A121571 Largest number that is not the sum of n-th powers of distinct primes.

6, 17163, 1866000

Offset

1,1

Comments

As stated by Sierpinski, H. E. Richert proved a(1) = 6. Dressler et al. prove a(2) = 17163.

Fuller & Nichols prove T. D. Noe's conjecture that a(3) = 1866000. They also prove that 483370 positive numbers cannot be written as the sum of cubes of distinct primes. - Robert Nichols, Sep 08 2017

Noe conjectures that a(4) = 340250525752 and that 31332338304 positive numbers cannot be written as the sum of fourth powers of distinct primes. - Charles R Greathouse IV, Nov 04 2017

References

W. Sierpinski, Elementary Theory of Numbers, Warsaw, 1964, p. 143-144.

Links

Table of n, a(n) for n=1..3.

R. E. Dressler, Addendum to "A stronger Bertrand’s postulate with an application to partitions", Proc. Am. Math. Soc., 38 (1973), 667.

Robert E. Dressler, Louis Pigno and Robert Young, Sums of squares of primes, Nordisk Mat. Tidskr. 24 (1976), 39-40.

C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18, (2015), #15.10.5.

H. E. Richert, Über Zerfällungen in ungleiche Primzahlen, Math. Z. 52 no. 1 (1948), 342-343.

Formula

a(1) = A231407(3), a(2) = A121518(2438). - Jonathan Sondow, Nov 26 2013

Example

a(1) = 6 because only the numbers 1, 4 and 6 are not the sum of distinct primes.

Crossrefs

Cf. A231407 (numbers that are not the sum of distinct primes).

Cf. A121518 (numbers that are not the sum of squares of distinct primes).

Cf. A213519 (numbers that are the sum of cubes of distinct primes).

Cf. A001661 (integers instead of primes).

Sequence in context: A007702 A112642 A130434 * A123659 A079192 A278369

Adjacent sequences: A121568 A121569 A121570 * A121572 A121573 A121574

Keyword

nonn,hard,more,bref

Author

T. D. Noe, Aug 08 2006

Status

approved