# what defines A113495?

Jack Brennen jb at brennen.net
Wed Feb 20 20:30:51 CET 2008

```Then shouldn't A113495 go like this:

1, 4, 9, 16, 27, 64, 125, 196, ...

rather than the listed:

1, 4, 9, 16, 27, 64, 125, 256, ...

???

After all, 196-125=71 is a prime smaller than
256-125=131.

Max Alekseyev wrote:
> Richard,
>
> I've got an impression that A113495 corresponds to a sequence of 1
> followed by odd primes:
> P: 1 < p1 < p2 < p3 < ...
> such that all partial sums of this sequence are perfect powers and
> each of the primes is the smallest possible (hence, the sequence P can
> be viewed as lexicographically smallest of this type). The sequence
> A113495 is simply the partial sums of P.
>
> In other words, A113495 is the lexicographically smallest subsequence
> of A001597, such that the sequence of first differences of A113495 is
> a subsequence of A000040.
>
> Regards,
> Max
>
> On Wed, Feb 20, 2008 at 10:13 AM, Richard Mathar
> <mathar at strw.leidenuniv.nl> wrote:
>>  A113495 is defined as 1 plus the sum of some first k
>>  odd primes, if the sum equals a perfect power. The examples
>>  for a(6)-a(8) seem to indicate that this is not to be taken
>>  literally, because 11, 37, 67, 131 are certainly not a
>>  sequence of consecutive primes. So my expectation of just taking
>>  all A001597(i)=1+A071148(j), which are very few as it seems, fails.
>>
>>  - is this defined more liberally? How?
>>     Can we partition the perfect powers A001597 into any sum of
>>     1+odd primes to make it into the sequence?
>>  - is this a modified version of A110997 ?
>>  - how does this affect A113759 ?
>>
>>  Richard
>>
>
>

jb> From jb at brennen.net  Wed Feb 20 20:30:59 2008
jb> Date: Wed, 20 Feb 2008 11:30:51 -0800
jb> From: Jack Brennen <jb at brennen.net>
jb> To: seqfan at ext.jussieu.fr
jb> CC: Max Alekseyev <maxale at gmail.com>,
jb>         Richard Mathar <mathar at strw.leidenuniv.nl>
jb> Subject: Re: what defines A113495?
jb>
jb> Then shouldn't A113495 go like this:
jb>
jb>    1, 4, 9, 16, 27, 64, 125, 196, ...
jb>
jb> rather than the listed:
jb>
jb>    1, 4, 9, 16, 27, 64, 125, 256, ...
jb>
jb>
jb> ???
jb>
jb> After all, 196-125=71 is a prime smaller than
jb> 256-125=131.
jb>

We might use only the subset A025475 of powers ("pure prime powers")
instead A001597, to exclude 196 and to explain why 256 appears instead. This
gives a sequence consistent to the current OEIS (although this might
not be easy to extend with my finite computing power):

As a waste product, I could come up with a list of "ways of partitioning n
into 1 plus sums of distinct odd primes", if anyone is interested:
4=[3] +1
6=[5] +1
8=[7] +1
9=[3, 5] +1
11=[3, 7] +1
13=[5, 7] +1
14=[13] +1
15=[3, 11] +1
16=[3, 5, 7] +1
17=[3, 13] +1
17=[5, 11] +1
19=[5, 13] +1
19=[7, 11] +1
21=[3, 17] +1
21=[7, 13] +1
22=[3, 5, 13] +1
22=[3, 7, 11] +1
23=[3, 19] +1
23=[5, 17] +1
25=[5, 19] +1
25=[7, 17] +1
25=[11, 13] +1
26=[3, 5, 17] +1
26=[5, 7, 13] +1
27=[3, 5, 7, 11] +1
27=[3, 23] +1
27=[7, 19] +1
29=[3, 5, 7, 13] +1
29=[5, 23] +1
29=[11, 17] +1
30=[3, 7, 19] +1
30=[5, 7, 17] +1
30=[5, 11, 13] +1
30=[29] +1
31=[7, 23] +1
31=[11, 19] +1
31=[13, 17] +1
32=[3, 5, 23] +1
32=[3, 11, 17] +1
32=[5, 7, 19] +1
32=[7, 11, 13] +1
32=[31] +1
33=[3, 5, 7, 17] +1
33=[3, 5, 11, 13] +1
33=[3, 29] +1
33=[13, 19] +1
...

Richard

Max Alekseyev wrote:
ma> Richard,
ma>
ma> I've got an impression that A113495 corresponds to a sequence of 1
ma> followed by odd primes:
ma> P: 1 < p1 < p2 < p3 < ...
ma> such that all partial sums of this sequence are perfect powers and
ma> each of the primes is the smallest possible (hence, the sequence P can
ma> be viewed as lexicographically smallest of this type). The sequence
ma> A113495 is simply the partial sums of P.
ma>
ma> In other words, A113495 is the lexicographically smallest subsequence
ma> of A001597, such that the sequence of first differences of A113495 is
ma> a subsequence of A000040.
ma>
ma> Regards,
ma> Max
ma>
ma> On Wed, Feb 20, 2008 at 10:13 AM, Richard Mathar
ma> <mathar at strw.leidenuniv.nl> wrote:
ma>rm>  A113495 is defined as 1 plus the sum of some first k
ma>rm>  odd primes, if the sum equals a perfect power. The examples
ma>rm>  for a(6)-a(8) seem to indicate that this is not to be taken
ma>rm>  literally, because 11, 37, 67, 131 are certainly not a
ma>rm>  sequence of consecutive primes. So my expectation of just taking
ma>rm>  all A001597(i)=1+A071148(j), which are very few as it seems, fails.
ma>rm>
ma>rm>  - is this defined more liberally? How?
ma>rm>     Can we partition the perfect powers A001597 into any sum of
ma>rm>     1+odd primes to make it into the sequence?
ma>rm>  - is this a modified version of A110997 ?
ma>rm>  - how does this affect A113759 ?
ma>rm>
ma>rm>  Richard
marm>>

```

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