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R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235-267.
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I have verified a(n) > 0 for all n = 2..10^9. See A303234 for numbers of the form x*(x+1)/2 + 2^y with x and y nonnegative integers. See also A303363 for a stronger conjecture.
See also A303363 for a stronger conjectureIn contrast, Crocker proved in 2008 that there are infinitely many positive integers not representable as the sum of two squares and at most two powers of 2.
R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235-267.
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I think the conjecture can be strengthened as follows: Any integer n > 1 can be written as the sum of two triangular numbers, a power of 2 and a power of 4. This has been verified for all n = 2..2*10^8. - Zhi-Wei Sun, Apr 22 2018
See also A303363 for a stronger conjecture.
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I think the conjecture can be strengthened as follows: Any integer n > 1 can be written as the sum of two triangular numbers, a power of 2 and a power of 4. This has been verified for all n = 2..2*10^8. - _Zhi-Wei Sun_, Apr 22 2018
We I have verified a(n) > 0 for all n = 2..3*10^89. See A303234 for numbers of the form x*(x+1)/2 + 2^y with x and y nonnegative integers.
I think the conjecture can be strengthened as follows: Any integer n > 1 can be written as the sum of two triangular numbers, a power of 2 and a power of 4. This has been verified for all n = 2..2*10^8.
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