A385956 -id:A385956 - cp's OEIS frontend

A386951 Intersection of A025487 and A007531.

6, 24, 60, 120, 210, 720, 3360, 9240, 166320, 970200, 43243200

Offset: 1

Ken Clements, Aug 10 2025

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.

			a(1) = 6 = 1*2*3 = 2^1 * 3^1.
a(2) = 24 = 2*3*4 = 2^3 * 3^1.
a(3) = 60 = 3*4*5 = 2^2 * 3^1 * 5^1.
a(4) = 120 = 4*5*6 = 2^3 * 3^1 * 5^1.
a(5) = 210 = 5*6*7 = 2^1 * 3^1 * 5^1 * 7^1.
a(6) = 720 = 8*9*10 = 2^4 * 3^2 * 5^1.

Cf. A055932, A025487, A007531, A385189, A385956.

Select[FactorialPower[Range[0, 1000], 3], (Max@ Differences[(f = FactorInteger[#])[[;; , 2]]] < 1 && f[[-1, 1]] == Prime[Length[f]]) &] (* Amiram Eldar, Aug 10 2025 *)

from sympy import prime, factorint
def is_Hardy_Ramanujan(n):
    factors = factorint(n)
    p_idx = len(factors)
    if list(factors.keys())[-1] != prime(p_idx):
        return False
    expos = list(factors.values())
    e = expos[0]
    for i in range(1, p_idx):
        if expos[i] > e:
            return False
        e = expos[i]
    return True
print([ n*(n+1)*(n+2) for n in range(1, 1000) if is_Hardy_Ramanujan(n*(n+1)*(n+2))])

cp's OEIS Frontend

A386951 Intersection of A025487 and A007531.

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