Question

Sea $\tau(n)$ el número de divisores enteros positivos de $n.$ Halle la suma de los seis menores enteros positivos $n$ que son soluciones de $\tau(n) + \tau(n + 1) = 7.$

Let $\tau(n)$ denote the number of positive integer divisors of $n.$ Find the sum of the six least positive integers $n$ that are solutions to $\tau(n) + \tau(n + 1) = 7.$

Respuesta

Solución:

Como $7 = 2 + 5 = 3 + 4,$ uno de $\tau(n), \tau(n+1)$ es igual a $3$ o $5,$ y $\tau = 3$ significa un cuadrado de primo $p^2$ mientras que $\tau = 5$ significa una cuarta potencia de primo $p^4.$ Así que uno de $n, n+1$ está en $\{4, 9, 25, 49, 121,$ $169, 289, 361, \ldots\}$ $\cup \{16, 81, 625, \ldots\},$ y su vecino debe tener $\tau = 4$ (para un cuadrado) o ser primo (para una cuarta potencia).

Verificando los vecinos en orden creciente: $n = 8$ funciona $(\tau(8) = 4,$ $\tau(9) = 3);$ $n = 9$ funciona $(\tau(10) = 4);$ $n = 16$ funciona $(\tau(16) = 5,$ $17$ primo $);$ $n = 25$ funciona $(\tau(26) = 4).$ Luego $49, 81, 169,$ y $289$ fallan todos: $\tau(48) = 10,$ $\tau(50) = 6,$ $\tau(80) = 10,$ $82$ no es primo, $\tau(168) = 16,$ $\tau(170) = 8,$ $\tau(288) = 18,$ $\tau(290) = 8.$ A continuación, $n = 121$ funciona $(\tau(122) = 4)$ y $n = 361$ funciona $(\tau(362) = 4).$

Las seis menores soluciones son $8, 9, 16, 25, 121, 361,$ con suma $540.$

Since $7 = 2 + 5 = 3 + 4,$ one of $\tau(n), \tau(n+1)$ equals $3$ or $5,$ and $\tau = 3$ means a prime square $p^2$ while $\tau = 5$ means a prime fourth power $p^4.$ So one of $n, n+1$ lies in $\{4, 9, 25, 49, 121,$ $169, 289, 361, \ldots\}$ $\cup \{16, 81, 625, \ldots\},$ and its neighbor must have $\tau = 4$ (for a square) or be prime (for a fourth power).

Checking neighbors in increasing order: $n = 8$ works $(\tau(8) = 4,$ $\tau(9) = 3);$ $n = 9$ works $(\tau(10) = 4);$ $n = 16$ works $(\tau(16) = 5,$ $17$ prime $);$ $n = 25$ works $(\tau(26) = 4).$ Then $49, 81, 169,$ and $289$ all fail: $\tau(48) = 10,$ $\tau(50) = 6,$ $\tau(80) = 10,$ $82$ is not prime, $\tau(168) = 16,$ $\tau(170) = 8,$ $\tau(288) = 18,$ $\tau(290) = 8.$ Next, $n = 121$ works $(\tau(122) = 4)$ and $n = 361$ works $(\tau(362) = 4).$

The six least solutions are $8, 9, 16, 25, 121, 361,$ with sum $540.$

2019 AIME I Problema 9

Solución:

El Problema 9 en otros años