least prime factor(720)? => 2*360.

[Don McDonald]

> Message-ID: <00bd01c304af$a77d4ae0$858239d2 at computer>
> From: "y.kohmoto" <zbi74583 at boat.zero.ad.jp>
> To: <seqfan at ext.jussieu.fr>
> Subject: names of /functions
> Date: Thu, 17 Apr 2003 16:04:55 +0900
> 
>     Hello, seqfans
>     I wonder if the following functions have their names.
> 
>     a function which chooses p power in the factorization of n :
>     n=product p_i^r_i    ->    f_p (n)=p^r , where p=p_i, r=r_i
>     ex.    f_3 (720)=f_3 (2^4*3^2*5)=3^2 , f_7 (15)=1

Yasutoshi,

Pari-gp free program has function --  factor(720).
However, you desire something very specialised.
for example, the power of 41 in 20030417.

What is K-sequence, perhaps kohmoto?

>     a function which deletes  p power in the factorization of n :
>     n=product p_i^r_i    ->    g_p (n)=n/p^r , where p=p_i, r=r_i
>     ex.    g_3 (720)=g_3 (2^4*3^2*5)=2^4*5 , g_7 (15)=15
> 
>     Anyone who knows their names, please tell me them.
>     If we use these functions, the definition of
>     K-sequence becomes much
> easier.
>     a(n)=g_p ([A*a(n-1)+B])
>
don: Submitted new sequence
a(n+1) = greatest prime factor of (a(n)^2+2.), a(1)=2.

>     Yasutoshi
>     http://boat.zero.ad.jp/~zbi74583/another02.htm

Yasutoshi, Does the following help?

My program sericalc4s.bas by don.mcdonald.

I gave it your number (720) and then pressed key "f" repeatedly.

each press of F finds the least prime factor remaining
and (stores) the quotient n deflated by factor x.
 that is..

BEGIN, ENTER START NUMBER (Expression),  Q. END
?720
(result 1st) 720  (Press f) LEAST PRIME FACTOR

2  *  (result 2nd) 360  LEAST PRIME FACTOR
2  *  (3) 180  LEAST PRIME FACTOR
2  *  (4) 90  LEAST PRIME FACTOR
2  *  (5) 45  LEAST PRIME FACTOR
(repeated factor)
3  *  (6) 15  LEAST PRIME FACTOR
3  *  (7) 5  LEAST PRIME FACTOR
PRIME  (8) 5  


my program also prints divisors(720).

my prog also prints continued fraction convergents, powers, roots,...
and about 40 number theory or statistics functions?

(9) 720  ROUND TO NEAREST (+ve) INTEGER  option table D.
DIVISORS OF INTEGER, R% = 720
1 * 720       DSUM= 721 DD%= 2
2 * 360       DSUM= 1083 DD%= 4
3 * 240       DSUM= 1326 DD%= 6
4 * 180       DSUM= 1510 DD%= 8
5 * 144       DSUM= 1659 DD%= 10
DSUM/R% = 2.30416667 ƒCONTINU  
6 * 120       DSUM= 1785 DD%= 12
8 * 90        DSUM= 1883 DD%= 14
9 * 80        DSUM= 1972 DD%= 16
10 * 72       DSUM= 2054 DD%= 18
12 * 60       DSUM= 2126 DD%= 20
15 * 48       DSUM= 2189 DD%= 22
16 * 45       DSUM= 2250 DD%= 24
18 * 40       DSUM= 2308 DD%= 26
20 * 36       DSUM= 2364 DD%= 28
24 * 30       DSUM= 2418 DD%= 30
NO. OF DIVISORS , DD% = 30
  DSUM / number R% = 3.35833333
ABUNDANT.

I think I may have a sequence in oeis
"increasing recursive factorisations
of number 360."

(10) 720  +?9            (add xx originally 1-line calculator)
(11) 729  Optn Y : disp Roots x^(1/y),& powers x^y.

1        729     729
2        27     531441         squareroot and square..
3        9     387420489
index y |  x^(1/y)  |    x^y     
: x=729   CONT <ESC>. 
4        5.19615242     2.82429536E11
5        3.73719282     2.05891132E14
6        3 **(^6th. )    1.50094635E17
7        2.5642542     1.09418989E20
8        2.27950706     7.97664431E22
9        2.08008382     5.8149737E25
10        1.93318204     4.23911583E28
11        1.8207401     3.09031544E31
ƒindex y |  x^(1/y)  |    x^y     
ƒ: x=729   CONT <ESC>.

Q. PROGRAM  SERIcalc4S  E N D.

/ don.mcdonald.
> .Calc.Profile.eisintegsq.Seqfan.YasutoshiK.leastpfact
> .Calc.Profile.eisintegsq.Seqfan.YasutoshiK.LNgFrac

New sequence  2 3 11 41 17 97 3137 13499 60741001 ..

my file > *ram.sumryfact.creditcd1..  Thu,17 Apr 2003.23:18:34
by don.mcdonald.pgm.c22Bc. BBC Basic64.Risc os.Acorn Archimedes comp 1990.

(prob) -- #formula,--  value, --   FACTORS -- ,  (centiseconds).
---   
2  2..=2= 2*all 1cs

             (n) is last/greatest prime factor of previous factorisation.

3  n^2+2   ..=6= 2*3*all 1cs
4  n^2+2..=11= 11*all 2cs
5  n^2+2..=123= 3*41*all 2cs
6  n^2+2..=1683= 3*3*11*17*all 3cs
7  n^2+2..=291= 3*97*all 3cs
8  n^2+2..=9411= 3*3137*all 4cs
9  n^2+2..=9840771= 3*3*3*3*3*3*13499*all 6cs
10  n^2+2..=182223003= 3*60741001*all 50cs
11  n^2+2..=3689469202482003= 3*4211*20627*14158633*all 3s.
12  n^2+2..=200466888428691= 3*3*3*7424699571433prime 3mn

e n d. prog c241Q  24.2.03   close *spool
/don.