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A111151 Numbers not of the form a^2+b^3+c^4+d^5 for a,b,c,d >= 0. 7
15, 23, 55, 62, 71, 471, 478, 510, 646, 806, 839, 879, 939, 1023, 1063, 1287, 2127, 5135, 6811, 7499, 9191, 26471 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

It is conjectured that this list is complete.

Comments from Richard C. Schroeppel:

"I can't put my hand on the book, but I think Vaughn has shown

that every sufficiently large number is of the form a^2+b^3+c^5.

The key is that the sum 1/2 + 1/3 + 1/5 = 31/30 > 1, so the expected

number of representations for a number N is, on average, K * N^1/30.

K is some mess of gamma(fractions). [It's also necessary to check

that there's no modular exclusion.] I once tried to estimate how

large an N was "sufficiently large" and if I remember correctly got around 10^60.

The results for a^2+b^3+c^4+d^5 and a^2+b^3+c^4+d^5+e^6 would

follow immediately, although proving an upper bound is a big

question. I'm not even sure if Vaughn's proof is constructive.

Chris Landauer worked on the 2...6 problem long long ago;

I think his program got up to a million or so.

There was a paper in Math. Comp. about five years ago, about the

problem of 4 cubes. Beeler & I tried to find the empirical last-

unrepresentable number, but couldn't reach it. The Math. Comp. paper found

a likely candidate. It was around 10^13 or 14."

No other n < 10^8. - T. D. Noe, Mar 10 2008

LINKS

Table of n, a(n) for n=1..22.

MATHEMATICA

Complement[Range[10^6], Flatten[Table[a^2 + b^3 + c^4 + d^5, {a, 0, 1000}, {b, 0, 100}, {c, 0, 31}, {d, 0, 15}]]] (* Robert G. Wilson v, Oct 19 2005 *)

CROSSREFS

Cf. A135911 (number of 4-tuples (x, y, z, t) of nonnegative integers such that x^2+y^3+z^4+t^5 = n).

Sequence in context: A171167 A242412 A195036 * A166657 A059144 A274339

Adjacent sequences:  A111148 A111149 A111150 * A111152 A111153 A111154

KEYWORD

nonn,fini

AUTHOR

David W. Wilson, Oct 19 2005

STATUS

approved

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Last modified April 4 20:53 EDT 2020. Contains 333229 sequences. (Running on oeis4.)