# Questions on sequence: "3-analogue of A035930"

Jonathan Post jvospost3 at gmail.com
Sun Jul 6 03:12:27 CEST 2008

Is this (base 10) sequence worth submitting?

Maximal product of three numbers whose concatenation is n.

Example: a(12345) = MAX{1*2*345 = 345, 1*23*45 = 1035, 12*3*45 = 1260,
12*34*5 = 2040, 123*4*5 = 2460} = 2460.

What should the offset be?  Offset 100,12?  Offset 1000,1?

Comments: 3-analogue of A035930. Different from A007954.
First differs from A007954 at 1111 where 1*1*1*1 = 1 but 1 * 1 * 11 = 11.

Cf. A035930  Maximal product of two numbers whose concatenation is n.

jvp> From seqfan-owner at ext.jussieu.fr  Sun Jul  6 03:13:32 2008
jvp> Date: Sat, 5 Jul 2008 18:12:27 -0700
jvp> From: "Jonathan Post" <jvospost3 at gmail.com>
jvp> To: SeqFan <seqfan at ext.jussieu.fr>
jvp> Subject: Questions on sequence: "3-analogue of A035930"
jvp>
jvp> Is this (base 10) sequence worth submitting?
jvp>
jvp> Maximal product of three numbers whose concatenation is n.
jvp>
jvp> Example: a(12345) = MAX{1*2*345 = 345, 1*23*45 = 1035, 12*3*45 = 1260,
jvp> 12*34*5 = 2040, 123*4*5 = 2460} = 2460.
jvp> ...

This example is also a member of the sequence
"Maximal product of three numbers whose concatenation is A007908(n)"
which starts with offset 3 as
3 6
4 144
5 2460
6 37020
7 518490
8 6913536
9 88888824
10 8987647760
11 898873417896
12 89887460308032
13 8988746159198168
14 898874616058088304
15 89887461605956978440
16 8988746160595855868576
17 898874616059585754758712
18 89887519853076216631671504
19 10112345983471074380063043462
20 1033706478310376492184222220560
21 103394243304999081725309035824078
22 10339449049567867768490499292069596
23 1033944930799448734608462233812515114
24 103394493106911129416769088394752960632
25 10339449310719202791630995157197693406150
26 1033944931071949492607051765490200633851668
27 103394493107194979597743126962243203574297186
28 10339449310719497991234944644800996206514742704
29 1033944931071949799156078690426839749209455188222
30 105653995126999815457253410195664152040356694799060

with examples
a(3)=3*2*1
a(4)=4*3*12
a(5)=5*4*123
a(6)=6*5*1234
a(7)=7*6*12345
a(8)=8*7*123456
a(9)=9*8*1234567
a(10)=910*8*1234567
a(11)=91011*8*1234567
a(12)=9101112*8*1234567
a(13)=910111213*8*1234567
a(14)=91011121314*8*1234567
a(15)=9101112131415*8*1234567
a(16)=910111213141516*8*1234567
a(17)=91011121314151617*8*1234567
a(18)=8*910111213141516171*12345678
a(19)=9*91011121314151617181*12345678
a(20)=920*91011121314151617181*12345678
a(21)=92021*91011121314151617181*12345678
a(22)=9202122*91011121314151617181*12345678
a(23)=920212223*91011121314151617181*12345678
a(24)=92021222324*91011121314151617181*12345678
a(25)=9202122232425*91011121314151617181*12345678
a(26)=920212223242526*91011121314151617181*12345678
a(27)=92021222324252627*91011121314151617181*12345678
a(28)=9202122232425262728*91011121314151617181*12345678
a(29)=920212223242526272829*91011121314151617181*12345678
a(30)=930*92021222324252627282*1234567891011121314151617181

The 2-factor analog is
"Maximal product of two numbers whose concatenation is A007908(n)"
a(n)= A035930(A007908(n)) offset 2, b-file like
2 2
3 36
4 492
5 6170
6 74070
7 864192
8 9876536
9 111111102
10 11234566980
11 1123592500458
12 112359398193936
13 11235939979887414
14 1123593998161580892
15 112359399816343274370
16 11235939981634524967848
17 1123593998163452706661326
18 112359399816345270888354804
19 11235939981634527089070048282
20 1135802459730231609019487806520
21 113606171898734394449545964612801
22 11360644350367041689623507796858082
23 1136064463431765662218141005173003363
24 113606446372806195606081012056939148644
25 11360644637311483757883379238547705293925
26 1136064463731180474553504213008938471439206
27 113606446373118080788683478601169329237584487
28 11360644637311808113436248808428329720003729768
29 1136064463731180811379427349682155490110769875049
30 114814813864034282216100397918579736761554943372260

with examples
a(2)=2*1
a(3)=3*12
a(4)=4*123
a(5)=5*1234
a(6)=6*12345
a(7)=7*123456
a(8)=8*1234567
a(9)=9*12345678
a(10)=910*12345678
a(11)=91011*12345678
a(12)=9101112*12345678
a(13)=910111213*12345678
a(14)=91011121314*12345678
a(15)=9101112131415*12345678
a(16)=910111213141516*12345678
a(17)=91011121314151617*12345678
a(18)=9101112131415161718*12345678
a(19)=910111213141516171819*12345678
a(20)=920*1234567891011121314151617181
a(21)=92021*1234567891011121314151617181
a(22)=9202122*1234567891011121314151617181
a(23)=920212223*1234567891011121314151617181
a(24)=92021222324*1234567891011121314151617181
a(25)=9202122232425*1234567891011121314151617181
a(26)=920212223242526*1234567891011121314151617181
a(27)=92021222324252627*1234567891011121314151617181
a(28)=9202122232425262728*1234567891011121314151617181
a(29)=920212223242526272829*1234567891011121314151617181
a(30)=930*123456789101112131415161718192021222324252627282