A140972 Numbers k such that arithmetic mean of squares of first k Lucas numbers is an integer.

1, 10, 12, 24, 36, 48, 60, 72, 96, 108, 120, 144, 168, 180, 192, 216, 240, 250, 288, 300, 324, 336, 360, 384, 432, 442, 480, 504, 540, 550, 552, 576, 600, 612, 624, 648, 660, 672, 684, 720, 768, 840, 864, 900, 960, 972, 1008, 1068, 1080, 1104, 1152, 1176, 1200

Offset

1,2

Comments

The root mean square RMS(L(0),...,L(k-1)) is firstly an integer for k = 36.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Formula

k such that (L(0)^2+ L(1)^2+ ... + L(k-1)^2) / k is an integer, where L(i) is the i-th Lucas number.

Example

k = 10: (L(0)^2+...+L(9)^2)/10 = 935.

Mathematica

With[{nn=1200}, Transpose[Select[Thread[{Range[nn], Accumulate[ LucasL[ Range[0, nn-1]]^2]}], IntegerQ[Last[#]/First[#]]&]][[1]]] (* Harvey P. Dale, Jul 15 2012 *)

Prog

(PARI) list(lim) = {my(L1 = 2, L2 = 1, s = 5); print1(1, ", "); for(k = 3, lim, L3 = L1 + L2; s += L3^2; if(!(s % k), print1(k, ", ")); L1 = L2; L2 = L3); } \\ Amiram Eldar, Jul 04 2025

Crossrefs

Cf. A000032, A140480, A140971.

Sequence in context: A235686 A087697 A241177 * A384969 A108901 A372488

Adjacent sequences: A140969 A140970 A140971 * A140973 A140974 A140975

Keyword

easy,nonn

Author

Ctibor O. Zizka, Jul 27 2008

Extensions

a(1) inserted and a(48) onwards added by R. J. Mathar, Aug 04 2008

Status

approved