A121571 - OEIS

%I #45 Feb 13 2022 04:47:27

%S 6,17163,1866000

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

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

%C 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

%C 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

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

%H R. E. Dressler, <a href="https://doi.org/10.1090/S0002-9939-1973-0309842-8">Addendum to "A stronger Bertrand’s postulate with an application to partitions"</a>, Proc. Am. Math. Soc., 38 (1973), 667.

%H Robert E. Dressler, Louis Pigno and Robert Young, <a href="http://www.jstor.org/stable/43774462">Sums of squares of primes</a>, Nordisk Mat. Tidskr. 24 (1976), 39-40.

%H C. Fuller and R. H. Nichols Jr., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Fuller/fuller2.html">Generalized Anti-Waring Numbers</a>, J. Int. Seq. 18, (2015), #15.10.5.

%H H. E. Richert, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002381303">Über Zerfällungen in ungleiche Primzahlen</a>, Math. Z. 52 no. 1 (1948), 342-343.

%F a(1) = A231407(3), a(2) = A121518(2438). - _Jonathan Sondow_, Nov 26 2013

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

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

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

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

%Y Cf. A001661 (integers instead of primes).

%K nonn,hard,more,bref

%O 1,1

%A _T. D. Noe_, Aug 08 2006