A385956 Intersection of A025487 and A002378.

2, 6, 12, 30, 72, 210, 240, 420, 1260, 6480, 50400, 147840, 510510, 4324320

Offset: 1

Author

Ken Clements, Aug 10 2025

Keywords

• nonn
• fini
• full

Comments

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.

Examples

			a(1) = 2 = 1*2 = 2^1.
a(2) = 6 = 2*3 = 2^1 * 3^1.
a(3) = 12 = 3*4 = 2^2 * 3^1.
a(4) = 30 = 5*6 = 2^1 * 3^1 * 5^1.
a(5) = 72 = 8*9 = 2^3 * 3^2.
a(6) = 210 = 14*15 = 2^1 * 3^1 * 5^1 * 7^1.
		

Crossrefs

Cf. A055932, A025487, A002378, A385189, A386951.

Programs

Mathematica

Select[FactorialPower[Range[0, 3000], 2], (Max@Differences[(f = FactorInteger[#])[[;; , 2]]] < 1 && f[[-1, 1]] == Prime[Length[f]]) &] (* Amiram Eldar, Aug 10 2025 *)

Python

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) for n in range(1, 10_000) if is_Hardy_Ramanujan(n*(n+1))])