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

6, 17163, 1866000

Offset: 1

Author

T. D. Noe, Aug 08 2006

Keywords

• nonn
• hard
• more
• bref

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

Examples

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

References

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

Links

• 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.

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).

Formula

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