[seqfan] Re: Knife numbers

[Eric Angelini]

Merci Maximilian -- much more clear
like this!

Catapulté de mon aPhone

> Le 21 sept. 2014 à 17:35, "M. F. Hasler" <oeis at hasler.fr> a écrit :
> 
>> On Sat, Sep 20, 2014 at 4:58 PM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
>> Hello SeqFans,
>> If I'm not wrong, the "Knife numbers"
>> (as we might call them) are not in the
>> database.
>> 
>> A "knife number" K can be divided
>> in two chunks P and Q such that P+Q
>> divide K.
>> 
>> K = 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 105, 108, 110, 120, 126, 130, 132,...
>> 
>> Examples:
>> 12 is divisible by 1+2 (result = 4)
>> 126 is divisible by 12+6 (result = 7)
>> 132 is divisible by 1+32 (result = 4)
> 
> Hello Eric & SeqFans,
> 
> I think there are at least 4 possible definitions :
> 
> 1. n=concat(a,b), a,b>0, and (a+b) | n
> (here it is understood that none of a,b,n has leading zeros)
> This is a subsequence of all the following:
> 
> 2. n=concat(a,b), a>0, and (a+b) | n
> (here a must be nonzero and b cannot have leading zeros but may be zero)
> e.g. all n = concat(k,0) = 10k = 0 (mod k+0) are included
> but 102 is not included.
> 
> 3. (n%p + n\p) | n  where 1 < p=10^k <= n and b>0
> ("b may have leading zeros" but must be nonzero)
> (% = mod ; \ = div)
> e.g. 102 = 0 (mod 1+02) is, but not all multiples of 10 are included
> (10,20,..100 are excluded, 110 is allowed because =0 (mod 1+10))
> 
> 4. (n%p + n\p) | n  where 1 < p=10^k <= n
> ("b" may have leading zeros and even be zero)
> (% = mod ; \ = div)
> This includes all of the preceding.
> 
> (PARI)
> is4(n)=n&&for(k=1,#Str(n)-1,n%([1,1]*divrem(n,10^k)||return(1))
> is3(n)=n&&for(k=1,#Str(n)-1,n%([1,1]*divrem(n,10^k))==0&&(n%10^k)&&return(1))
> is2(n)=n&&for(k=1,#Str(n)-1,n%([1,1]*divrem(n,10^k))==0&&(k==1||n%10^k>=10^(k-1))&&return(1))
> is1(n)=n&&for(k=1,#Str(n)-1,n%([1,1]*divrem(n,10^k))==0&&(n%10^k>=10^(k-1))&&return(1))
> 
> for(n=1,999,is4(n)&&print1(n",")) \\ https://oeis.org/draft/A247624
> 12,18,21,24,27,36,42,45,48,54,63,72,81,84,105,108,110,126,132,144,147,162,168,189,198,216,220,231,243,264,288,297,324,330,396,405,...
> 
> for(n=1,999,is3(n)&&print1(n","))\\https://oeis.org/draft/A247625
> 12,18,21,24,27,36,42,45,48,54,63,72,81,84,102,105,108,110,126,132,144,147,162,168,189,198,201,204,207,209,216,220,231,243,264,288,...
> 
> for(n=1,999,is1(n)&&print1(n","))\\https://oeis.org/draft/A2476246
> 10,12,18,20,21,24,27,30,36,40,42,45,48,50,54,60,63,70,72,80,81,84,90,100,102,105,108,110,120,126,130,132,140,144,147,150,160,162,...
> 
> M.
> 
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