[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).
Jack Brennen
jfb at brennen.net
Thu Aug 12 21:07:41 CEST 2010
A[12,12] == 165009150
A[13,13] == 673932798
Members of the twelfth row:
12164094, 65098240, 68068350, 70390782,
72936448, 82707966, 95025150, 98690560,
112941054, 123884800, 159529984, 165009150
Members of the thirteenth row:
65071998,
110261248, 169674750, 262986750, 270011392,
274184190, 330477568, 397662208, 501798400,
542922750, 582419968, 592497664, 673932798
Member of the fourteenth row (only one so far):
652963840
Jack Brennen 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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>
>
>
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