cp's OEIS Frontend

A380448 Least primes which can be represented as the sum of distinct positive cubes in exactly n ways, or 0 if no such prime exists.

%I A380448 #38 Jul 02 2025 11:34:55
%S A380448 73,757,953,2521,1801,3257,2953,4013,4139,4789,5347,4481,5669,4663,
%T A380448 6427,6659,5867,6301,6841,7867,7687,7741,10169,7057,7723,7561,9631,
%U A380448 8443,8191,8387,9883,10079,10313,10891,10729,10009,9109,10711,9829,11161,10457,12547,11699,10513,10333,11159,13007
%N A380448 Least primes which can be represented as the sum of distinct positive cubes in exactly n ways, or 0 if no such prime exists.
%C A380448 From _David A. Corneth_, Jun 22 2025: (Start)
%C A380448 a(234) = 0. We have if a(234) > 0 then a(234) > 10^6.
%C A380448 Proof:
%C A380448 Let Q(k, u) be the number of ways to write k as a sum of distinct cubes c where c <= u^3. Then for all m we have Q(k, m+1) >= Q(k, m).
%C A380448 for 39031 <= k <= 633371 we have Q(k, 40) >= 235.
%C A380448 Those are 633371 - 39031 + 1 = 594341 > 41^3 consecutive values for k.
%C A380448 Therefore Q(k, 41) >= 235 for 39031 <= k <= 633371 + 26^3.
%C A380448 We have n^3 + (n+1)^3 > (n+2)^3 for n >= 6. So via induction we have Q(k, u) > 234 for some u for any k >= 39031 which completes the proof.
%C A380448 Similarly Q(k, 40) > 10000 for 100000 <= k <= 500000 which is more than enough to confirm zero's found in the first 10000 terms in a search up to 10^6. (End)
%H A380448 David A. Corneth, <a href="/A380448/b380448.txt">Table of n, a(n) for n = 1..10000</a> (first 233 terms from Zhining Yang)
%e A380448 a(4) = 2521 because 2521 = 4^3 + 9^3 + 12^3 = 1^3 + 4^3 + 5^3 + 10^3 + 11^3= 1^3 +4^3+ 6^3 + 8^3 + 12^3 =4^3 + 6^3 + 8^3 + 9^3 + 10^3 and 2521 is the least prime that can be written as the sum of distinct positive cubes in 4 different ways.
%t A380448 s = CoefficientList[Series[Product[(1 + x^(r^3)), {r, 20}], {x, 0, 8000}], x];
%t A380448 Table[SelectFirst[Flatten@Position[s, k] - 1, PrimeQ], {k, 20}]
%Y A380448 Cf. A003997, A275154, A385094.
%K A380448 nonn
%O A380448 1,1
%A A380448 _Zhining Yang_, Jun 21 2025