A075052 Write down squares and cubes in order, as in A002760. The sequence gives the first differences between terms.

1, 3, 4, 1, 7, 9, 2, 9, 13, 15, 17, 19, 21, 4, 19, 25, 27, 20, 9, 31, 33, 35, 19, 18, 39, 41, 43, 28, 17, 47, 49, 51, 53, 55, 57, 59, 61, 39, 24, 65, 67, 69, 71, 35, 38, 75, 77, 79, 81, 47, 36, 85, 87, 89, 91, 81, 12, 95, 97, 99, 101, 103, 40, 65, 107, 109, 111, 113, 115, 11

Offset: 1

Author

Zak Seidov, Oct 07 2002

Keywords

• nonn

Comments

Ordered sequence of squares and cubes is as in A002760.

Examples

			First sorted squares and cubes are: 0, 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100, hence a(1) = 1 - 0 = 1, a(10) = 64 - 49 = 15, a(12) = 100 - 81 = 19.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A002760.

Programs

Mathematica

Module[{nn=70,sq,cb},sq=Range[0,nn]^2;cb=Select[Range[nn]^3,#<=sq[[-1]]&];Join[sq,cb]//Union//Differences] (* Harvey P. Dale, Jun 11 2024 *)

PARI

lista(nn) = {vec = vector(nn, i, i-1); pp = select(i->(issquare(i) || (ispower(i, 3))), vec); for (i=1, #pp-1, print1(pp[i+1] - pp[i], ", "););} \\ Michel Marcus, Oct 03 2013

Python

from math import isqrt
from sympy import integer_nthroot
def A075052(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 min((a+1)**2,(b+1)**3)-m # Chai Wah Wu, Aug 09 2024

Formula

a(n) = A002760(n+1) - A002760(n).