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• ## Re: Sphenic tree by factor concatenation on 114

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• ... If my client-server method did not screw up, the sequence of counts continues 23366 33947 48816 69547 97473 135489 186328 252326 338513 448911 589071 Might
Message 1 of 16 , Jun 12, 2011
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> Here are the counts for the first 21 iterations:
> 2 2 2 1 3 7 13 28 39 73 141 264 459 757
> 1220 1937 3054 4700 7173 10766 15803

If my client-server method did not screw up,
the sequence of counts continues

23366 33947 48816 69547 97473 135489
186328 252326 338513 448911 589071

Might James confirm this, from his single-processor data?

David
• ... Here are my counts for the first 37 iterations: 2, 2, 2, 1, 3, 7, 13, 28, 39, 73, 141, 264, 459, 757, 1220, 1937, 3054, 4700, 7173, 10766, 15803, 23366,
Message 1 of 16 , Jun 13, 2011
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> 186328 252326 338513 448911 589071

Here are my counts for the first 37 iterations:

2, 2, 2, 1, 3, 7, 13, 28, 39, 73, 141, 264, 459,
757, 1220, 1937, 3054, 4700, 7173, 10766, 15803,
23366, 33947, 48816, 69547, 97473, 135489,
186328, 252326, 338513, 448911, 589071,
766390, 984903, 1253696, 1578502, 1966106,

with ratios of successive counts, from the last 10 entries,
1.354, 1.342, 1.326, 1.312, 1.301, 1.285, 1.273, 1.259, 1.246,
indicating that we are still well below the shrinking point.

It is clear that James was very generous in allowing
6 possible decimal concatenations of the 3 distinct
prime factors of a sphenic number.

David
• Yes, suspected my one computer was not adequate in any case. I would have had to do some hands-on measurements to be sure, and probably would not have. So I
Message 1 of 16 , Jun 13, 2011
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Yes, suspected my one computer was not adequate in any case. I would have had to do some hands-on measurements to be sure, and probably would not have. So I am glad for this particular response. My one PARI/GP window is done without the dividing line between iterations and has just the first 1.696Million nodes so far (with four 43-digit numbers and current mode at 33 digits with just over 230000).

On Mon Jun 13th, 2011 6:34 PM EDT djbroadhurst wrote:

>
>
>
>> 186328 252326 338513 448911 589071
>
>Here are my counts for the first 37 iterations:
>
>2, 2, 2, 1, 3, 7, 13, 28, 39, 73, 141, 264, 459,
>757, 1220, 1937, 3054, 4700, 7173, 10766, 15803,
>23366, 33947, 48816, 69547, 97473, 135489,
>186328, 252326, 338513, 448911, 589071,
>766390, 984903, 1253696, 1578502, 1966106,
>
>with ratios of successive counts, from the last 10 entries,
>1.354, 1.342, 1.326, 1.312, 1.301, 1.285, 1.273, 1.259, 1.246,
>indicating that we are still well below the shrinking point.
>
>It is clear that James was very generous in allowing
>6 possible decimal concatenations of the 3 distinct
>prime factors of a sphenic number.
>
>David
>
• Definition: A number is sphenic iff it is the product of 3 distinct primes. A sphenic chain is a sequence of sphenic numbers such that each except the first
Message 1 of 16 , Jun 14, 2011
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Definition: A number is sphenic iff it is the product of 3
distinct primes. A "sphenic chain" is a sequence of sphenic
numbers such that each except the first is one of the 6
decimal concatenations of the primes dividing its predecessor.

Example: Here is a chain of length 10:
1: 114 = 3*2*19
2: 3219 = 37*29*3
3: 37293 = 401*31*3
4: 401313 = 11*3*12161
5: 11312161 = 7*53*30491
6: 75330491 = 2887*97*269
7: 288797269 = 23*187409*67
8: 2318740967 = 1097*61*34651
9: 10976134651 = 1163*229*41213
10: 116322941213 [one may continue this chain]
which we may conveniently denote by
[114, 3, 6, 6, 3, 1, 5, 2, 3, 3]
where 114 is the first number and then we indicate which of
the permutations {123, 132, 213, 231, 312, 321} were used.

Here is how generate a chain of length 80, using Pari-GP:

{sphen80 = [114,
3, 6, 6, 3, 1, 5, 2, 3, 3, 4, 5, 3, 5, 5, 6, 3, 6, 3, 1, 3,
3, 4, 5, 3, 4, 4, 6, 3, 1, 1, 1, 1, 2, 1, 1, 5, 6, 6, 2, 5,
2, 2, 6, 3, 3, 1, 4, 1, 5, 3, 6, 3, 5, 4, 3, 2, 6, 5, 2, 4,
3, 2, 3, 5, 3, 4, 6, 1, 1, 3, 5, 2, 6, 4, 4, 3, 1, 6, 2];}

{ischain(s)=local(f,n=s[1],P,t);
P=[[1,2,3],[1,3,2],[2,1,3],[2,3,1],[3,1,2],[3,2,1]];
for(k=1,#s,print1(k": ");n=eval(n);f=factor(n)[,1];
if(n!=f[1]*f[2]*f[3],print("fail");break,
if(k==#s,print(n" has factors "f~),print1(n " = ");
n="";for(j=1,3,t=f[P[s[k+1]][j]];print1(t);
if(j<3,print1("*"),print());n=concat(n,t)))));}

ischain(sphen80);

with takes less than a minute to generate the output in
Sadly, none of the 6 concatenations the 3 primes
from the 80th member yields a sphenic number.

Puzzle: Find a sphenic chain with more than 80 members.

economical format that I used for "sphen80".

• Hello David, ... Many GHz hours later..... {sphen81 = [114, 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6, 6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5,
Message 1 of 16 , Jun 15, 2011
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Hello David,

At 03:19 AM 15/06/2011, djbroadhurst wrote:

>Definition: A number is sphenic iff it is the product of 3
>distinct primes. A "sphenic chain" is a sequence of sphenic
>numbers such that each except the first is one of the 6
>decimal concatenations of the primes dividing its predecessor.
>
>[ snip ]
>Here is how generate a chain of length 80, using Pari-GP:
>
> {sphen80 = [114,
> 3, 6, 6, 3, 1, 5, 2, 3, 3, 4, 5, 3, 5, 5, 6, 3, 6, 3, 1, 3,
> 3, 4, 5, 3, 4, 4, 6, 3, 1, 1, 1, 1, 2, 1, 1, 5, 6, 6, 2, 5,
> 2, 2, 6, 3, 3, 1, 4, 1, 5, 3, 6, 3, 5, 4, 3, 2, 6, 5, 2, 4,
> 3, 2, 3, 5, 3, 4, 6, 1, 1, 3, 5, 2, 6, 4, 4, 3, 1, 6, 2];}
>
>[ snip]
>Puzzle: Find a sphenic chain with more than 80 members.

Many GHz hours later.....

{sphen81 = [114,
3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6,
6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5, 6, 2, 4, 5, 5, 1, 4,
6, 4, 6, 4, 5, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 4, 2, 3, 4, 2,
3, 2, 1, 5, 4, 5, 6, 3, 5, 6, 3, 5, 6, 6, 3, 4, 4, 2, 4, 4];}

Best Regards,

Kevin.

[Non-text portions of this message have been removed]
• ... And then of course, just a few minutes later: {sphen84a = [114, 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6, 6, 4, 2, 6, 2, 2, 4, 4, 3, 2,
Message 1 of 16 , Jun 15, 2011
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At 08:09 PM 15/06/2011, Kevin Acres wrote:
>Hello David,
>
>At 03:19 AM 15/06/2011, djbroadhurst wrote:
>
> >Definition: A number is sphenic iff it is the product of 3
> >distinct primes. A "sphenic chain" is a sequence of sphenic
> >numbers such that each except the first is one of the 6
> >decimal concatenations of the primes dividing its predecessor.
> >
> >[ snip ]
> >Here is how generate a chain of length 80, using Pari-GP:
> >
> > {sphen80 = [114,
> > 3, 6, 6, 3, 1, 5, 2, 3, 3, 4, 5, 3, 5, 5, 6, 3, 6, 3, 1, 3,
> > 3, 4, 5, 3, 4, 4, 6, 3, 1, 1, 1, 1, 2, 1, 1, 5, 6, 6, 2, 5,
> > 2, 2, 6, 3, 3, 1, 4, 1, 5, 3, 6, 3, 5, 4, 3, 2, 6, 5, 2, 4,
> > 3, 2, 3, 5, 3, 4, 6, 1, 1, 3, 5, 2, 6, 4, 4, 3, 1, 6, 2];}
> >
> >[ snip]
> >Puzzle: Find a sphenic chain with more than 80 members.
>
>Many GHz hours later.....
>
> {sphen81 = [114,
> 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6,
> 6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5, 6, 2, 4, 5, 5, 1, 4,
> 6, 4, 6, 4, 5, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 4, 2, 3, 4, 2,
> 3, 2, 1, 5, 4, 5, 6, 3, 5, 6, 3, 5, 6, 6, 3, 4, 4, 2, 4, 4];}

And then of course, just a few minutes later:

{sphen84a = [114,
3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6,
6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5, 6, 2, 4, 5, 5, 1, 4,
6, 4, 6, 4, 5, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 4, 2, 3, 4, 2,
3, 2, 1, 5, 4, 5, 6, 3, 5, 6, 3, 5, 6, 6, 3, 4, 4, 2, 4, 4,
4, 3, 4];}

{sphen84b = [114,
3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6,
6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5, 6, 2, 4, 5, 5, 1, 4,
6, 4, 6, 4, 5, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 4, 2, 3, 4, 2,
3, 2, 1, 5, 4, 5, 6, 3, 5, 6, 3, 5, 6, 6, 3, 4, 4, 2, 4, 4,
4, 3, 3];}

[Non-text portions of this message have been removed]
• ... Congrats! I m now in the low 90s, but do not expect to make it to length 100. David
Message 1 of 16 , Jun 15, 2011
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Kevin Acres <research@...> wrote:

> {sphen84a = [114,
> 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6,
> 6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5, 6, 2, 4, 5, 5, 1, 4,
> 6, 4, 6, 4, 5, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 4, 2, 3, 4, 2,
> 3, 2, 1, 5, 4, 5, 6, 3, 5, 6, 3, 5, 6, 6, 3, 4, 4, 2, 4, 4,
> 4, 3, 4];}
>
> {sphen84b = [114,
> 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6,
> 6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5, 6, 2, 4, 5, 5, 1, 4,
> 6, 4, 6, 4, 5, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 4, 2, 3, 4, 2,
> 3, 2, 1, 5, 4, 5, 6, 3, 5, 6, 3, 5, 6, 6, 3, 4, 4, 2, 4, 4,
> 4, 3, 3];}

Congrats!

I'm now in the low 90s, but do not expect to make it to length 100.

David
• ... In fact, I was able to reach length 105: {sphen105 = [114, 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6, 1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1,
Message 1 of 16 , Jun 15, 2011
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> do not expect to make it to length 100

In fact, I was able to reach length 105:

{sphen105 = [114,
3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6,
1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1, 2, 5, 4, 3, 5, 2, 5, 2,
4, 1, 5, 1, 6, 5, 3, 3, 5, 5, 2, 1, 5, 5, 1, 6, 4, 5, 5, 3,
5, 2, 4, 4, 4, 3, 4, 5, 6, 4, 6, 3, 6, 4, 2, 4, 5, 2, 6, 1,
1, 4, 6, 1, 6, 1, 2, 3, 2, 4, 6, 3, 3, 1, 6, 3, 3, 3, 1, 3,
1, 5, 3, 1];}

contains 8 helpers, obtained by ECM, and then the chain of
is generated in less than 4 minutes.

David
• ... The best I can do at present is length 108: {sphen108 = [114, 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6, 1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6,
Message 1 of 16 , Jun 16, 2011
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> able to reach length 105

The best I can do at present is length 108:

{sphen108 = [114,
3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6,
1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1, 2, 5, 4, 3, 5, 2, 5, 2,
4, 1, 5, 1, 6, 5, 3, 3, 5, 5, 2, 1, 5, 5, 1, 6, 4, 5, 5, 3,
5, 2, 4, 4, 4, 3, 4, 5, 6, 4, 6, 3, 6, 4, 2, 4, 5, 2, 6, 1,
1, 4, 6, 1, 6, 1, 2, 3, 2, 4, 6, 3, 3, 1, 6, 3, 3, 3, 1, 3,
5, 1, 4, 5, 2, 5, 2];}

with input and output in

David
• ... Length 109 is achieved by {sphen109 = [114, 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6, 1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1, 2, 5, 4, 3, 5,
Message 1 of 16 , Jun 20, 2011
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> The best I can do at present is length 108

Length 109 is achieved by

{sphen109 = [114,
3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6,
1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1, 2, 5, 4, 3, 5, 2, 5, 2,
4, 1, 5, 1, 6, 5, 3, 3, 5, 5, 2, 1, 5, 5, 1, 6, 4, 5, 5, 3,
5, 2, 4, 4, 4, 3, 4, 5, 6, 4, 6, 5, 6, 2, 1, 1, 6, 6, 5, 6,
6, 4, 6, 6, 2, 4, 4, 6, 6, 2, 5, 3, 1, 2, 2, 1, 2, 2, 4, 3,
1, 4, 3, 1, 2, 5, 3, 5];}

with input and output in

David
• Hi David, ... Well done for that. Its been windy here and wind = power outage where I live :-) One day I ll get a decent size UPS. Best Regards, Kevin.
Message 1 of 16 , Jun 20, 2011
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Hi David,

At 02:20 PM 21/06/2011, djbroadhurst wrote:

>
> > The best I can do at present is length 108
>
>Length 109 is achieved by
>
> {sphen109 = [114,
> 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6,
> 1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1, 2, 5, 4, 3, 5, 2, 5, 2,
> 4, 1, 5, 1, 6, 5, 3, 3, 5, 5, 2, 1, 5, 5, 1, 6, 4, 5, 5, 3,
> 5, 2, 4, 4, 4, 3, 4, 5, 6, 4, 6, 5, 6, 2, 1, 1, 6, 6, 5, 6,
> 6, 4, 6, 6, 2, 4, 4, 6, 6, 2, 5, 3, 1, 2, 2, 1, 2, 2, 4, 3,
> 1, 4, 3, 1, 2, 5, 3, 5];}

Well done for that. Its been windy here and "wind = power outage"
where I live :-)

One day I'll get a decent size UPS.

Best Regards,

Kevin.
• ... The current record is length 112, achieved by {sphen112a = [114, 3, 6, 6, 3, 2, 2, 1, 3, 3, 2, 2, 1, 5, 2, 3, 2, 1, 6, 4, 4, 2, 4, 4, 2, 5, 4, 2, 3, 4, 2,
Message 1 of 16 , Jun 23, 2011
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Kevin Acres <research@...> wrote:

> > {sphen109 = [114,
....
> Well done for that. Its been windy here and "wind = power outage"
> where I live :-)

The current record is length 112, achieved by

{sphen112a = [114,
3, 6, 6, 3, 2, 2, 1, 3, 3, 2, 2, 1, 5, 2, 3, 2, 1, 6, 4, 4,
2, 4, 4, 2, 5, 4, 2, 3, 4, 2, 1, 3, 3, 6, 5, 4, 3, 1, 5, 4,
2, 5, 1, 1, 3, 4, 5, 5, 5, 5, 6, 3, 6, 1, 6, 1, 4, 3, 1, 6,
4, 6, 3, 1, 3, 2, 1, 2, 2, 6, 3, 2, 3, 5, 4, 6, 6, 5, 2, 2,
2, 3, 4, 6, 5, 2, 3, 3, 2, 6, 5, 1, 2, 6, 1, 4, 3, 5, 3, 3,
3, 4, 1, 4, 2, 2, 1, 2, 6, 3, 2];}

{sphen112b = [114,
3, 6, 6, 3, 2, 2, 1, 3, 3, 2, 2, 1, 5, 2, 3, 2, 1, 6, 4, 4,
2, 4, 4, 2, 5, 4, 2, 3, 4, 2, 1, 3, 3, 6, 5, 4, 3, 1, 5, 4,
2, 5, 1, 1, 3, 4, 5, 5, 5, 5, 6, 3, 6, 1, 6, 1, 4, 3, 1, 6,
4, 6, 3, 1, 3, 2, 1, 2, 2, 6, 3, 2, 3, 5, 4, 6, 6, 5, 2, 2,
2, 3, 4, 6, 5, 2, 3, 3, 2, 6, 5, 1, 2, 6, 1, 4, 3, 2, 3, 3,
4, 2, 2, 3, 2, 2, 1, 4, 5, 1, 1];}

with input and output in