# More sequences related to RMS Numbers (A140480)

Andrew Weimholt andrew at weimholt.com
Tue Jul 8 00:53:02 CEST 2008

```I've submitted the following 5 sequences related to A140480.

A141812: RMS Values of the RMS Numbers (A140480)
A141813: Primitive RMS Numbers (not a product of smaller RMS Numbers)
A141814: RMS Values of Primitive RMS Numbers (A141813)
A141815: RMS Numbers with non-unique RMS-Values
A141816: RMS Values for A141815

The b-files for the first three are attached.
I didn't create b-files for the last two, because for A141815, if you
try to include too many terms from the list of known terms of A140480,
you run the risk of erroneously excluding terms. For example, a term
in A140480 may appear to have a unique RMS Value if you only look at
the RMS Values of the first X terms of A140480, but may in fact share
an RMS Value with A140480(X+k).

%I A141812
%S A141812 1, 5, 29, 169, 145, 845, 1105, 2405, 3445, 4901, 2665,
5525, 9425, 12325, 12025,
17225, 24505, 13325, 32045, 55205, 47125, 61625, 69745, 101065, 99905,
77285, 124501,
160225, 186745, 204425, 239425, 160225, 273325, 276025, 292825,
226525, 446165, 456025,
357425, 406445, 456025, 348725, 801125, 582205, 450385, 637325,
493025, 505325, 499525
%N A141812 RMS Values of the RMS Numbers: a(n) is the Root Mean Square
of the divisors of A140480(n)
%e A141812 a(5)=145, because A140480(5)=287, with divisors 1,7,41,287
and RMS(1,7,41,287) = 145.
%Y A141812 Cf. A140480, A141813, A141814, A141815, A141816.
%O A141812 1
%K A141812 ,nonn,
%A A141812 Andrew Weimholt (andrew at weimholt.com), Jul 07 2008

%I A141813
%S A141813 1, 7, 41, 239, 3055, 6665, 9545, 9855, 26095, 34697,
155287, 380511, 421655, 627215,
814463, 823537, 1166399, 1204281, 1256489, 1289441, 1815073, 2265353,
2544697, 2627343,
3132935, 3188809, 3762639, 4647985, 4730879, 4963127, 4995569,
5054015, 5143945, 6542705,
6956927, 9369319, 9963071, 10286649, 10359689, 10707903, 11369969,
11538505, 12158135
%N A141813 Primitive RMS Numbers: RMS Numbers which are not the
product of two smaller RMS Numbers.
%C A141813 RMS Numbers (see A140480) are numbers such that the RMS
(Root Mean Square) of their divisors is an integer. If A and B both
appear in A140480 and GCD(A,B)=1, then A*B is also in A140480. This
sequence contains only those RMS numbers that
are not a product smaller RMS numbers.
%e A141813 The RMS Number 287 is not in the sequence because 287=7*41
and both 7 and 41 are RMS Numbers.
%Y A141813 Cf. A140480, A141812, A141814, A141815, A141816.
%O A141813 1
%K A141813 ,nonn,
%A A141813 Andrew Weimholt (andrew at weimholt.com), Jul 07 2008

%I A141814
%S A141814 1, 5, 29, 169, 1105, 2405, 3445, 2665, 9425, 12325, 55205,
101065, 124501, 160225, 204425,
239425, 292825, 226525, 446165, 456025, 456025, 801125, 637325,
493025, 801125, 801125,
706225, 1185665, 1185665, 1759925, 1770305, 1291225, 1313845, 1185665,
1743625, 6625109,
2497625, 1932125, 3663925, 2010025, 4032145, 2953925, 3112525,
4032145, 4254445, 6326125
%N A141814 RMS Values of the Primitive RMS Numbers: a(n) is the Root
Mean Square of the divisors of A141813(n)
%e A141814 a(5)=1105, because A141813(5)=3305, with divisors
1,5,13,47,65,235,611,3055
and RMS(1,5,13,47,65,235,611,3055) = 1105.
%Y A141814 Cf. A140480, A141812, A141813, A141815, A141816.
%O A141814 1
%K A141814 ,nonn,
%A A141814 Andrew Weimholt (andrew at weimholt.com), Jul 07 2008

%I A141815
%S A141815 627215, 876785, 1289441, 1815073, 2265353, 3132935,
3188809, 4390505, 4647985, 4730879, 6542705, 9026087, 11369969,
12705511, 15203889, 15857471, 17888153, 18253913, 18578719, 20871649,
21026655, 21930545, 22321663, 23630711,
24738935, 27857871, 29160361, 32535895, 32810935, 33116153, 33392983,
34357905, 35378105, 36007615, 38598255,
45761033, 45798935, 49375521, 52867081, 53926695, 60447311, 62336569,
63286535, 74417993, 79589783, 83597345,
86369945, 92879473, 104475337, 106427223
%N A141815 RMS Numbers with non-unique RMS Values
%e A141815 627215 is an RMS Number with an RMS value of 160225.
876785 is an RMS Number with an RMS value of 160225.
Since these two RMS Numbers have the same RMS Value, their RMS Values are
non-unique, and therefore they belong to the sequence.
%Y A141815 Cf. A140480, A141812, A141813, A141814, A141816.
%O A141815 1
%K A141815 ,nonn,
%A A141815 Andrew Weimholt (andrew at weimholt.com), Jul 07 2008

%I A141816
%S A141816 160225, 160225, 456025, 456025, 801125, 801125, 801125,
801125, 1185665, 1185665, 1185665, 2280125, 4032145,
2280125, 4032145, 4005625, 6326125, 6456125, 6569225, 5226065,
4032145, 4005625, 4005625, 5928325, 6326125,
5226065, 10310885, 5928325, 5928325, 5928325, 5928325, 6569225,
6456125, 6569225, 5226065, 16182725, 5928325,
6569225, 13224725, 10310885, 21376225, 22079005, 16182725, 13224725,
20160725, 21376225, 22079005, 23232625,
26130325, 20160725
%N A141816 RMS Values of the RMS Numbers with non-unique RMS Values:
a(n) is the Root Mean Square of the divisors of A141815(n).
%e A141816 a(1)=160225, because A141815(1)=627215, and the Root Mean
Square of the divisors of 627215 is 160225.
%Y A141816 Cf. A140480, A141812, A141813, A141814, A141815.
%O A141816 1
%K A141816 ,nonn,
%A A141816 Andrew Weimholt (andrew at weimholt.com), Jul 07 2008

Andrew
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