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·a
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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”的和为
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
∵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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