# [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 20:24:17 CEST 2010

```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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```