¿Cuál es el máximo de $n$ tal que $\sum_{i=1}^n\frac{1}{a_i}=1$ donde $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$ ?

Question

¿Cuál es el máximo de $n$ tal que $$\sum_{i=1}^n\frac{1}{a_i}=1$$ donde $a_{i}\ (i=1,2,\cdots,n)$ son números enteros que satisfacen $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$ ?

Además, necesito cómo probar que el $n$ que se obtiene es el máximo.

Mi enfoque :

Voy a representar a $\sum_{i=1}^n\frac{1}{a_i}$ como $(a_1,a_2,\cdots,a_n).$

$(2,3,6)\rightarrow(4,5,7,9, 12, 15,18,30,42,45,90)$ $\rightarrow(8,9,10, 12, 14, 15, 16, 18,22,27, 30, 35, 40, 42, 45, 48, 54, 56, 60, 72, 90, 99)$

(aquí he utilizado $(3)=(4,12), (6)=(7,42)$ etc.)

Esta es la $n=22$ caso.

Actualización : Acabo de recibir lo siguiente $n=42$ caso. No sé si esto es el máximo.

$(8,9,10, 12, 14, 15, 16, 18,22,27, 30, 35, 40, 42, 45, 48, 54, 56, 60, 72, 90, 99)\rightarrow(15,17,20,21,22,26,27,30,32,33,34,35,36,38,39,40,42,44,45,48,50,52,54,55,56,60,63,66,70,75,76,77,78,80,84,85,88,90,91,95,96,99)$

Aquí he utilizado $(10)=(17,34,85), (14)=(28,44,77)$ etc.

Actualización 2 : Acabo de recibir otro $n=42$ caso.

$(17,18,20,21,22,24,26,27,32,33,34,35,36,38,39,40,42,44,45,48,50,52,54,55,56,60,63,66,70,72,75,76,77,78,80,84,85,88,91,95,96,99)$

Aquí he utilizado $(15,30,90)=(18,24,72)$ .

Actualización 3 : He hecho un crossposting en MO.

https://mathoverflow.net/questions/142300/what-is-the-max-of-n-such-that-sum-i-1n-frac1a-i-1-where-2-le-a-1-l

Answer 1

[ EDITADO para incluir otras potencias principales y dar la lista de 27 soluciones máximas ]

El máximo es $42$ , alcanzado en $27$ formas (enumeradas a continuación); hay $566$ con el subcampeón $41$ y luego $6747$ con $40$ , otro $52078$ con $39$ , "etc.".

Los recuentos se obtienen mediante programación dinámica. Simplificamos el cálculo comprobando que el $a_i$ no puede incluir ningún múltiplo de una potencia prima mayor que $27$ y si los múltiplos de $11$ , $13$ , $16$ , $17$ , $19$ , $25$ o $27$ aparecen entonces deben combinarse para eliminar ese factor del denominador, lo que sólo puede hacerse en $46,\phantom. 9,\phantom. 7,\phantom., 1, \phantom. 2,\phantom. 1,\phantom. 1$ maneras, respectivamente (los cuatro últimos son $(17,34,85)$ , $(19,57,76)$ o $(38,76,95)$ , $(50,75)$ y $(25,54)$ respectivamente; $23$ no se produce). Esto reduce el denominador a $D = 2^3 3^2 5 \phantom. 7 = 2520$ , lo suficientemente pequeño como para hacer una tabla del número de veces que cada par de enteros surge como $(n, D\sum_{i=1}^n 1/a_i)$ con $\sum_i 1/a_i \leq 1$ , y al final extraer los recuentos de $(n,D)$ .

Este enfoque no da inmediatamente la lista de $27$ soluciones máximas, pero se puede modificar para calcular esta lista en su lugar: en cada etapa, en lugar de registrar el número de representaciones de cada fracción, se registra la(s) representación(es) con el mayor número de términos. La lista de $42$ es el siguiente, en orden lexicográfico; cada uno utiliza el primo 17, y todos utilizan el 99 excepto dos que tienen $\max_i a_i = 96$ .

[12, 17, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 18, 21, 22, 24, 26, 27, 32, 33, 34, 35, 38, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 18, 22, 24, 26, 27, 28, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 20, 21, 22, 24, 26, 27, 32, 33, 34, 35, 36, 38, 40, 42, 44, 48, 50, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 20, 21, 22, 24, 26, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 52, 55, 56, 60, 63, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 20, 21, 22, 26, 27, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 65, 66, 70, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 20, 21, 24, 26, 27, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96]
[13, 17, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 38, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[14, 17, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 44, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 18, 21, 22, 24, 26, 27, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 18, 22, 24, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 20, 21, 22, 24, 26, 27, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 20, 21, 22, 24, 26, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 20, 21, 22, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 20, 21, 24, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96]
[15, 17, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 19, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 57, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 96, 99]
[17, 18, 19, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 57, 60, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 96, 99]
[17, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 38, 39, 40, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 48, 50, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 76, 77, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 20, 21, 22, 24, 26, 27, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 91, 95, 96, 99]
[17, 18, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 54, 55, 56, 60, 65, 66, 70, 72, 75, 76, 77, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 66, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 19, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 36, 39, 40, 42, 44, 48, 50, 52, 54, 55, 56, 57, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 96, 99]

Answer 2

Su $22$ no es el máximo porque $(12)=(19,57,95)$ . Y después de eso $(9)=(12,36)$ .