unsquares in A004215

[Wouter Meeussen]

hi,
numbers that can not be written as x^2+y^2+z^2 with x,y,z integer (incl 0)

7,15,23,28,31,39,47,55,60,63,71,79,87,92,95,103,111,112,119,124,127,135,143,
151,156,159,167,175,183,188,191,199,207,215,220,223,231,239,240,247,252,255,
263,271,279,284,287,295,303,311,316,319,327,335,343,348,351,359,367,368,375,
380,383,391,399,407,412,415,423,431,439,444,447,448,455,463,471,476,479,487,
495,496,503,508,511,519,527,535,540,543,551,559,567,572,575,583,591,599,604,
607,615,623,624,631,636,639,647,655,663,668,671,679,687,695,700,703,711,719,
727,732 etc...
is in EIS as:
Matches (up to a limit of 10) found for  7 15 23 28 31 39 47 55 60 63 71 79
87  :
----------------------
%I A004215 M4349
%S A004215 7,15,23,28,31,39,47,55,60,63,71,79,87,92,95,103,111,112,
%T A004215 119,124,127,135,143,151,156,159,167,175,183,188,191,199,
%U A004215 207,215,220,223,231,239,240,247,252,255,263,271,279,284
%N A004215 Sum of 4 but no fewer nonzero squares.%R A004215 D1 2 261.
%O A004215 1,1%K A004215 nonn%P A004215 /usr/njas/gauss/hisdir/paul48.f
%E A004215 extended 12/95%A A004215 njas, jhc%F A004215 Cf. A000378.
their successive differences can be 2,4,7,8 or 13 :
8,8,5,3,8,8,8,5,3,8,8,8,5,3,8,8,1,7,5,3,8,8,8,5,3,8,8,8,5,3,8,8,8,5,3,8,8,1,7,
5,3,8,8,8,5,3,8,8,8,5,3,8,8,8,5,3,8,8,1,7,5,3,8,8,8,5,3,8,8,8,5,3,1,7,8,8,5,3,
8,8,1,7,5,3,8,8,8,5,3,8,8,8,5,3,8,8,8,5,3,8,8,1,7,5,3,8,8,8,5,3,8,8,8,5,3,8,8,
8,5,3,8,8,1,7,5,3,8,8,8,5,3,8,8,8,5,3,8,8,8,5,3,8,8,1,7,5,3,8,8,8,5,3,8,8,8,5,
3,1,7,8,8,5,3,8,8,1,7,5,3,8,8,8,5,3,8,8,8,5,3,8,8,8,5,3,8,8,1,7,5,3,8,8,8

if we prepend a "8",
change each sequence 1,7  into 8 and
   then
we get a simple repeating of the pattern 8,8,8,5,3

if look at the positions where the "1" of the 1,7 sequences are :

17,38,59,73,81,102,123,144,158,166,187,208,229,243,251,272,293,297,315,329,
337,358,379,400,414,422,443,464,485,499,507,528,549,570,584,592,613,634,638,
656,670,678,699,720,741,755,763,784,805,826,840,848,869,890,911,925,933,954,
975,979,997,1011,1019,1040,1061,1082

so the "1" s are seperated by
21,21,14,8,21,21,21,14,8,21,21,21,14,8,21,21,4,18,14,8,21,21,21,14,8,21,21,21,
14,8,21,21,21,14,8,21,21,4,18,14,8,21,21,21,14,8,21,21,21,14,8,21,21,21,14,8,
21,21,4,18,14,8,21,21,21

in THIS sequence again, it is sufficient to replace the 4,18 sequence by 21,
and we get a simple repeat of 21,21,14,8,21
The positions where these  4 's occur is at :
17,38,59 ...
just like the positions of the "1" earlier.
This should (in principle, and irregularities not withstanding) be
sufficient to generate all unsquare numbers efficiently.
proofs or disproofs (or pointer to..) welcomed


wouter.

w.meeussen.vdmcc at vandemoortele.be
tel  +32 (0) 51 33 21 11
fax +32 (0) 51 33 21 75



Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be