A106322 -id:A106322 - cp's OEIS frontend

A106318 Bhaskara twins: n such that 2n^2 = X^3 and 2n^3 = Y^2.

2, 128, 1458, 8192, 31250, 93312, 235298, 524288, 1062882, 2000000, 3543122, 5971968, 9653618, 15059072, 22781250, 33554432, 48275138, 68024448, 94091762, 128000000, 171532242, 226759808, 296071778, 382205952, 488281250, 617831552

Offset: 1

Lekraj Beedassy, Apr 29 2005

S. S. Gupta, 'Bhaskara Pairs' in 'Science Today' (subsequently renamed '2001'), January 1988, pp. 68, Times of India, Mumbai.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Richard J. Mathar, Construction of Bhaskara Pairs, arXiv:1703.01677 [math.NT], 2017.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A106319, A106320, A106321, A106322.

Cf. A001014.

a106318 = (* 2) . (^ 6)  -- Reinhard Zumkeller, May 27 2015

LinearRecurrence[{7,-21,35,-35,21,-7,1},{2,128,1458,8192,31250,93312,235298},30] (* Harvey P. Dale, May 11 2017 *)

a(n)=2*n^6 \\ Charles R Greathouse IV, Feb 09 2012

a(n) = 2*n^6 = 2*A001014(n).
G.f.: 2*(1+x)*(1+56*x+246*x^2+56*x^3+x^4)/(1-x)^7. - Colin Barker, Apr 18 2012
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Wesley Ivan Hurt, Apr 23 2021

A106320 Values b of a Bhaskara pair (a,b), a<=b, sorted on values of b. A Bhaskara pair (a,b) is such that a^2 + b^2 = X^3 and a^3 + b^3 = Y^2.

Original entry on oeis.org

2, 128, 1250, 1458, 8192, 31250, 80000, 93312, 235298, 524288, 911250, 1062882, 2000000

Offset: 1

Lekraj Beedassy, Apr 29 2005

For the corresponding values of a see A106319. This sequence is guaranteed to not be missing any terms. - Jud McCranie, Jul 04 2013

S. S. Gupta, 'Bhaskara Pairs' in 'Science Today' (subsequently renamed '2001') January 1988 pp. 68, Times of India, Mumbai.

Richard J. Mathar, Construction of Bhaskara Pairs, arXiv:1703.01677 [math.NT], 2017.

Cf. A106319, A106318, A106320, A106321.

A106318 UNION A106322. - R. J. Mathar, Jan 27 2017

Corrected and extended by Jud McCranie, Jul 04 2013

A106319 Values a of a Bhaskara pair (a,b), a<=b, sorted by value of b. A Bhaskara pair (a,b) is such that a^2 + b^2 = X^3 and a^3 + b^3 = Y^2.

Original entry on oeis.org

2, 128, 625, 1458, 8192, 31250, 40000, 93312, 235298, 524288, 455625, 1062882, 2000000

Offset: 1

Lekraj Beedassy, Apr 29 2005

Contains Bhaskara twins (A106318). For corresponding values of b see A106320.

S. S. Gupta, 'Bhaskara Pairs' in 'Science Today' (subsequently renamed '2001'), January 1988, pp. 68, Times of India, Mumbai.

Richard J. Mathar, Construction of Bhaskara pairs, arXiv:1703.01677 [math.NT], 2017.

Cf. A106320, A106322. Non-sorted union of A106318 and A106321.

Corrected and extended by Jud McCranie, Jul 04 2013

A106321 Smaller member of a Bhaskara pair (excluding Bhaskara twins, that is, include only a < b); a Bhaskara pair (a,b) is such that a^2 + b^2 = X^3 and a^3 + b^3 = Y^2.

Original entry on oeis.org

625, 40000, 455625

Offset: 1

Lekraj Beedassy, Apr 29 2005

Ordered by the larger of the pair. For corresponding larger value see A106322.
The sequence is infinite because it contains (as a subsequence) all numbers of the form a = 625*s^6 where s is a positive integer. The pairs are (a,b=2*a) =(625*s^6, 1250*s^6) associated with (X,Y) = (125*s^4, 46875*s^9). - R. J. Mathar, Jan 26 2017
The sequence also contains all numbers of the form a = 3430000*s^6 where s is a positive integer. The pairs are (a,b=3*a) associated with (X,Y) = (49000*s^4, 33614000000*s^9). The sequence also contains all numbers of the form a = 22936954625 *s^6 with b=4*a, all numbers of the form a = 19592846 *s^6 with b=5*a, all numbers of the form a = 19150763710393 *s^6 with b=6*a, all numbers of the form a= 3975350 *s^6 with b=7*a, all numbers of the form a = 3305810795625 * s^6 with b=8*a. - R. J. Mathar, Jan 26 2017

S. S. Gupta, 'Bhaskara Pairs' in 'Science Today' (subsequently renamed '2001') Jan 1988 pp. 68, Times of India, Mumbai.

Cf. A106322, A106318, A106319, A106320.

Corrected and extended Jud McCranie, Jul 04 2013

A282639 Greater value of a coprime pair (x,y) satisfying x^3+y^3=z^2.

Original entry on oeis.org

2, 37, 65, 112, 312, 433, 877, 1064, 1177, 1201, 1617, 1969, 2345, 2480, 2543, 3081, 3482, 4019, 4440, 5160, 5985, 6337, 6489, 7729, 8663, 8848, 9265, 10274, 10753, 11821, 12071, 12688, 13937, 14345, 15520, 15962, 16296, 16352, 17761, 21923, 22753, 26208

Offset: 1

R. J. Mathar, Feb 20 2017

nonn

A permutation of A099533 sorted by increasing y. 1<=x<=y, GCD(x,y)=1.

			(x,y,z) = (11,37,228), (56,65,671), (57,112,1261), (217,312,6371), (242,433,9765), (851,877,35928), (305,1064,35113), (23,1177,40380), (122,1201,41643), (592,1617,66599), (1706,1969,112245), (1376,2345,124501), (1001,2480,127499), (2257,2543,167160), (1960,3081,191771)... - _R. J. Mathar_, Jan 22 2025

R. J. Mathar, Table of n, a(n) for n = 1..325

Cf. A099533, A106322, A099426 (values of z). Subsequence of A320662.

A282639 := proc(n)
    option remember ;
    local x,y ;
    if n =1 then
        2
    else
        for y from procname(n-1)+1 do
            for x from 1 to y do
                if igcd(x,y) = 1 then
                    if issqr(x^3+y^3) then
                        return y;
                    end if;
                end if;
            end do:
        end do:
    end if;
end proc:
seq(A282639(n),n=1..17) ; # R. J. Mathar, Jan 22 2025

cp's OEIS Frontend

A106318 Bhaskara twins: n such that 2*n^2 = X^3 and 2*n^3 = Y^2.

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A106320 Values b of a Bhaskara pair (a,b), a<=b, sorted on values of b. A Bhaskara pair (a,b) is such that a^2 + b^2 = X^3 and a^3 + b^3 = Y^2.

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A106319 Values a of a Bhaskara pair (a,b), a<=b, sorted by value of b. A Bhaskara pair (a,b) is such that a^2 + b^2 = X^3 and a^3 + b^3 = Y^2.

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A106321 Smaller member of a Bhaskara pair (excluding Bhaskara twins, that is, include only a < b); a Bhaskara pair (a,b) is such that a^2 + b^2 = X^3 and a^3 + b^3 = Y^2.

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A282639 Greater value of a coprime pair (x,y) satisfying x^3+y^3=z^2.

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A106318 Bhaskara twins: n such that 2n^2 = X^3 and 2n^3 = Y^2.