A234334 -id:A234334 - cp's OEIS frontend

A234335 Numbers k such that distances from k to three nearest squares are three perfect squares.

0, 5, 65, 160, 325, 1025, 2501, 5185, 5525, 7200, 9605, 16385, 26245, 40001, 40885, 58565, 82945, 93925, 97920, 114245, 153665, 160225, 187200, 202501, 204425, 219385, 262145, 334085, 419905, 430625, 521285, 640001, 707200, 777925, 781625, 869465, 937025, 972725

Offset: 1

Alex Ratushnyak, Dec 23 2013

nonn

A subsequence of A234334.

			5 is in the sequence because the following three are perfect squares: 5-4=1, 5-1=4, 9-5=4.
65 is in the sequence because the following three are perfect squares: 65-64=1, 65-49=16, 81-65=16, where 49, 64, 81 are the three squares nearest to 65.

Cf. A000290, A234334.

#include 
#include 
typedef unsigned long long U64;
U64 isSquare(U64 a) {
  U64 r = sqrt(a);
  return r*r==a;
}
int main() {
  for (U64 n=0; ; ++n) {
    U64 r = sqrt(n);
    if (r*r==n && n)  --r;
    if (isSquare(n-r*r) && isSquare((r+1)*(r+1)-n)) {
      U64 rp = (r+2)*(r+2)-n;
      r = n-(r-1)*(r-1);
      if (n<=1 || rp

ps3Q[n_]:=AllTrue[Take[Sort[Abs[n-(Floor[Sqrt[n]]+{-2,-1,0,1,2})^2]],3],IntegerQ[Sqrt[#]]&]; Join[ {0},Select[Range[2,10^6],ps3Q]] (* Harvey P. Dale, Jul 03 2024 *)

A234348 Numbers k such that both distances from k to two nearest cubes are perfect cubes.

Original entry on oeis.org

0, 1, 152, 189, 513, 728, 5859, 6832, 64008, 68913, 150605, 155736, 345744, 355167, 1062936, 1090999, 1481571, 1520848, 6653933, 6742008, 7665056, 7742709, 9667693, 9796248, 15253056, 15438185, 16582104, 16592023, 16766568, 16776487, 26201448, 26460217, 28672299

Offset: 1

Alex Ratushnyak, Dec 24 2013

nonn

Except a(1)=0, a(n) are numbers k such that both k-x and y-k are perfect cubes, where x and y are two nearest to k cubes: x < k <= y.

			152 is in the sequence because the following are cubes: 152-125=27 and 216-152=64, where 125 and 216 are the nearest to 152 cubes.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000578, A234334.

#include 
#include    // gcc -O3 A234348.c -lgmp
int main() {
  long long in=0;
  mpz_t n, r, i;
  mpz_init(r);
  mpz_init(i);
  mpz_init_set_ui(n, in);
  while (in < (1ULL<<32)) {
    if (mpz_root(r, n, 3) && in)  mpz_sub_ui(r, r, 1);
    mpz_mul(i, r, r);
    mpz_mul(i, i, r);
    mpz_sub(i, n, i);
    if (mpz_root(i, i, 3)) {
      mpz_add_ui(r, r, 1);
      mpz_mul(i, r, r);
      mpz_mul(i, i, r);
      mpz_sub(i, i, n);
      if (mpz_root(i, i, 3))  printf("%llu, ", in);
    }
    mpz_add_ui(n, n, 1);
    if ((++in&0xfffff)==0)  printf(".");
  }
  return 0;
}

A234336 Triangular numbers t such that both distances from t to two nearest squares are perfect squares.

Original entry on oeis.org

0, 1, 45, 153, 325, 10440, 1385280, 2530125, 145462096, 253472356000, 896473314291600, 18598323060963360, 4923539323344237960, 27021247523935843321, 1779312917089890560241, 2355054824151326520405, 21328127890911040269960, 124797500891024855239125

Offset: 1

Alex Ratushnyak, Dec 23 2013

nonn

Triangular numbers in A234334.
Except a(1)=0, a(n) are triangular numbers t such that both t-x and y-t are perfect squares, where x and y are two nearest to k squares: x < t <= y.
The sequence of k's such that triangular(k) is in A234334 begins: 0, 1, 9, 17, 25, 144, 1664, 2249, 17056, 712000, ...

			Triangular(9) = 45 is in the sequence because both 45-36=9 and 49-45=4 are perfect squares, where 36 and 49 are the two squares nearest to 45.

Cf. A000217, A000290, A229909, A234334.

#include 
#include 
typedef unsigned long long U64;
U64 isSquare(U64 a) {
  U64 r = sqrt(a);
  return r*r==a;
}
int main() {
  for (U64 i=0; i<(1ULL<<32); ++i) {
    U64 n = i*(i+1)/2, r = sqrt(n);
    if (r*r==n && n)  --r;
    if (isSquare(n-r*r) && isSquare((r+1)*(r+1)-n))
      printf("%llu, ", n);
  }
  return 0;
}

A280965 Nonsquares whose distances to the two nearest squares are squares.

Original entry on oeis.org

5, 8, 40, 45, 65, 80, 153, 160, 200, 221, 325, 360, 416, 425, 493, 520, 680, 725, 925, 936, 1025, 1040, 1073, 1088, 1305, 1360, 1768, 1800, 1813, 1845, 1961, 2000, 2320, 2385, 2501, 2600, 2925, 3016, 3185, 3200, 3400, 3445, 3848, 3869, 3944, 3965, 4640, 4745, 5185, 5248, 5265, 5328, 5525, 5576, 5785, 5920, 6120

Offset: 1

Emmanuel Vantieghem, Feb 27 2017

nonn

The sequence is infinite because there are terms of it between n^2 and (n+1)^2 whenever 2n+1 is a sum of two squares.

			a(3) = 40 because the two nearest squares are 36 and 49 and 40 - 36 = 4, 49 - 40 = 9 are both squares.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A057653, A234334.

Select[Range[6120], IntegerQ[Sqrt[# - (Floor[Sqrt[#]])^2]] && IntegerQ[Sqrt[(Ceiling[Sqrt[#]])^2 - #]] &]

is(n)=my(k=sqrtint(n)); issquare(n-k^2) && issquare((k+1)^2-n) && n>k^2 \\ Charles R Greathouse IV, Feb 27 2017

list(lim)=my(v=List(),k2,K2,n); for(k=2,sqrtint(lim\1)-1, k2=k^2; K2=(k+1)^2; for(s=1,sqrtint(K2-k2-1), n=k2+s^2; if(issquare(K2-n), listput(v,n)))); k2=sqrtint(lim\1)^2; K2=(sqrtint(lim\1)+1)^2; for(n=k2+1,lim, if(issquare(n-k2) && issquare(K2-n), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Feb 27 2017

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A234335 Numbers k such that distances from k to three nearest squares are three perfect squares.

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A234348 Numbers k such that both distances from k to two nearest cubes are perfect cubes.

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A234336 Triangular numbers t such that both distances from t to two nearest squares are perfect squares.

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C

A280965 Nonsquares whose distances to the two nearest squares are squares.

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