A106318 -id:A106318 - cp's OEIS frontend

A003358 Numbers that are the sum of 2 nonzero 6th powers.

2, 65, 128, 730, 793, 1458, 4097, 4160, 4825, 8192, 15626, 15689, 16354, 19721, 31250, 46657, 46720, 47385, 50752, 62281, 93312, 117650, 117713, 118378, 121745, 133274, 164305, 235298, 262145, 262208, 262873, 266240, 277769, 308800, 379793, 524288

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
10069120217 is in the sequence as 10069120217 = 29^6 + 46^6.
139314070233 is in the sequence as 139314070233 = 3^6 + 72^6.
404680615040 is in the sequence as 404680615040 = 22^6 + 86^6. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Samuel S. Wagstaff, Jr., Equal Sums of Two Distinct Like Powers, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.

Cf. A088677 (2 distinct 6th). Supersequence of A106318.
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

With[{k = 6}, Union@ Map[(#[[1]]^k + #[[2]]^k) &, Tuples[Range[8], {2}]]] (* Michael De Vlieger, Sep 09 2022, after Harvey P. Dale at A004999  *)

Removed incorrect program. David A. Corneth, Aug 01 2020

A106322 Larger member of a Bhaskara pair (excluding Bhaskara twins, i.e., include only a < b), sorted on the larger value; 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

1250, 80000, 911250

Offset: 1

Lekraj Beedassy, Apr 29 2005

This sequence does not miss any terms since it is sorted by the larger member of the pair. - Jud McCranie, Jul 05 2013

The sequence likely continues 5120000, 10290000, 19531250, 27827450, 52743945, 58320000, 75357100, ... - R. J. Mathar, Jan 27 2017

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

R. J. Mathar, Numerical Construction of Bhaskara Pairs
R. J. Mathar, Construction of Bhaskara Pairs, arXiv:1703.01677 [math.NT] (2017).

Cf. A106321, A106318, A106319, A106320.

A106320 SETMINUS A106318. - R. J. Mathar, Jan 27 2017

Corrected and extended by Jud McCranie, Jul 04 2013

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

A269792 a(n) = 5*n^4.

Original entry on oeis.org

0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805, 50000, 73205, 103680, 142805, 192080, 253125, 327680, 417605, 524880, 651605, 800000, 972405, 1171280, 1399205, 1658880, 1953125, 2284880, 2657205, 3073280, 3536405, 4050000, 4617605, 5242880, 5929605

Offset: 0

Ilya Gutkovskiy, Mar 31 2016

More generally, the ordinary generating function for the sequences of the form k*n^m, is k*Sum_{j>=1}x^j*j^m (when abs(x)<1).

More generally, the ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r, is (r + (p + q + k + m - 4*r)*x + (11*p + 3*q - k - 3*m + 6*r)*x^2 + (11*p - 3*q - k + 3*m - 4*r)*x^3 + (p - q + k - m + r)*x^4)/(1 - x)^5.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quartic polynomial
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. similar sequences of the form k*n^m, for k = 1...5, m = 1...10: A001477(k = 1, m = 1), A005843 (k = 2, m = 1), A008585 (k = 3, m = 1), A008586 (k = 4, m = 1), A008587 (k = 5, m = 1), A000290 (k = 1, m = 2), A001105 (k = 2, m = 2), A033428 (k = 3, m = 2), A016742 (k = 4, m = 2), A033429 (k = 5, m = 2), A000578 (k = 1, m = 3), A033431 (k = 2, m = 3), A117642 (k = 3, m = 3), A033430 (k = 4, m = 3), A244725 (k = 5, m = 3), A000583 (k = 1, m = 4), A244730 (k = 2, m = 4), A219056 (k = 3, m = 4), A141046 (k = 4, m = 4), this sequence(k = 5, m = 4), A000584 (k = 1, m = 5), A001014 (k = 1, m = 6), A106318 (k = 2, m = 6), A001015 (k = 1, m = 7), A001016 (k = 1, m = 8), A001017 (k = 1, m = 9), A008454 (k = 1, m = 10).

A269792:=n->5*n^4: seq(A269792(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[5 n^4, {n, 0, 33}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 80, 405, 1280}, 34]

x='x+O('x^99); concat(0, Vec(5*x*(1+11*x+11*x^2+x^3)/(1-x)^5)) \\ Altug Alkan, Mar 31 2016

G.f.: 5*x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5.
E.g.f.: 5*exp(x)^x*x*(1 + 7*x + 6*x^2 + x^3).
a(n) = 5*a(n-1) - 10*(9n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 5*A000583(n) = A008587(n)*A000578(n).
Sum_{n>=1} 1/a(n) = Pi^4/450 = (1/450)*A092425 = 0.216464646742...

cp's OEIS Frontend

A003358 Numbers that are the sum of 2 nonzero 6th powers.

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A106322 Larger member of a Bhaskara pair (excluding Bhaskara twins, i.e., include only a < b), sorted on the larger value; 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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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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A269792 a(n) = 5*n^4.

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