[seqfan] Re: Is this right? Main diagonal A[n.n] of array A[k, n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity).

Jonathan Post jvospost3 at gmail.com
Thu Aug 12 20:48:48 CEST 2010


Thank you, Jack!

Once again proving that it's better to ask on seqfan if one is unsure
that to prematurely submit to OEIS and have to have corrections made
later.

On Thu, Aug 12, 2010 at 11:24 AM, Jack Brennen <jfb at brennen.net> wrote:
> I get a different value for A[5,5]; I get 918.  Pretty sure that 464
> doesn't belong in Row 5.
>
> I get 3, 33, 42, 196, 918, 6640, 24750, 246078, 781248, 6565374,
> 25227774, ...
>
> That's from a PARI/GP script that just iterates through all numbers
> taking bigomega() of each one and looking for duplicates of the same
> value at n and n+2.  Note that beyond A[11,11], as far as I got, the
> brute force approach probably gets beaten badly by some sort of
> sieving approach.  Note that in the range around 10^8 to 10^9, if
> a number has 12 or more prime factors, a pretty good chunk of those
> factors have to be single digit factors, and if two numbers n and
> n+2 both have 12 or more prime factors, the only prime factor they
> can share is 2, and one of the numbers has to have no more than a
> single factor of 2.  So coming up with a sieve that dumps out the
> numbers with bigomega(C) >= 12 and not divisible by 4 would be a
> worthwhile exercise, but one that I don't have time for right now.  :)
>
> I will let the script run for a while and see if it gets me A[12,12]
> or A[13,13]; knowing those would be useful at least to check a
> sieve-based implementation for correctness.
>
>
>
>   Jack
>
> Jonathan Post wrote:
>> 3, 33, 42, 196, 750, ...?
>>
>> n-th natural number m such that m and m+2 are both divisible by
>> exactly n primes (with multiplicity).
>>
>> Is this right?  Main diagonal A[n,n]  of  A[k,n] = n-th natural number
>> m such that m and m+2 are both divisible by exactly k primes (counted
>> with multiplicity).
>>
>> Row 1 = A001359 = the lesser of twin primes.
>> 3, 5, 11, 17, 29, 41, 59, 71, 101, 107
>>
>> Row 2 = A092207  = Numbers n such that n and n+2 are semiprimes. .
>> 4, 33, 49, 55, 85, 91, 93, 119, 121, 141
>>
>> Row 3 = A180117 = m and m+2 are both divisible by exactly 3 primes
>> (counted with multiplicity).
>> 18,28,42,50,66,68,76,114,170,172,186,188,236,242
>>
>> Row 4 = A180150 = m and m+2 are both divisible by exactly 4 primes
>> (counted with multiplicity).
>> 54,88,150,196,232,248,294,306,328,340,342,348,460,488,490,568,570,664
>>
>> Row 5 = A180151 = m and m+2 are both divisible by exactly 5 primes
>> (counted with multiplicity).
>> 270,464,592,700,750,918,1240,1638,1648,1672
>>
>> I would appreciate, if anyone is interested, checking each of the rows
>> as above (I see that A092207 Numbers n such that n and n+2 are
>> semiprimes. replaced a defective recent seq that claimed to be the
>> same but missed the value 49).
>>
>> Then perhaps extend any row(s), add rows for k>5, and checking thr
>> main diagonal as I have tried to generate "by hand" from unedited
>> data.
>>
>>
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>
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