(factors.keys())[-1] != prime(p_idx):
%o A385956         return False
%o A385956     expos = list(factors.values())
%o A385956     e = expos[0]
%o A385956     for i in range(1, p_idx):
%o A385956         if expos[i] > e:
%o A385956             return False
%o A385956         e = expos[i]
%o A385956     return True
%o A385956 print([ n*(n+1) for n in range(1, 10_000) if is_Hardy_Ramanujan(n*(n+1))])
%Y A385956 Cf. A055932, A025487, A002378, A385189, A386951.
%K A385956 nonn,fini,full
%O A385956 1,1
%A A385956 _Ken Clements_, Aug 10 2025