A374754 a(n) is the difference between the sum of the squares and the sum of the cubes for the n first terms of A002760.

0, 0, 4, -4, 5, 21, 46, 19, 55, 104, 104, 185, 285, 406, 281, 425, 594, 790, 574, 799, 1055, 1344, 1668, 1325, 1686, 2086, 2527, 3011, 2499, 3028, 3604, 4229, 4905, 4905, 5689, 6530, 7430, 8391, 7391, 8415, 9504, 10660, 11885, 13181, 11850, 13219, 14663, 16184

Offset: 1

Author

Felix Huber, Jul 28 2024

Keywords

• sign

Comments

For A002760(n) <= k < A002760(n+1), the difference between the sum of the squares and the sum of the cubes in the first k nonnegative integers is a(n).

Examples

			a(7) = a(6) + A002760(7) = 21 + 1*25 = 46, since 25 is a square but not a cube.
a(8) = a(7) - A002760(8) = 46 + (-1)*27 = 19, since 27 is a cube but not a square.
a(11) = a(10) + A002760(11) - A002760(11) = 104 + 0*64 = 104, since 64 is a square and a cube.
The difference between the sum of the squares and the sum of the cubes in the first 24 nonnegative integers is a(6) = 21, because A002760(6) = 16 <= 24 < A002760(7) = 25.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000330 (sum of squares), A000537 (sum of cubes), A001014 (sixth powers), A002760 (squares and cubes), A061023, A087285, A087286.

Programs

Maple

isA374754:=proc(k)
   option remember;
   if k=0 then 0
   elif issqr(k) and not type(root(k,3),integer) then procname(k-1)+k;
   elif type(root(k,3),integer) and not issqr(k) then procname(k-1)-k;
   else procname(k-1)
   fi;
end proc;
A374754:=k->
   if k=0 then 0
   elif isA374754(k)<>isA374754(k-1) or type(root(k,6),integer) then isA374754(k)
   fi;
seq(A374754(k),k=0..1521);

PARI

lista(nn) = my(v = select(x->issquare(x) || ispower(x, 3), [0..nn]), s=0, w = vector(#v)); for (i=1, #v, if (issquare(v[i]), s += v[i]); if (ispower(v[i], 3), s -= v[i]); w[i] = s;); w; \\ Michel Marcus, Aug 04 2024

Python

from math import isqrt
from sympy import integer_nthroot
def A374754(n):
    def f(x): return n-1+x+integer_nthroot(x,6)[0]-(b:=integer_nthroot(x,3)[0])-(a:=isqrt(x)), a, b
    m = n-1
    k, a, b = f(n-1)
    while m != k:
        m = k
        k, a, b = f(k)
    return a*(a+1)*((a<<1)+1)//3-((b*(b+1))**2>>1)>>1 # Chai Wah Wu, Aug 09 2024

Formula

a(1) = 0. For n >= 2, a(n) = a(n-1) + f*A002760(n) where f = 1 if A002760(n) is a square but not a cube, f = -1 if A002760(n) is a cube but not a square and f = 0 if A002760(n) is a square and a cube.