A269569 Numbers k such that k+s+c is a square, where s is the nearest square to k and c is the nearest cube to k.

0, 2, 4, 8, 29, 37, 45, 56, 68, 80, 92, 99, 115, 235, 257, 279, 413, 593, 751, 791, 831, 909, 955, 1001, 1396, 1450, 1504, 1572, 1961, 2019, 2087, 2500, 2576, 3093, 3634, 3720, 4129, 4223, 4317, 4522, 4624, 4726, 5816, 5928, 6040, 7110, 7232, 7354, 7535, 7665, 8368, 8500

Offset

1,2

Links

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

Example

The square nearest to 8 is 9, the cube nearest to 8 is 8. Because 8+8+9 is a square, 8 is in the sequence.
The square nearest to 29 is 25, the cube nearest to 29 is 27. Because 29+25+27 is a square, 29 is in the sequence.

Maple

filter:= proc(k) local s, ss, sp, c, cc, cp;
  s:= isqrt(k); ss:= s^2;
  if ss < k then sp:= (s+1)^2; if k - ss > sp - k then ss:= sp fi
  elif ss > k then sp:= (s-1)^2; if ss - k > k - sp then ss:= sp fi
  fi;
  c:= iroot(k, 3); cc:= c^3;
  if cc < k then cp:= (c+1)^3; if k - cc > cp - k then cc:= cp fi
  elif cc > k then cp:= (c-1)^3; if cc - k > k - cp then cc:= cp fi
  fi;
  issqr(k+ss+cc)
end proc:
select(filter, [$0..10000]); # Robert Israel, Apr 29 2026

Prog

(Python)
from itertools import count, islice
from math import isqrt
from sympy import integer_nthroot
from sympy.ntheory.primetest import is_square
def A269569_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:is_square(n+(c if (2*n<(c:=(a:=integer_nthroot(n, 3)[0])**3)+(d:=(a+1)**3)) else d)+((m:=isqrt(n))+int(n>m*(m+1)))**2), count(max(startvalue, 0)))
A269569_list = list(islice(A269569_gen(), 50)) # Chai Wah Wu, Apr 30 2026

Crossrefs

Cf. A000290, A000578, A053187, A201053, A238456, A238489.

Sequence in context: A134316 A049461 A103791 * A366080 A018346 A006398

Adjacent sequences: A269566 A269567 A269568 * A269570 A269571 A269572

Keyword

nonn

Author

Alex Ratushnyak, Feb 29 2016

Status

approved