RMS - sequence

[Andrew Weimholt]

Ctibor's "RMS numbers" inspired me to contemplate an entirely different kind
of RMS sequence...

a(n) is the smallest unused number such that the RMS of a(0) thru a(n)
is an integer...

1, 7, 5, 11, 28, 14, 182, 70, 2, 66,
1518, 462, 1540, 616, 296, 600, 1950, 750, 10, 730,
2336, 876, 2436, 996, 3154, 1162, 5698, 210, 1554, 3234,
1638, 5382, 1872, 23088, 4368, 5934, 201, 4359,

The actual RMS values are...

1, 5, 5, 7, 14, 14, 70, 70, 66, 66,
462, 462, 616, 616, 600, 600, 750, 750, 730, 730,
876, 876, 996, 996, 1162, 1162, 1582, 1554, 1554, 1638,
1638, 1872, 1872, 4368, 4368, 4419, 4359, 4359,

Another version requires that the sequence always increase.
a(n) is the smallest number, larger than the previous, such that the
RMS of a(0) thru a(n) is an integer...

1, 7, 25, 33, 84, 294, 462, 750, 1155, 1705,
2431, 3017, 18130, 19684, 22052, 28996, 36907,

The actual RMS values of the second version are...

1, 5, 15, 21, 42, 126, 210, 330, 495, 715,
1001, 1295, 5180, 7252, 9028, 11356, 14195,

Are these acceptable under the Summer Rules?

Andrew