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 A144642 Positive numbers that are not the sum of a triangular number, a square and a Fibonacci number. 4

%I

%S 3970902,39022919,102132857,110468517,368495972,391099413,395147912,

%T 421129348,452808398,776218485,1005771844,1485470432,3038310485,

%U 3263773338,3485976107,3640901241,3758331463,3784200441,3944795435

%N Positive numbers that are not the sum of a triangular number, a square and a Fibonacci number.

%C Prompted by a conjecture of Zhi-Wei SUN (now known not to be true) that every number can be written in the stated form.

%C A number n = x^2 + y*(y+1)/2 + F_m if and only if 8*n+1 = 8*x^2 + (2*y+1)^2 + 8*F_m, if and only if 8*n + 1 - 8*F_m = 8*x^2 + (2*y+1)^2 for some m >= 0. Now if M == 1 (mod 8) (e.g. M = 8*n + 1 -8*F_m), then M = 8*x^2 + (2*y+1)^2 if and only if the exact prime-power exponent of each prime p == 5 or 7 (mod 8) which divides M is even.

%C a(10) was found by D. S. McNeil in 12/2008.

%H Donovan Johnson, <a href="/A144642/b144642.txt">Table of n, a(n) for n = 1..101</a> (terms < 4*10^10)

%H Zhi-Wei SUN, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;2446667f.0812">Posting to Number Theory List, Dec 21 2008</a>

%Y See A115171 for another version.

%K nonn

%O 1,1

%A Kurt Foster (drsardonicus(AT)earthlink.net), Jan 25 2009

%E Definition corrected and a(3)-a(19) from _Donovan Johnson_, Oct 24 2009

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Last modified June 3 05:29 EDT 2020. Contains 334798 sequences. (Running on oeis4.)