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

1,2

Comments

Ordered sequence of squares and cubes is as in A002760.

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A002760.

Sequence in context: A053289 A076412 A053707 * A111516 A210636 A116392

Adjacent sequences: A075049 A075050 A075051 * A075053 A075054 A075055

Keyword

nonn

Author

Zak Seidov, Oct 07 2002

Status

approved