已知数列{an}满足a1=1,an=log n为底数(n+1)为真数(n≥2,n属于正整数).定义:使乘积a1·a2·a3······ak为正整数的k(k属于正整数)叫做“和谐数”,则在区间【1,2010】内所有的“和谐数”的和
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![已知数列{an}满足a1=1,an=log n为底数(n+1)为真数(n≥2,n属于正整数).定义:使乘积a1·a2·a3······ak为正整数的k(k属于正整数)叫做“和谐数”,则在区间【1,2010】内所有的“和谐数”的和](/uploads/image/z/7164752-32-2.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3a1%3D1%2Can%3Dlog+n%E4%B8%BA%E5%BA%95%E6%95%B0%EF%BC%88n%2B1%EF%BC%89%E4%B8%BA%E7%9C%9F%E6%95%B0%EF%BC%88n%E2%89%A52%2Cn%E5%B1%9E%E4%BA%8E%E6%AD%A3%E6%95%B4%E6%95%B0%EF%BC%89.%E5%AE%9A%E4%B9%89%EF%BC%9A%E4%BD%BF%E4%B9%98%E7%A7%AFa1%C2%B7a2%C2%B7a3%C2%B7%C2%B7%C2%B7%C2%B7%C2%B7%C2%B7ak%E4%B8%BA%E6%AD%A3%E6%95%B4%E6%95%B0%E7%9A%84k%EF%BC%88k%E5%B1%9E%E4%BA%8E%E6%AD%A3%E6%95%B4%E6%95%B0%EF%BC%89%E5%8F%AB%E5%81%9A%E2%80%9C%E5%92%8C%E8%B0%90%E6%95%B0%E2%80%9D%2C%E5%88%99%E5%9C%A8%E5%8C%BA%E9%97%B4%E3%80%901%2C2010%E3%80%91%E5%86%85%E6%89%80%E6%9C%89%E7%9A%84%E2%80%9C%E5%92%8C%E8%B0%90%E6%95%B0%E2%80%9D%E7%9A%84%E5%92%8C)
已知数列{an}满足a1=1,an=log n为底数(n+1)为真数(n≥2,n属于正整数).定义:使乘积a1·a2·a3······ak为正整数的k(k属于正整数)叫做“和谐数”,则在区间【1,2010】内所有的“和谐数”的和
已知数列{an}满足a1=1,an=log n为底数(n+1)为真数(n≥2,n属于正整数).定义:使乘积a1·a2·a3······ak为正整数的k(k属于正整数)叫做“和谐数”,则在区间【1,2010】内所有的“和谐数”的和为
已知数列{an}满足a1=1,an=log n为底数(n+1)为真数(n≥2,n属于正整数).定义:使乘积a1·a2·a3······ak为正整数的k(k属于正整数)叫做“和谐数”,则在区间【1,2010】内所有的“和谐数”的和
k=1时满足
∵an=logn(n+1)=lg(n+1)/lgn
∴a1*a2*a3…*ak
=lg3/lg2*lg4/lg3*lg5/lg4…*lgk/lg(k-1)*lg(k+1)/lgk
=lg(k+1)/lg2
∵使其为正整数
∴k+1=2^n→k(n)=2^n-1
∵k€[1,2010]
∴n(max)=10
∴Sk=k1+k2…+k10
=(2-1)+(2^2-1)…+(2^10-1)
=(2+2^2…+2^10)-10
=2^11-12
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