求和

123456789

S=1/4+1/8+1/9+1/16+1/25+1/27+1/32+1/36+1/49+1/64+1/81+1/100+1/121+1/125+1/144+1/169+……………………

esuruekinz

无穷

evuojad

显然收敛

uzmetub

@mathe,我总结的规律是可以表达成单个数的幂的形式的所有整数的倒数和。而这个存在多因子的合数的幂的情况，应该大于所有单因子数的幂的倒数和，而质数是无穷多，所以这个是发散的。

aavucopuyu

$S<(1/2^2+1/2^3+1/2^4+……)+(1/3^2+1/3^3+1/3^4+……)+(1/4^2+1/4^3+1/4^4+……)+……
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+……
=1.$

Geraldufas

奥对对，首项应该是从平方开始，所以， $S < \sum_{n=2}^{+\infty}{1/n^2}/{1-1/n}  = 1 $

esuruekinz

谢谢《数学研发论坛》！谢谢各路大侠！这些年进步是很大。

111

S=1/4+1/8+1/9+1/16+1/25+1/27+1/32+1/36+1/49+1/64+1/81+1/100+1/121+1/125+1/144+1/169+……………………
=0.874464368404944866694351320597373165935338431924214...

A072102
{8, 7, 4, 4, 6, 4, 3, 6, 8, 4, 0, 4, 9, 4, 4, 8, 6, 6, 6, 9, 4, 3, 5, 1, 3, 2, 0, 5, 9, 7, 3, 7, 3, 1, 6, 5, 9, 3, 5, 3, 3, 8, 4, 3, 1, 9, 2, 4, 2, 1,
4, 5, 7, 7, 6, 2, 5, 7, 8, 8, 2, 5, 3, 5, 0, 9, 3, 7, 0, 0, 6, 4, 1, 2, 9, 7, 2, 3, 6, 7, 6, 5, 9, 9, 3, 3, 2, 2, 6, 1, 7, 8, 5, 7, 5, 8, 0, 1, 6, 2,
8, 7, 7, 0, 6, 3, 4, 1, 9, 3, 6, 2, 5, 5, 9, 0, 5, 3, 0, 1, 0, 3, 8, 6, 3, 2, 8, 5, 5, 3, 6, 0, 8, 7, 5, 3, 1, 1, 8, 7, 2, 9, 5, 6, 8, 5, 4, 3, 1, 0}

RealDigits[Total[Block[{$MaxExtraPrecision = 10^3}, N[#, 150] & /@ Table[MoebiusMu[k] (1 - Zeta[k]), {k, 2, 10^3}]]]][[1]]


A001597
1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400,
441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521,
1600, 1681, 1728, 1764, 1849, 1936, 2025, 2048, 2116, 2187, 2197, 2209, 2304, 2401, 2500, 2601, 2704, 2744, 2809, 2916,
3025, 3125, 3136, 3249, 3364, 3375, 3481, 3600, 3721, 3844, 3969, 4096, 4225, 4356, 4489, 4624, 4761, 4900, 4913, ......

Join[{1}, Select[Range[5000], GCD@@FactorInteger[#][[All, 2]]>1&]]       (* Harvey P. Dale, Apr 30 2018 *)

ivamvanibumiq

谢谢 mathe!   谢谢 天山草！

N\(\bigg[\D\sum_{k=2}^{166}\big(1-Zeta[k]\big)*MoebiusMu[k], 50\bigg]\)

0.87446436840494486669435132059737316593533843192421

166可以达到50位(165还不行)，331可以达到100位(330还不行)，

0.87446436840494486669435132059737316593533843192421
   45776257882535093700641297236765993322617857580163

Sherryufas

谢谢 mathe!   谢谢 天山草！谢谢 Nicolas2050!

太好了！多年的心事总算了了！给出前20项，大家一起分享！谢谢大家！

s(02)=(1-1.6449340668482264365...)×(-1)=+0.6449340668482264365...
s(03)=(1-1.2020569031595942854...)×(-1)=+0.2020569031595942854...
s(04)=(1-1.0823232337111381915...)×(00)=00
s(05)=(1-1.0369277551433699263...)×(-1)=+0.0369277551433699263...
s(06)=(1-1.0173430619844491397...)×(+1)=-0.0173430619844491397...
s(07)=(1-1.0083492773819228268...)×(-1)=+0.0083492773819228268...
s(08)=(1-1.0040773561979443394...)×(00)=00
s(09)=(1-1.0020083928260822144...)×(00)=00
s(10)=(1-1.0009945751278180853...)×(+1)=-0.0009945751278180853...
s(11)=(1-1.0004941886041194646...)×(-1)=+0.0004941886041194646...
s(12)=(1-1.0002460865533080483...)×(00)=00
s(13)=(1-1.0001227133475784891...)×(-1)=+0.0001227133475784891...
s(14)=(1-1.0000612481350587048...)×(+1)=-0.0000612481350587048...
s(15)=(1-1.0000305882363070205...)×(+1)=-0.0000305882363070205...
s(16)=(1-1.0000152822594086519...)×(00)=00
s(17)=(1-1.0000076371976378998...)×(-1)=+0.0000076371976378998...
s(18)=(1-1.0000038172932649998...)×(00)=00
s(19)=(1-1.0000019082127165539...)×(-1)=+0.0000019082127165539...
s(20)=(1-1.0000009539620338728...)×(00)=00
s(21)=(1-1.0000004769329867878...)×(+1)=-0.0000004769329867878...

s(02)+s(03)+s(04)+ .... +s(19)+s(20)+s(21)=+0.87446449947854614427..