# Array generalizing A031165

Alexander Povolotsky apovolot at gmail.com
Fri Aug 15 00:10:19 CEST 2008

```In regards to references to A115401, listed in both A031165 and also in A115400;
this A115401 can not be found, is it deleted/merged ?

On Thu, Aug 14, 2008 at 5:37 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
> Since I see that A031165 has just been (re)edited, and comments
> already read: "This sequence is the k=3 case of the family of
> sequences a(k,n) = prime(n+k) - prime(n). See A001223 and A031131 for
> k = 1 and 2."  it seems to me worth looking at the array a(k,n) =
> prime(n+k) - prime(n) by value (rather than just by reference). Is
> there anbything worth saying about it, or worth entering it as a seq
> by antidiagonals?
>
> .......|n=1.|.n=2.|.n=3.|.n=4.|.n=5.|.n=6.|.n=7.|.n=8.|.n=9.|.n=10.|.n=11.|.n=12.|.n=13.|.n=14.|.n=15.|.n=16.|.n=17.|.n=18.|.n=19.|.n=20.|.
> k=1.|...1..|...2...|...2...|...4..|...2...|...4...|...2...|...4...|...6..|.....2...|....6....|....4...|....2...|.....4...|....6...|...6.....|....2...|....6....|....4...|...2..|.A001223
> k=2.|...4..|...6...|...6...|...6..|...6...|...6...|..10..|...8...|...8...|...10..|....6....|....6...|..10...|...12...|....8...|...8.....|..10...|.....6....|....8...|..10..|.A031131
> k=3.|...5..|...8...|...8...|..10.|...8...|..10..|..12...|..12..|..14..|...12..|..12....|...10...|..12..|...16...|...14..|...14...|...12..|...12.....|...12...|..12..|.A031165
> k=4.|...9..|..10..|..12...|..12..|.12..|..16..|..14..|...18..|..18..|...14..|..16...|....16...|..18..|...18...|...20..|...18...|...14..|...18.....|...16...|..18..|.new
> k=5.|..11.|..14..|..14...|..16..|.18..|..18..|..20...|..22..|..20...|..18..|...22..|....22...|..20..|...24....|..24..|...20...|...20..|...22......|..22...|..26..|.new
> k=6.|..15.|..16..|..18...|..22..|.20.|...24..|..24...|..24..|..24...|..24..|...28..|....24...|..26..|...28....|..26,.|...26...|...24..|..28.......|..30...|..30.|.new
>
> and so forth.
> The n=1 column is 1,4,5,9,11,15,... which is not in OEIS.
> The main diagonal is 1, 6, 8, 12, 18, 24,... which is not in OEIS
>
> For instance, we get the k=4 case this way
>
> In[3]:= t = Array[Prime, 75]; Drop[t, 4] - Drop[t, -4]
>
> Out[3]= {9, 10, 12, 12, 12, 16, 14, 18, 18, 14, 16, 16, 18, 18, 20,
> 18, 14, 18, 16, 18, 24, 22, 20, 18, 12, 12, 24,
>
>>    24, 28, 26, 22, 20, 20, 24, 18, 22, 22, 18, 24, 20, 18, 18, 20, 30, 30, 30, 22, 16, 14, 22, 24, 24, 28, 20,
>
>>    20, 18, 14, 22, 30, 30, 30, 24, 24, 26, 34, 32, 22, 22, 20, 24, 26}
>
> and the k=5 case
>
> In[4]:= t = Array[Prime, 75]; Drop[t, 5] - Drop[t, -5]
>
> Out[4]= {11, 14, 14, 16, 18, 18, 20, 22, 20, 18, 22, 22, 20, 24, 24,
> 20, 20, 22, 22, 26, 28, 24, 24, 20, 16, 26,
>
>>    28, 30, 30, 36, 24, 26, 26, 28, 24, 28, 24, 28, 26, 24, 20, 30, 32, 34, 32, 34, 28, 18, 24, 28, 30, 30, 30,
>
>>    26, 24, 20, 24, 36, 34, 32, 34, 38, 30, 36, 36, 36, 28, 30, 26, 30}
>
> and n=6
> In[5]:= t = Array[Prime, 75]; Drop[t, 6] - Drop[t, -6]
>
> Out[5]= {15, 16, 18, 22, 20, 24, 24, 24, 24, 24, 28, 24, 26, 28, 26,
> 26, 24, 28, 30, 30, 30, 28, 26, 24, 30, 30,
>
>>    34, 32, 40, 38, 30, 32, 30, 34, 30, 30, 34, 30, 30, 26, 32, 42, 36, 36, 36, 40, 30, 28, 30, 34, 36, 32, 36,
>
>>    30, 26, 30, 38, 40, 36, 36, 48, 44, 40, 38, 40, 42, 36, 36, 32}
>

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