A141326 -id:A141326 - cp's OEIS frontend

A145383 Indices in A050791 for the subsequence A141326.

1, 3, 6, 10, 13, 17, 20, 23, 30, 37, 38, 42, 44, 48, 51, 55, 58, 63, 68, 72, 73, 77, 83, 84, 87, 90, 94, 97, 99, 102, 106, 108, 111, 115, 121, 123, 125, 130, 133, 134, 136, 140, 142, 146, 150, 153, 156, 159, 164, 167, 169, 172

Offset: 1

Lewis Mammel (l_mammel(AT)att.net), Oct 10 2008

The first difference of this sequence is used to define the sequence A145384, which is of particular interest.

Lewis Mammel, Table of n, a(n) for n = 1..122

Cf. A050791, values of z where 1+z^3 = x^3 + y^3.
Cf. A141326, a subsequence of A050791 defined by a simple formula.
Cf. A145384, a sequence based on the first difference of this sequence.

The defining equation is A050791(a(n)) = A141326(n).

A145384 The number of terms of A050791 bracketed by successive terms of A141326.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 2, 2, 6, 6, 0, 3, 1, 3, 2, 3, 2, 4, 4, 3, 0, 3, 5, 0, 2, 2, 3, 2, 1, 2, 3, 1, 2, 3, 5, 1, 1, 4, 2, 0, 1, 3, 1, 3, 3, 2, 2, 2, 4, 2, 1, 2, 4, 2, 0, 1, 2, 3, 1, 1, 1, 3, 0, 3, 1, 0, 3, 1, 1, 4, 2, 2, 1, 3, 3, 1, 2, 0, 3, 2, 5, 1, 1, 3, 6, 2, 4, 1, 0, 5, 2, 2, 2, 2, 3, 2, 3, 3, 0, 1

Offset: 1

Lewis Mammel (l_mammel(AT)att.net), Oct 10 2008

nonn

A141326 is a simply generated subsequence of A050791 and by observation it forms a natural measure of the parent sequence. The first several hundred terms of the parent sequence not belonging to A141326 are bracketed into groups with a small integral number of terms ( including 0 ) by the successive terms of the subsequence, A141326.
a(107),a(108) are the first occurrence of 2 consecutive 0's and a(119),a(120),a(121) are the first occurrence of 3 consecutive 0's. This leads to the following conjecture:
 -> 0 as n ->inf
where  = ( sum m=1,n of a(m) )/n

			0 = number of terms of A050791 preceding the first term of A141326
1 = number of terms of A050791 between the first and 2nd terms of A141326
2 = number of terms of A050791 between the 2nd and 3rd terms of A141326

Lewis Mammel, Table of n, a(n) for n = 1..122

Cf. A145383, A141326, A050791

a(1) = A145383(1) - 1

a(n) = A145383(n) - A145383(n-1) - 1 ; n>1

A143936 Subsequence of A050791, "Fermat near misses", generated by iteration of a linear form derived from Ramanujan's parametric formula for equal sums of two pairs of cubes.

Original entry on oeis.org

5262, 2262756, 972979926, 418379105532, 179902042398942, 77357459852439636

Offset: 1

Lewis Mammel (l_mammel(AT)att.net), Sep 05 2008

nonn

The formulas give an approximately geometric progression of values, z, such that 1 + z^3 = x^3 + y^3, along with the values for x and y. Iteration yields large values of x,y and z presumably unobtainable by exhaustive search.

			1 + 5262^3 = 4528^3 + 3753^3 = 145697644729
1 + 2262756^3 = 1947250^3 + 1613673^3 = 11585457155467377217
1 + 972979926^3 = 837313192^3 + 693875529^3 = 921110304262410135315034777

Charles Edward Sandifer, The Early Mathematics of Leonhard Euler, 2007, pp. 102-103.

Wolfram Mathworld, Diophantine Equation 3rd Powers

Cf. A050791, A141326.

/*
File: form.bc
Usage: bc form.bc
( In UNIX shell, e.g. bash on Cygwin )
*/
define a(x){ return( 321*x^2 + 216*x + 36 ); }
define b(x){ return( sqrt(a(x)) ); }
define n(z){ auto a,x; x=3; a = 215*z+12*b(z)+72 ;
a;b(a); return(v(a)); }
define v(z){ auto a,b,x,y,i,j,k,l;
a = z; b = ( a + b(a) )/2;
a = -a; x=3; y = 1-a*x;
i=a*x+y; j=b+x^2*y; k=b*x+y; l=a+x^2*y;
-a; b; i;j;k;l; i^3+j^3; k^3+l^3;
return ( -a ); }
z=144; v(z) ; z=n(z); z=n(z); z=n(z); /* ... etc. */

In Ramanujan's parametric formula:
(a*x+y)^3 + (b+x^2*y)^3 = (b*x+y)^3 + (a+x^2*y)^3
with
a^2 + a*b + b^2 = x*y^2,
we set x=3, ax+y=1 and obtain a quadratic equation for b in terms of a
( Since 'a' is always negative we write it explicitly as '-a' and solve for positive 'a' )
The surd of the quadratic formula then becomes:
sqrt(321*a^2 + 216*a + 36)
and we require that this be an integer. After finding an initial value of 'a' which satisfies this condition by inspection of the sequence A050791, we use Euler's method to find the bilinear recursion: ( with s_i == sqrt(321*a_i^2 + 216*a_i + 36) )
a_i+1 = 215*a_i + 12*s_i + 72
s_i+1 = 215*s_i + 3852*a_i + 1296
and these yield the values of x,y and z from Ramanujan's formula.

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A145383 Indices in A050791 for the subsequence A141326.

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A145384 The number of terms of A050791 bracketed by successive terms of A141326.

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A143936 Subsequence of A050791, "Fermat near misses", generated by iteration of a linear form derived from Ramanujan's parametric formula for equal sums of two pairs of cubes.

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