A167224 -id:A167224 - cp's OEIS frontend

A167222 Irregular triangle read by rows: T(n,k) = n^3 - k^2 with 0 <= k <= A077121(n).

0, 1, 0, 8, 7, 4, 27, 26, 23, 18, 11, 2, 64, 63, 60, 55, 48, 39, 28, 15, 0, 125, 124, 121, 116, 109, 100, 89, 76, 61, 44, 25, 4, 216, 215, 212, 207, 200, 191, 180, 167, 152, 135, 116, 95, 72, 47, 20, 343, 342, 339, 334, 327, 318, 307, 294, 279, 262, 243, 222, 199, 174

Offset: 0

Reinhard Zumkeller, Oct 31 2009

			Triangle begins:
  0;
  1, 0;
  8, 7, 4;
  27, 26, 23, 18, 11, 2;
  64, 63, 60, 55, 48, 39, 28, 15, 0;
  125, 124, 121, 116, 109, 100, 89, 76, 61, 44, 25, 4;
  216, 215, 212, 207, 200, 191, 180, 167, 152, 135, 116, 95, 72, 47, 20;
  ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

Cf. A000290, A000578, A077121.

For primes see A167224.

row(n) = my(c=n^3); vector(1+sqrtint(c), i, c-(i-1)^2); \\ Michel Marcus, May 28 2024

A161681 Primes that are the difference between a cube and a square (conjectured values).

Original entry on oeis.org

2, 7, 11, 13, 19, 23, 47, 53, 61, 67, 71, 79, 83, 89, 107, 109, 127, 139, 151, 167, 191, 193, 199, 223, 233, 239, 251, 271, 277, 293, 307, 359, 431, 433, 439, 463, 487, 499, 503, 547, 557, 587, 593, 599, 631, 647, 673, 683, 719, 727, 769, 797, 859, 887, 919

Offset: 1

Cino Hilliard, Jun 16 2009

nonn

The primes found among the differences are sorted in ascending order and unique primes are then extracted. I call this a "conjectured" sequence since I cannot prove that somewhere on the road to infinity there will never exist an integer pair x,y such that x^3-y^2 = 3,5,17,..., missing prime. For example, testing x^3-y^2 for x,y up to 10000, the count of some duplicates are:
duplicate,count
7,2
11,2
47,3
431,7
503,7
1999,5
28279,11
Yet for 3,5,17,29,... I did not find even one.
[Comment from Charles R Greathouse IV, Nov 03 2009: 587 = 783^3 - 21910^2, 769 = 1025^3 - 32816^2, and 971 = 1295^3 - 46602^2 were skipped in the original.]
Conjecture: The number of primes in x^3-y*2 is infinite.
Conjecture: The number of duplicates for a given prime is finite. Then there is the other side - the primes that are not in the sequence 3, 5, 17, 29, 31, 37, 41, 43, 59, 73, 97, 101, 103, ... Looks like a lot of twin components here. Do these have an analytical form? Is there such a thing as a undecidable sequence?
Range of A167224. - Reinhard Zumkeller, Oct 31 2009

			3^3 - 4^2 = 15^3 - 58^2 = 11.

R. Zumkeller, Some Examples [From _Reinhard Zumkeller_, Oct 31 2009]

Cf. A000040.

diffcubesq(n) =
{
  local(a,c=0,c2=0,j,k,y);
  a=vector(floor(n^2/log(n^2)));
  for(j=1,n,
    for(k=1,n,
      y=j^3-k^2;
      if(ispseudoprime(y),
        c++;
        a[c]=y;
      )
    )
  );
  a=vecsort(a);
  for(j=2,c/2,
    if(a[j]!=a[j-1],
      c2++;
      print1(a[j]",");
      if(c2>100,break);
    )
  );
}

Integers x,y such that x^3-y^2 = p where p is prime. The generation bound is 10000.

Extended and edited by Charles R Greathouse IV, Nov 03 2009

Further edits by N. J. A. Sloane, Nov 09 2009

A167223 Number of primes of the form n^3 - k^2, 0<=k<=A077121(n).

Original entry on oeis.org

0, 0, 1, 3, 0, 3, 3, 3, 7, 1, 3, 6, 9, 6, 7, 11, 1, 20, 13, 5, 19, 11, 8, 15, 15, 0, 17, 22, 11, 22, 16, 7, 39, 28, 8, 29, 1, 12, 31, 22, 16, 46, 33, 13, 32, 30, 13, 58, 43, 0, 47, 22, 28, 49, 39, 20, 47, 51, 18, 44, 32, 21, 84, 63, 0, 70, 38, 28, 113, 45, 23, 43, 66, 46, 52, 63, 28, 78

Offset: 0

Reinhard Zumkeller, Oct 31 2009

nonn

Number of terms per row in the table of A167224;

a(n) <= A077121(n).

			a(3) = #{27-4, 27-16, 27-25} = #{23, 11, 2} = 3;
a(4) = #{} = 0;
a(5) = #{125-16, 125-36, 125-64} = #{109, 89, 61} = 3.

R. Zumkeller, Table of n, a(n) for n = 0..500

cp's OEIS Frontend

A167222 Irregular triangle read by rows: T(n,k) = n^3 - k^2 with 0 <= k <= A077121(n).

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A161681 Primes that are the difference between a cube and a square (conjectured values).

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A167223 Number of primes of the form n^3 - k^2, 0<=k<=A077121(n).

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