Research on Smarandache functions and other related problems

2011·5.2 MB·English

by  Ma Rong

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Peer Reviewer:
Prof. Wenpeng Zhang, Department of Mathematics, Northwest University,
Xi’an, Shaanxi, P. R. China.
Prof. Xingsen Li, Ningbo Institute of Technology, Zhejiang University, Ningbo,
P. R. China.
Prof. Chunyan Yang and Weihua Li, Guangdong University of Technology,
Institute of Extenics and Innovative Methods, Guangzhou, P. R. China.
Prof. Qiaoxing Li, Lanzhou University, Lanzhou, P. R. China.
Ph. D. Xiaomei Li, Dept. of Computer Science, Guangdong University of
Technology, P. R. China.

Copyright 2012 by The Educational Publisher, translators, editors and authors
for their papers

Many books can be downloaded from the following Digital Library of Science:
http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm

ISBN: 9781599731902
Words: 175,000
Standard Address Number: 297-5092
Printed in the United States of America
8 ¹
8 ¹
c ó................................................................................. III
1 {0.................................................................................. 1
2 Smarandache¼ê†–Smarandache¼ê.......................................... 3
2.1 'uSmarandache¼ê(cid:27)˜‡ßŽ .............................................. 3
2.2 ˜a(cid:157)¹Smarandache¼êÚEuler¼ê(cid:27)(cid:144)§I ............................... 5
2.3 ˜a(cid:157)¹Smarandache¼êÚEuler¼ê(cid:27)(cid:144)§II .............................. 9
2.4 'u–Smarandache¼ê(cid:27)ߎ................................................. 12
2.5 'uSmarandache¼ê†–Smarandache¼ê(cid:27)(cid:144)§I ......................... 16
2.5.1 (cid:18)µ9yG.................................................................... 16
2.5.2 ˜(cid:10)ƒ'¯K ................................................................. 19
2.6 'uSmarandache¼ê†–Smarandache¼ê(cid:27)(cid:144)§II ........................ 20
2.7 'uSmarandache¼ê†ƒê ................................................... 23
3 †Smarandache¼ê9–Smarandache¼êƒ'(cid:27)˜(cid:10)Ù§¼ê................ 27
3.1 ˜a†–Smarandache¼êƒ'(cid:27)¼ê(cid:144)§.................................... 27
3.2 'uSmarandachep‡¼ê†–Smarandache¼ê(cid:27)(cid:144)§..................... 29
3.3 'u¹–Smarandache¼ê9Ùéó¼ê(cid:27)(cid:144)§............................... 34
3.4 'u¹Smarandachep‡¼ê†–Smarandache¼ê(cid:27)(cid:144)§ .................. 39
3.5 'uSmarandacheV(cid:30)¦¼ê ................................................... 42
3.6 ˜a2–Smarandache¼ê ................................................... 45
3.7 ˜‡(cid:157)¹Smarandache¼ê91(cid:19)a–Smarandache¼ê(cid:27)(cid:144)§ ............. 48
3.8 'uSmarandache LCM¼ê†Smarandache¼ê(cid:27)þŠ ...................... 51
3.9 'uSmarandache LCM¼ê9Ùéó¼ê..................................... 53
4 SmarandacheS(cid:15)ïÄ.............................................................. 57
4.1 Smarandache LCM 'ÇS(cid:15)I................................................... 57
4.2 Smarandache LCM 'ÇS(cid:15)II.................................................. 61
4.3 Smarandache 1(cid:15)ª ............................................................. 64
4.3.1 SmarandacheÌ‚1(cid:15)ª ..................................................... 64
4.3.2 Smarandache Vé¡1(cid:15)ª.................................................. 67
4.4 Smarandache (cid:17)(cid:28)ê ............................................................. 69
4.5 Smarandache 3n êiS(cid:15)....................................................... 71
I
8 ¹
4.6 Smarandache kn êiS(cid:15)....................................................... 74
4.7 Smarandache ²(cid:144)ê(cid:15) .......................................................... 78
5 Ù§êدK ....................................................................... 82
5.1 ¹kFibonacciê†Lucasê(cid:27)ð(cid:31)ª............................................ 82
5.2 Dirichlet L-¼ê†n(cid:14)Ú........................................................ 91
5.3 2ÂDirichlet L-¼ê.............................................................104
5.3.1 'u½n5.4 ...................................................................107
5.3.2 'u½n5.5 ...................................................................113
ë(cid:127)©z.................................................................................128
II
c ó
c ó
êØq¡Š(cid:18)êØ, ´ïÄê(cid:27)5Æ, AO´ïÄ(cid:18)ê5Ÿ(cid:27)êÆ©|. êØ/¤
˜€Õá(cid:27)Ɖ(cid:0), ‘XÙ¦êÆ©|(cid:27)uÐ, ïÄêØ(cid:27)(cid:144){(cid:143)3Øä(cid:27)uÐ, y“
êØ®²(cid:29)\(cid:20)êÆ(cid:27)Nõ©|, 3¥I, êØ(cid:143)´uÐ(cid:129)@(cid:27)êÆ©|ƒ˜. (cid:20)F1
<Ú¥I<(cid:31)é@Òk(cid:10)êØ(cid:127)£.
êØ3êÆ¥(cid:27)/ ´ÕA(cid:27), pdQ²`L/êÆ´‰Æ(cid:27)•(cid:0), êØ´êÆ¥
(cid:27)•)0. Ïd, êÆ[ÑU(cid:129)rêØ¥˜(cid:10)](cid:13)™û(cid:27)¦J(cid:23)‰/•)þ(cid:27)²¾0,
±(cid:19)y<‚(cid:22)/Á(cid:18)0. êØfm©(cid:27)žÿ´^ȃ(cid:27)ín(cid:144){ïÄ(cid:18)ê(cid:27)5Ÿ, =Ð
(cid:31)êØ. (cid:0)5úúuÐÑy)ÛêØ, “êêØ, |ÜêØ(cid:31). §‚Ï(cid:143)ïÄ(cid:144){(cid:27)ØÓ
(cid:13)ˆ(cid:143)Ù¶, (cid:2)ѱÓ{Š(cid:143)Ù(cid:129)Ä(cid:29)(cid:27)nØ(cid:157)â.
(cid:29)Ö̇éSmarandache¼ê9Ùƒ'¯K(cid:27)CÏïÄ(J?1(cid:18)nnã,
̇0(cid:11)(cid:10)Smarandache¼ê9†Ù§ƒ'¼ê(Ü(cid:27)ïÄ?Ð, Óž„0(cid:11)(cid:10)
–Smarandache ¼ê9†Ù§ƒ'¼êéX(cid:27)˜(cid:10)(cid:129)#ïÄ(J, (cid:129)(cid:0)0(cid:11)A‡)Ûê
Ø¥(cid:27)¯K¿Jј(cid:10)#(cid:27)¯K. äN5`, (cid:29)Ö̇ŠXeSü:
1˜Ù̇{ü0(cid:11)Smarandache¼ê9–Smarandache¼ê(cid:27)(cid:18)µÚ½Â; 1(cid:19)
Ù̇0(cid:11)Smarandache¼ê†–Smarandache¼ê(cid:27)ïÄ(J9˜(cid:10)#(cid:27)¯K; 1n
Ù̇0(cid:11)†Smarandache¼ê9–Smarandache¼êƒ'(cid:27)˜(cid:10)Ù§¼ê(cid:27)ïÄ(
J; 1oÙ̇0(cid:11)SmarandacheS(cid:15)(cid:27)˜(cid:10)ïĤJ; 1ÊÙ̇0(cid:11)êإ٧S
(cid:15)9Dirichlet L-¼ê(cid:27)A‡ïĤJ.
(cid:29)Ö´ÄuSmarandache¼ê9Ùƒ'˜X(cid:15)êدK(cid:27)#(cid:27)ïĤJ(cid:27)®o. 3
(cid:17)¤L§¥, É(cid:20)(cid:10)Ü©+(cid:19)Ç(cid:27)õg(cid:141)(cid:19), édŠöL«š~a(cid:28). ÓžŠö(cid:143)a
(cid:28)Ü(cid:16)ó’ŒÆÄ:ïÄÄ7é(cid:29)Ö(cid:27)]Ï. (cid:129)(cid:0)•¤k(cid:143)(cid:29)ÖŠÑ(cid:0)z(cid:27)P“ÚÆ
)!Ó1Ú?6L«©%(cid:27)a(cid:28). duŠöY²k(cid:129), Ö¥J(cid:157)Ñy†Ø, (cid:129)H2ŒÖ
ö1µ(cid:141)(cid:20).
III
c ó
IV
1˜Ù {0
1˜Ù {0
35(cid:144)k¯K, vk)‰6˜Ö¥, {7ÛêZæͶêØ;[F.Smarandache(cid:19)
ÇJÑ(cid:10)108‡ÿ™)û(cid:27)êدK, ùÚå(cid:10)¯õêØ;[(cid:27),(cid:21). (cid:29)Ö̇0(cid:11)(cid:10)
©¥k'Smarandache¼ê†–Smarandache¼ê9Ùƒ'¯K(cid:27)ïÄÚ#?Ð.
é?¿(cid:20)(cid:18)ênͶ(cid:27)Smarandache¼êS(n)½Â(cid:143)(cid:129)(cid:2)(cid:27)(cid:20)(cid:18)êm¦(cid:26)n m!, =
|
S(n) = min m : m N,n m! .
{ ∈ | }
lS(n)(cid:27)½Â<‚N´íÑXJn = pα1pα2 pαr L«n(cid:27)IO©)ª, @o
1 2 ··· r
S(n) = max S(pαi) .
1 i r{ i }
≤ ≤
ddØJOŽÑS(1) = 1, S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) = 3, S(7) =
7, S(8) = 4, S(9) = 6, S(10) = 5, S(11) = 1, S(12) = 4, S(13) = 13, S(14) = 7, S(15) =
5, S(16) = 6 .
···
é?¿(cid:20)(cid:18)ênͶ(cid:27)–Smarandache¼êZ(n)½Â(cid:143)(cid:129)(cid:2)(cid:27)(cid:20)(cid:18)êm¦(cid:26)n(cid:18)
Øm(m+1), =
2
m(m+1)
Z(n) = min m : m N,n .
∈ 2
(cid:26) (cid:12) (cid:27)
(cid:12)
lZ(n)(cid:27)½ÂN´íÑZ(n)(cid:27)cA‡Š(cid:143): Z(1(cid:12)) = 1,Z(2) = 3,Z(3) = 2,Z(4) =
(cid:12)
7,Z(5) = 4,Z(6) = 3,Z(7) = 6,Z(8) = 15,Z(9) = 8,Z(10) = 4,Z(11) = 10,Z(12) =
8,Z(13) = 12,Z(14) = 7,Z(15) = 5,Z(16) = 31, .
···
'uS(n)ÚZ(n)(cid:27)Žâ5Ÿ, NõÆöÑ?1(cid:10)ïÄ, ¼(cid:26)(cid:10)Ø(cid:8)k(cid:21)(cid:27)(
J. (cid:29)Ö̇Œ7ùüa¼ê9Ùƒ'¼ê(cid:27)(cid:129)#ïÄ?Љ˜‡XÚ(cid:27)8Bo
(. äN/`, (cid:29)Ö̇(cid:157))Smarandache¼ê†–Smarandache¼ê(cid:27)ïÄ(J!
†Smarandache¼ê9–Smarandache¼êƒ'¯KïÄ(JÚSmarandacheS(cid:15)ïÄ
(J(cid:31), (cid:143)êØOÐöJøë(cid:127)](cid:14), (cid:143)(cid:143)ïÄSmarandache¯K(cid:27)ÆöJø(cid:6)(cid:29)©z
(cid:27)(cid:144)B.
1
Smarandache ¼ê9Ùƒ'¯KïÄ
2
1(cid:19)Ù Smarandache¼ê†–Smarandache¼ê
1(cid:19)Ù Smarandache¼ê†–Smarandache¼ê
2.1 'uSmarandache¼ê(cid:27)˜‡ßŽ
é?¿(cid:20)(cid:18)ên, Ͷ(cid:27)F. Smarandache¼êS(n)½Â(cid:143)(cid:129)(cid:2)(cid:27)(cid:20)(cid:18)êm¦(cid:26)n m!.
|
~XS(n)(cid:27)cA‡ŠS(1) = 1, S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) =
3, S(7) = 7, S(8) = 4, S(9) = 6, S(10) = 10, S(11) = 11, S(12) = 4, . 'uS(n)(cid:27){
···
üŽâ5Ÿ, žë(cid:29)©z[1,7], ùpØ2 E.
'uS(n)(cid:27)(cid:141)(cid:29)(cid:143)5Ÿ, NõÆö?1(cid:10)ïÄ, ¼(cid:26)(cid:10)Ø(cid:8)k(cid:21)(cid:27)(J, ~XF.
Luca(cid:19)Ç3©z[2]¥?Ø(cid:10)¼ê
1
A(x) = S(n)
x
n6x
X
(cid:27)þe.(cid:15)O¯K, ‰Ñ(cid:10)A(x)(cid:27)˜‡(cid:22)r(cid:27)þ.(cid:15)O. Óž¦„3©z[3]¥y²(cid:10)?
ê
nα
S(1) S(2) S(n)
n>1 · ···
X
´ýéÂñ(cid:27), ¿JÑ(cid:10)e¡(cid:27)ßÿ:
ßßߎŽŽ: é?¿¢êx > 1, kìCúª
LS(x) lnS(n) = lnx lnlnx+O(1).
≡ −
n6x
X
'uù˜ßŽ, –8q(cid:2)vk<ïÄ, –(cid:8)3yk(cid:27)©z¥„vw(cid:20)k'(Ø. F.
Luca (cid:19)Ç@(cid:143)kŒUy²lnx lnlnx´LS(x)(cid:27)þ., (cid:2)´éJy²LS(x)(cid:27)e.. ù
−
˜¯K´k¿Â(cid:27), –(cid:8)Œ±‡NÑS(n)¼ê(cid:27)éêþŠ©Ù5Ÿ. (cid:143)d, (cid:2)éù˜¯
K, |^Ð(cid:31)9)Û(cid:144){?1ïÄ, Œ±(cid:26)Ñ(cid:31),ØÓ(cid:27)(Ø, =Xe½n:
½½½nnn 2.1 éu?¿¢êx > 1, kìCúª
LS(x) lnS(n) = lnx+O(1).
≡
n6x
X
w,d½nØ=`²ßŽ(cid:27)1(cid:19)‡Ì‘lnlnx´Ø(cid:127)3(cid:27), (cid:143)Ò´`,©z[3]¥(cid:27)ß
ÿ´†Ø(cid:27), Óž(cid:143)‰Ñ(cid:10)LS(x)(cid:27)(cid:20)(L«/ª. (cid:8),XJ|^ƒê©Ù¥(cid:27)(cid:29)(cid:143)(
J, „Œ±(cid:26)(cid:20)(cid:141)°((cid:27)ìCúª, =Ò´
3
Smarandache ¼ê9Ùƒ'¯KïÄ
é?¿(cid:20)(cid:18)êk > 1, k
1 1
LS(s) lnS(n) = lnx+C +O ,
≡ x lnkx
n6x (cid:18) (cid:19)
X
Ù¥C(cid:143)ŒOŽ(cid:27)~ê.
(cid:143)(cid:10)y²½n2.1, I‡e¡(cid:27)A‡Ún.
ÚÚÚnnn 2.1 é?¿ƒêp, kìCúª
lnp
= lnx+O(1),
p
p6x
X
x
lnp = x+O .
lnx
Xp6x (cid:16) (cid:17)
y²:ë(cid:29)©z[4 6].
−
e¡|^dÚn†(cid:26)‰Ñ½n(cid:27)y². Äk‰ÑLS(x)(cid:27)þ.(cid:15)O.
¯¢þ, dS(n)(cid:27)Ð(cid:31)5Ÿ(cid:127), é?¿(cid:20)(cid:18)ênkS(n) 6 n, ¤±dEuler¦Úú
ª(ë(cid:29)©z[6]¥½n3. 1), k
1 1
LS(x)= lnS(n) 6 lnn
x x
n6x n6x
X X
1 x 1 x ln(x+1)
= lnndy 6 lnydy +
x x x
n6xZ1 Z1
X
1 ln(x+1)
=lnx 1+ + .
− x x
=
1
LS(x) = lnS(n) 6 lnx+O(1). (2-1)
x
n6x
X
Ùg, Ó(cid:24)|^þ¡(cid:27)Ún5(cid:15)OS(x) 1 LnS(n) (cid:27)e.. é?¿(cid:20)(cid:18)
≡ x
n6x
ên > 1, w,n–(cid:8)k˜‡ƒÏfp, Ø”(cid:23)n = p n P, u´dS(n)(cid:27)5Ÿ
1
·
S(n) > p
(cid:127)
lnS(n) = lnS(p n ) > lnp.
1
·
l(cid:13)dÙ(cid:127)(cid:27)©(cid:13)ð(cid:31)ª(ë(cid:29)©z[6]½n3.17), k
LS(x)
1
= lnS(n)
x
n6x
X
4