[seqfan] Re: Differences of consecutive primes

[Maximilian Hasler]

If you mean a(n) = the gap/2 when a "hole" (definition?) is filled, the
sequence is different (I think - even if the following list contains some
more cases, maybe your sequence is a subsequence of the following):

? {o=2;g=0;l=1;forprime(p=3,10^7, bittest(g,(-o+o=p)\2) & next;
a=[p,(p-precprime(p-1))/2]); g+=1<<((p-precprime(p-1))\2); l==a[2] | next;
while(bittest(g,l++),); print1( "p",[primepi(a[1])]":"a"->", l-1,", ")) }

p[3]:[5, 1]->1, p[5]:[11, 2]->2, p[10]:[29, 3]->3, p[25]:[97, 4]->4,
p[35]:[149, 5]->5,
p[47]:[211, 6]->7, p[283]:[1847, 8]->12, p[368]:[2503, 13]->13,
p[430]:[2999, 14]->14,
p[591]:[4327, 15]->15, p[739]:[5623, 16]->17, p[1184]:[9587, 18]->18,
p[3303]:[30631, 19]->22, p[7971]:[81509, 23]->27, p[8029]:[82129, 28]->31,
p[8689]:[89753, 32]->32, p[14863]:[162209, 33]->34, p[15784]:[173429,
35]->36,
p[34203]:[404671, 37]->39, p[44774]:[542683, 40]->43, p[44904]:[544367,
44]->45,
p[73322]:[927961, 46]->46, p[85788]:[1101071, 47]->50, p[110225]:[1444411,
51]->53, p[165327]:[2238931, 54]->57, p[402885]:[5845309, 58]->61,
p[460884]:[6752747, 62]->66, p[474030]:[6958801, 67]->69,
p[515911]:[7621399, 70]->70,

(The 2nd element of the list [P,G] is the gap/2 that has been found, and
after the arrow, up to where the list of gaps is then complete, so the next
G will be this + 1.)
My p[739] and p[1184] above correspond to your K=738 and K=1183.

Maximilian

On Fri, Jan 14, 2011 at 4:46 AM, Veikko Pohjola <veikko at nordem.fi> wrote:

> Maybe I was not clear enough in presenting what I am searching for.
>
> I give two more examples
>
> 738<=K<=1182 -> {1,2,3,...,15,16,17} -> a(6)=17
> 1183<=K<=1830 -> {1,2,3,...,16,17,18} -> a(7)=18
>
>
>
> The Mathematica program of Harvey is ok but gives actually the results
> above. Thus, for instance, Range[n], {n, 31, 34} gives {1,2,3,4,7} and
> Range[n], {n, 35, 46} gives {1,2,3,4,5,7}, which means that there is a gap
> in the K-values from 30 to 45 which does not produce perfect sequences of
> small integers, which are what I am looking for. Also the existence of the
> gaps from 99 to 737 and from 1831 to somewhere are supported by the program.
>
> I am still wondering would there be a continuation for the sequence 1, 2,
> 3, 4, 7, 17, 18,.
> Veikko
>
>
>
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