Peter Pein petsie at dordos.net
Thu Jan 3 07:31:56 CET 2008

```Hi Neil, seqfans!

first I wish all of you a healthy and successfull new year full of love :-)

No answer to this question appeared in my mailbox (yet).

I used the definition, the first 10 million primes and Mathematica to get via:
In[1]:=
p = Prime[Range[10^7]];
pg = Rest[p] - Most[p];
Print["Record-gaps:"];
pgmax = Rest at Fold[If[Last[#1] < #2, Append[#1, #2], #1] & , {1}, pg]
px = Cases[(Flatten[Position[pg, #1, 1, 2]] & ) /@ pgmax, {x_Integer, y_}];
Print["A053686:"];
Pick[Take[pgmax, Length[px] - 1], Apply[#1[[2]] < #2[[1]] & ,
Partition[px, 2, 1], {1}]]
From In[1]:=
"Record-gaps:"
Out[4]=
{2, 4, 6, 8, 14, 18, 20, 22, 34, 36, 44, 52, 72, 86, 96, 112, 114, 118, 132, 148,
154, 180, 210, 220, 222}
From In[1]:=
"A053686:"
Out[7]=
{2, 4, 6, 14, 34, 36, 52, 86, 132, 154}

which is the start of the below mentioned sequence.

I'd say: "looks good"

Peter

N. J. A. Sloane schrieb:
> Dear Friends,  I was trying to understand this entry.
> I have made a guess at the definition - is it correct? (See the
> first Comment line).  Thanks!  Neil
>
>
> %I A053686
> %S A053686 2,4,6,14,34,36,52,86,132,154,250,252,258
> %N A053686 Record gaps between consecutive primes that repeat at least once before a new record occurs.
> %C A053686 I assume this means: scan the sequence of prime differences (A001223) looking for new records, but append the record difference to the present sequence only if the difference appears at least twice in A001223 before it is beaten by a new record.  - njas, Dec 30 2007
> %C A053686 If this interpretation is correct, then the sequence of primes where these gaps first appear, 3, 7, 23, 113, ..., should be added to the OEIS.
> %C A053686 Next gap occurs between two primes greater than 10000000000. - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 13 2006
> %Y A053686 Cf. A001223, A053695.
> %K A053686 more,nonn
> %O A053686 1,1
> %A A053686 Jeff Burch (gburch(AT)erols.com), Mar 23 2000
> %E A053686 More terms from Naohiro Nomoto (6284968128(AT)geocities.co.jp), Jul 23 2001
> %E A053686 Corrected by Jorge Coveiro (jorgecoveiro(AT)yahoo.com), Jul 24 2006
> %E A053686 More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 13 2006
>
>

```