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

3970902, 39022919, 102132857, 110468517, 368495972, 391099413, 395147912, 421129348, 452808398, 776218485, 1005771844, 1485470432, 3038310485, 3263773338, 3485976107, 3640901241, 3758331463, 3784200441, 3944795435

Offset: 1

Author

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

Keywords

• nonn

Comments

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

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.

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

Links

• Donovan Johnson, Table of n, a(n) for n = 1..101 (terms < 4*10^10)
• Zhi-Wei SUN, Posting to Number Theory List, Dec 21 2008

Crossrefs

See A115171 for another version.

Extensions

Definition corrected and a(3)-a(19) from Donovan Johnson, Oct 24 2009