[seqfan] Re: Rank of a prime (Collatz/Syracuse)
Jacques Tramu
jacques.tramu at echolalie.com
Sun Feb 1 04:40:24 CET 2009
> Eric Angelini wrote:
> the idea behind my question was this one -- I'm trying to produce a
> little "number game" like the Collatz-one you all know:
> http://mathworld.wolfram.com/CollatzProblem.html
>
> a) start with an integer (here, "6")
> b) decompose it in prime factors (small --> big): "6" = 2.3
> c) write under each prime factor it's rank in the primes succession:
> 2.3
> 1 2 (because 2 is the 1st prime, and 3 is the 2nd prime)
> d) concatenate the ranks to produce a new integer: "12"
> e) start again the procedure from there
>
> REMARK: it may happen that the number obtained via concatenation
> is itself prime; simply replace it by it's own rank in the primes
> succession -- and go on with the procedure from there.
>
Let's call ea(n) this sequence - n= starter
I found the following :
->ea(4)
4 11 5 3 2 1
-> ea(5)
5 3 2 1
-> ea(6)
6 12 112 11114 1733 270 12223 7128 11122225
33991010 13913661 2107998 12222775 33910130
131212367 56113213 6837229 4201627 266366 112430 131359 7981
969 278 134 119 47 15 23 9 22 15
-> ea(7)
7 4 11 5 3 2 1
-> ea(8)
8 111 212 1116 112211 52626 124441 28192 11111152
111165448 1117261018 1910112963 252163429 42205629
2914219 454002 127605 231542 110938 15631 44510 13605 23155
3582 12246 12637 1509 296 11112 111290 131172 1127117
76613 9470 13161 21328 11
111114 14142115 3625334 1125035 348169 78151 11369 1373 220
1135 349 70 134 119 47 15 23 9 22
15
-> ea(9)
9 22 15 23 9
-> ea(10)
10 13 6 12 112 11114 1733 270 12223 7128
11122225 33991010 13
913661 2107998 12222775 33910130 131212367 56113213
6837229 4201627 266366 112430 131359 7981 969 278 134
119 47 15 23 9 22 15
-> ea(1000)
1000 111333 271217 23744 111111416 111668035 38123416
111333214 111134771 52
041017 4504142 1193600 11111113374 1291293111 210659941
11635952 111158582 112114318
13341506 1455641 512354 182534 151042 17438 11087 1344
11111124 11285010 122351376
11112431073 272684284 117283810 134146302 121044136
1118132200 111331051208 11
1654243638 14366699733 2225493779 81368636 112001924
112921719 25911084 11
2331838 128221488 111122262170 13566196845 222320108177
8863086101 405548474 116871062
145241063 5289904 111182002 14821986 12181067
902891 522175 332349 234139 20756 11
691 22284 1122114 1242932 1126858 1141559 95098 17407 6627
21515 3667 844 1147 1112 11
134 1862 1448 11142 122114 16152 1112122 155186 111368
1111646 145756 116409 24089 6729 23
34 1277 206 127 31 11 5 3 2 1
-> ea(23)
23 9 22 15 23
-> ea(57233) - can somebody compute it ?
->ea(2^100) - can somebody compute it ?
So the question is :
does any sequence terminates in "1" or with the" 15 23 9 22 15 loop" ?
or 25 - ea(25) is 25 33 25
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GBnums code:
void ea (n)
{
mpz u[] ; // factors
mpz tr[]; // sequence
print(n);
while(n > 1)
{
lfactors(u,n); // factorize into u
vmap(u,pi); // replace factors by rank
n = catv(u); // concatenate
print(n);
if(vsearch(tr,n) > 0) break; // loop found
vpush(tr,n); // remember n
}
println('');
}
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http://www.echolalie.com/gbnums
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