RMS Numbers (A140480)

[Ctibor.ZIZKA]

so there must be some general sets of numbers (A_1, A_2, ......, A_k) 
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Subject: Re: a discrepancy concerning A063979
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Date: Tue, 8 Jul 2008 09:36:41 -0700
From: "Jonathan Post" <jvospost3 at gmail.com>
To: SeqFan <seqfan at ext.jussieu.fr>
Subject: Partial sum of cab-taxi (or cabtaxi) numbers A047696
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Partial sum of cab-taxi (or cabtaxi) numbers A047696  Smallest
positive number that can be written in n ways as a sum of two (not
necessarily positive) cubes.

1, 92, 820, 2742076, 8759269, 1421534080, 12723732568,
137526572736064, 425047917053529064, 933953175803355750064

Is it just the Law of Small Numbers that only one of these, a(4) =
8759269, is square-free, and most are cube-full?

1 + 91 = 92 = 2^2 * 23

1 + 91 + 728 = 820 = 2^2 * 5 * 41

1 + 91 + 728 + 2741256 = 2742076 = 2^2 * 685519

1 + 91 + 728 + 2741256 + 6017193 =
8759269 = 37 * 236737

1 + 91 + 728 + 2741256 + 6017193 + 1412774811 =
1421534080 = 2^7 * 5 * 37 * 173 * 347

1 + 91 + 728 + 2741256 + 6017193 + 1412774811 + 11302198488 =
12723732568 = 2^3 * 37 * 359 * 119737

1 + 91 + 728 + 2741256 + 6017193 + 1412774811 + 11302198488 + 137513849003496 =
137526572736064 = 2^6 * 37 * 569 * 3803 * 26839

1 + 91 + 728 + 2741256 + 6017193 + 1412774811 + 11302198488 +
137513849003496 + 424910390480793000 =
425047917053529064 = 2^3 * 17 * 37 * 3416563 * 24723379

1 + 91 + 728 + 2741256 + 6017193 + 1412774811 + 11302198488 +
137513849003496 + 424910390480793000 + 933528127886302221000
= 933953175803355750064 = 2^4 * 11 * 37 * 143420327979630797

If we exclude the initial "1" from the sums, all sums are divisble by 3^2

See also the Boyer announcement (via mathpuzzle.com) confirming
cab-taxi(10). I've sent that in already as a comment to OEIS.