A318530 Numbers that can be written in more than one way as p^2 + q^3 + r^4 with p, q and r primes.

145, 210, 637, 754, 2317, 2530, 2917, 5218, 5437, 5890, 6447, 6997, 7469, 7653, 7738, 8650, 9333, 11818, 12417, 12796, 14770, 15178, 15197, 15295, 15513, 16349, 16501, 17367, 18389, 19709

Offset

1,1

Comments

Theorem: a number n can be written in at most n^(11/30 + o(1)) ways as p^2 + q^3 + r^4. Proof: a result of Pila (see Links) shows that there are at most x^(1/3 + o(1)) solutions to a cubic curve, hence fix r and there are at most n^(1/6 + o(1)) solutions with p,q <= sqrt(n); multiplying by the n^(1/5) choices for r yields the result. - Charles R Greathouse IV, Mar 09 2026

Links

Robert Israel, Table of n, a(n) for n = 1..10000

J. Pila, Density of integral and rational points on varieties, Astérisque, tome 228 (1995), pp. 183-187.

Formula

n << a(n) << n^(60/43 + o(1)). The exponent is 1/(1/2+1/3+1/4-11/30). - Charles R Greathouse IV, Mar 09 2026

Example

a(1) = 145 = 2^2 + 5^3 + 2^4 = 11^2 + 2^3 + 2^4 .
The first term which can be written in three different ways is 17367 = 23^2 + 13^3 + 11^4 = 113^2 + 13^3 + 7^4 = 131^2 + 5^3 + 3^4 .

Maple

N:= 10^5: # to get terms <= N
P:= select(isprime, [2, seq(i, i=3..floor(sqrt(N)))]):
Psq:= map(`^`, P, 2):
P3:= select(`<=`, map(`^`, P, 3), N):
P4:= select(`<=`, map(`^`, P, 4), N):
V:= Vector(N):
for a in Psq do for b in P3 do for c in P4 do
   s:= a+b+c;
   if s <= N then V[s]:= V[s]+1 fi
od od od:
select(t -> V[t]>=2, [$1..N]); # Robert Israel, Jan 30 2019

Crossrefs

Subsequence of A134657.

Sequence in context: A051414 A176699 A177223 * A158133 A094613 A207058

Adjacent sequences: A318527 A318528 A318529 * A318531 A318532 A318533

Keyword

nonn

Author

Philippe Guglielmetti, Aug 28 2018

Status

approved