Vladimir Pletser - cp's OEIS frontend

A387128 First numbers A = a(n) of two numbers (A, B) such that the sums 2*A^2 + B^2 = p == 3 mod 8, where p = A007520(n) and B = A387129(n).

1, 1, 3, 3, 5, 3, 1, 7, 5, 3, 9, 7, 9, 1, 11, 9, 3, 9, 13, 3, 13, 1, 11, 5, 15, 9, 3, 13, 15, 17, 15, 3, 17, 11, 9, 15, 9, 15, 7, 3, 21, 21, 19, 11, 17, 21, 1, 9, 19, 21, 7, 25, 23, 15, 17, 13, 19, 27, 27, 23, 1, 9, 5, 27, 7, 27, 17, 3, 21, 27, 23, 19, 3, 29, 31, 25, 27, 31, 9, 1, 27

Offset: 1

Vladimir Pletser, Aug 17 2025

Prime numbers p congruent to 3 mod 8 can be written as the sum of twice the square of an integer A and of the square of another integer B, i.e., 2*A^2 + B^2 = p, where A = a(n), B = A387129(n), and p = A007520(n) == 3 mod 8.
This representation is unique, i.e., for a given n, there are no other integer values of A(n) and B(n) such that p(n) = 2 * A(n)^2 + B(n)^2 where p(n) = A007520(n), the 3 mod 8 prime numbers.
For all n, A = a(n) and B = A387129(n) are always odd.
Terms are ordered according to increasing order of A007520(n).

			1 belongs to the sequence as 2 * 1^2 + 1^2 = 3.
5 belongs to the sequence as 2 * 5^2 + 21^2 = 491.

Cartier P. "An Introduction to Zeta Functions", Chap 1.2, in eds. M. Waldschmidt, P. Moussa, J.M., Luck, C. Itzykson “From Number Theory to Physics”, Springer-Verlag, Berlin, pp. 22-41, 1960.
Conway J.H. and Guy R.K. "The Book of Numbers", Chap. 5, Springer-Verlag, New York, pp. 127-149, 1996.
Hardy, G. H. and Wright, E. M. "Primes in k(i)" and "The Fundamental Theorem of Arithmetic in k(i)." 12.7 and 12.8 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 183-187, 1979.
Sierpinski W. "Elementary Theory of Numbers", Chap. 13.3 and 13.4, ed. A Schinzel, North Holland, Amsterdam, pp. 459-462, 1988.

Vladimir Pletser, Table of n, a(n) for n = 1..10000

Cf. A007520, A387129.

2 * a(n)^2 + A387129(n)^2 = A007520(n).

A387129 Second numbers B = a(n) of two numbers (A, B) such that the sums 2*A^2 + B^2 = p == 3 mod 8, where p = A007520(n) and A = A387128(n).

Original entry on oeis.org

1, 3, 1, 5, 3, 7, 9, 3, 9, 11, 1, 9, 7, 15, 3, 11, 17, 13, 3, 19, 9, 21, 15, 21, 7, 19, 23, 15, 11, 3, 13, 25, 9, 21, 23, 17, 25, 19, 27, 29, 1, 5, 15, 27, 21, 13, 33, 31, 21, 17, 33, 3, 15, 29, 27, 33, 27, 1, 5, 21, 39, 37, 39, 11, 39, 13, 33, 41, 29, 17, 27, 33, 43, 15, 3, 27, 23, 9

Offset: 1

Vladimir Pletser, Aug 17 2025

Prime numbers p congruent to 3 mod 8 can be written as the sum of twice the square of an integer A and of the square of another integer B, i.e., 2*A^2 + B^2 = p, where A = A387128(n), B = a(n) (this sequence), and p = A007520(n) == 3 mod 8.
This representation is unique, i.e., for a given n, there are no other integer values of A(n) and B(n) such that p(n) = 2 * A(n)^2 + B(n)^2 where p(n) = A007520(n), the 3 mod 8 prime numbers.
For all n, A = A387128(n) and B = a(n) are always odd.
Terms are ordered according to increasing order of A007520(n).

			1 belongs to the sequence as 2 * 1^2 + 1^2 = 3.
21 belongs to the sequence as 2 * 5^2 + 21^2 = 491.

Cartier P. "An Introduction to Zeta Functions", Chap 1.2, in eds. M. Waldschmidt, P. Moussa, J.M., Luck, C. Itzykson “From Number Theory to Physics”, Springer-Verlag, Berlin, pp. 22-41, 1960.
Conway J.H. and Guy R.K. "The Book of Numbers", Chap. 5, Springer-Verlag, New York, pp. 127-149, 1996.
Hardy, G. H. and Wright, E. M. "Primes in k(i)" and "The Fundamental Theorem of Arithmetic in k(i)." 12.7 and 12.8 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 183-187, 1979.
Sierpinski W. "Elementary Theory of Numbers", Chap. 13.3 and 13.4, ed. A Schinzel, North Holland, Amsterdam, pp. 459-462, 1988.

Vladimir Pletser, Table of n, a(n) for n = 1..10000

Cf. A007520, A387128.

2 * A387128(n)^2 + a(n)^2 = A007520(n).

A355753 a(n) = 3(2n - 1)( 3(2*n - 1)^3 - 1) / 2 for n > 0.

Original entry on oeis.org

3, 360, 2805, 10794, 29511, 65868, 128505, 227790, 375819, 586416, 875133, 1259250, 1757775, 2391444, 3182721, 4155798, 5336595, 6752760, 8433669, 10410426, 12715863, 15384540, 18452745, 21958494, 25941531, 30443328, 35507085, 41177730, 47501919, 54528036, 62306193, 70888230, 80327715, 90679944, 102001941, 114352458, 127791975

Offset: 1

Vladimir Pletser, Jul 15 2022

Numbers D > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 such that the difference C - D == 3 (mod 6), C - D = 3*(2*n - 1) for n > 1, and the difference of the positive cubes C^3 - D^3 is equal to centered cube numbers, with C > D > B > 0, and A > 0, A = 27*t^3 *(27*t^6+1)/4 with t = 2*n-1, and where A = A352759(n), B = A355751(n), C = A355752(n), and D = a(n) (this sequence).
There are infinitely many such numbers a(n) = D in this sequence.
Subsequence of A352136 and of A352223.

			a(1) = 3 belongs to the sequence as 6^3 - 3^3 = 4^3 + 5^3 = 189 and 6 - 3 = 3 = 3*(2*1 - 1).
a(2) = 360 belongs to the sequence as 369^3 - 360^3 = 121^3 + 122^3 = 3587409 and 369 - 360 = 9 = 3*(2*2 - 1).
a(3) = 3*(2*3 - 1)*( 3*(2*3 - 1)^3 - 1) / 2 = 2805.
a(4) = 3*a(3) - 3*a(2) + a(1) + 1728*2 = 3*2805 - 3*360 + 3 + 1728*2  = 10794.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352222, A352224, A352225, A352755, A352756, A352757, A352759, A352759, A355751, A355752.

restart; for n to 20 do (1/2)* 3*(2*n - 1)*(3*(2*n - 1)^3-1); end do;

A355752(n)^3 - a(n)^3 = A355751(n)^3 + (A355751(n) + 1)^3 = A352759(n) and A355752(n) - a(n) = 3*(2*n - 1).
a(n) = 3*(2*n - 1)*( 3*(2*n - 1)^3 - 1) / 2 for n > 0.
For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 1728*(n - 2), with a(1) = 3, a(2) = 360 and a(3) = 2805.
a(n) can be extended for negative n such that a(-n) = a(n+1) + (2n + 1).
G.f.: -3*x*(1+115*x+345*x^2+113*x^3+2*x^4) / (x-1)^5 . - R. J. Mathar, Aug 03 2022

A355752 a(n) = 3(2n - 1)( 3(2*n - 1)^3 + 1) / 2.

Original entry on oeis.org

6, 369, 2820, 10815, 29538, 65901, 128544, 227835, 375870, 586473, 875196, 1259319, 1757850, 2391525, 3182808, 4155891, 5336694, 6752865, 8433780, 10410543, 12715986, 15384669, 18452880, 21958635, 25941678, 30443481, 35507244, 41177895, 47502090, 54528213, 62306376, 70888419, 80327910, 90680145, 102002148, 114352671, 127792194

Offset: 1

Vladimir Pletser, Jul 15 2022

Numbers C > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 such that the difference (C - D) == 3 (mod 6), C - D = 3 (2n - 1) for n > 1, and the difference between the positive cubes C^3 - D^3 is equal to a centered cube number, C^3 - D^3 = B^3 + (B+1)^3, with C > D > B > 0, and A > 0, A = 27*t^3 *(27*t^6+1)/4 with t = 2*n-1, and where A = A352759(n), B = A355751(n), C = a(n) (this sequence), and D = A355753(n).

There are infinitely many such numbers a(n) = C in this sequence.

			a(1) = 6 belongs to the sequence as 6^3 - 3^3 = 4^3 + 5^3 = 189 and 6 - 3 = 3 = 3 (2*1 - 1).
a(2) = 369 belongs to the sequence as 369^3 - 360^3 = 121^3 + 122^3 = 3587409 and 369 - 360 = 9 = 3*(2*2 - 1).
a(3) = 3*(2*3 - 1)*( 3*(2*3 - 1)^3 + 1) / 2 = 2820.
a(4) = 3*a(3) - 3*a(2) + a(1) + 1728*2 = 3*2820 - 3*369 + 6 + 1728*2  = 10815.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352223, A352224, A352225, A352755, A352756, A352757, A352758, A352759, A355751, A355753.

restart; for n to 20 do (1/2)* 3*(2*n - 1)*(3*(2*n - 1)^3+1); end do;

a(n)=3*(2*n-1)*(3*(2*n-1)^3+1)/2 \\ Charles R Greathouse IV, Oct 21 2022

a(n)^3 - A355753(n)^3 = A355751(n)^3 + (A355751(n) + 1)^3 = A352759(n) and a(n) - A355753(n) = 3*(2*n - 1).
a(n) = 3*(2*n - 1)*( 3*(2*n - 1)^3 + 1) / 2.
For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 1728*(n - 2), with a(1) = 6, a(2) = 369 and a(3) = 2820.
a(n) can be extended for negative n such that a(-n) = a(n+1) - 3*(2*n + 1).
G.f.: -3*x*(2+113*x+345*x^2+115*x^3+x^4) / (x-1)^5 . - R. J. Mathar, Aug 03 2022

A355751 Positive numbers k such that the centered cube number k^3 + (k+1)^3 is equal to the difference of two positive cubes and to A352759(n).

Original entry on oeis.org

4, 121, 562, 1543, 3280, 5989, 9886, 15187, 22108, 30865, 41674, 54751, 70312, 88573, 109750, 134059, 161716, 192937, 227938, 266935, 310144, 357781, 410062, 467203, 529420, 596929, 669946, 748687, 833368, 924205, 1021414, 1125211, 1235812, 1353433

Offset: 1

Vladimir Pletser, Jul 15 2022

Numbers B > 0 such that the centered cube number B^3 + (B+1)^3 is equal to the difference of two positive cubes, i.e., A = B^3 + (B+1)^3 = C^3 - D^3 and such that C - D = 3 (2n - 1) == 3 (mod 6), with C > D > B > 0, and A > 0, A = 27*t^3 * (27*t^6 + 1) /4 with t = 2*n-1, and where A = A352759(n), B = a(n) (this sequence), C = A355752(n) and D = A355753(n).
There are infinitely many such numbers a(n) = B in this sequence.
Subsequence of A352134.

			a(1) = 4 is a term because 4^3 + 5^3 = 6^3 - 3^3 and 6 - 3 = 3 = 3*(2*1 - 1).
a(2) = 121 is a term because 121^3 + 122^3 = 369^3 - 360^3 and 369 - 360 = 9 = 3*(2*2 - 1).
a(3) = (9*(2*3 - 1)^3 - 1) / 2 = 562.
a(4) = 3*562 - 3*121 + 4 + 216 = 1543.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352222, A352223, A352224, A352225, A352755, A352756, A352757, A352758, A352759, A355752, A355753.

restart; for n to 20 do (1/2)*(9*(2*n - 1)^3-1); end do;

a(n)^3 + (a(n)+1)^3 = A355752(n)^3 - A355753(n)^3 and A355752(n) - A355753(n) = 3*(2*n - 1).
a(n) = (9*(2*n - 1)^3 - 1) / 2.
For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 216, with a(1) = 4, a(2) = 121 and a(3) = 562.
a(n) can be extended for negative n such that a(-n) = - a(n+1) - 1.
From Jianing Song, Jul 18 2022: (Start)
G.f.: x*(4+105*x+102*x^2+5*x^3)/(1-x)^4.
E.g.f.: 5 + exp(x)*(-5+9*x+54*x^2+36*x^3). (End)

A352758 a(n) = (3(2n - 1)^2((2n - 1)^2 + 2) - 2*n + 3)/2 for n > 0.

Original entry on oeis.org

5, 148, 1011, 3746, 10081, 22320, 43343, 76606, 126141, 196556, 293035, 421338, 587801, 799336, 1063431, 1388150, 1782133, 2254596, 2815331, 3474706, 4243665, 5133728, 6156991, 7326126, 8654381, 10155580, 11844123, 13734986, 15843721, 18186456, 20779895, 23641318, 26788581, 30240116, 34014931, 38132610

Offset: 1

Vladimir Pletser, Apr 02 2022

Numbers D > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 such that the difference C - D is odd, C - D = 2*n - 1, and the difference of the positive cubes C^3 - D^3 is equal to centered cube numbers, with C > D > B > 0, and A > 0, A = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1, and where A = A352755(n), B = A352756(n), C = A352757(n), and D = a(n) (this sequence).
There are infinitely many such numbers a(n) = D in this sequence.
Subsequence of A352136 and of A352223.

			a(1) = 5 belongs to the sequence as 6^3 - 5^3 = 3^3 + 4^3 = 91 and 6 - 5 = 1 = 2*1 - 1.
a(2) = 148 belongs to the sequence as 151^3 - 148^3 = 46^3 + 47^3 = 201159 and 151 - 148 = 3 = 2*2 - 1.
a(3) = (3*(2*3 - 1)^2*((2*3 - 1)^2 + 2) - 2*3 + 3)/2 = 1011.
a(4) = 3*a(3) - 3*a(2) + a(1) + 576*2 = 3*1011 - 3*148 + 5 + 576*2  = 3746.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352222, A352224, A352225, A352755, A352756, A352757, A352759, A355751, A355752, A355753.

restart; for n to 20 do (1/2)*(3*(2*n - 1)^2*((2*n - 1)^2 + 2) - 2*n + 3); end do;

def A352758(n): return n*(n*(n*(24*n - 48) + 48) - 25) + 6 # Chai Wah Wu, Jul 11 2022

A352757(n)^3 - a(n)^3 = A352756(n)^3 + (A352756(n) + 1)^3 = A352755(n) and A352757(n) - a(n) = 2*n - 1.
a(n) = (3*(2*n - 1)^2*((2*n - 1)^2 + 2) - 2*n + 3)/2.
For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 576*(n - 2), with a(1) = 5, a(2) = 148 and a(3) = 1011.
a(n) can be extended for negative n such that a(-n) = a(n+1) + (2n + 1).
G.f.: x*(5 + 123*x + 321*x^2 + 121*x^3 + 6*x^4)/(1 - x)^5. - Stefano Spezia, Apr 08 2022

A352756 Positive numbers k such that the centered cube number k^3 + (k+1)^3 is equal to the difference of two positive cubes and to A352755(n).

Original entry on oeis.org

3, 46, 197, 528, 1111, 2018, 3321, 5092, 7403, 10326, 13933, 18296, 23487, 29578, 36641, 44748, 53971, 64382, 76053, 89056, 103463, 119346, 136777, 155828, 176571, 199078, 223421, 249672, 277903, 308186, 340593, 375196, 412067, 451278, 492901, 537008, 583671, 632962, 684953, 739716, 797323, 857846, 921357

Offset: 1

Vladimir Pletser, Apr 02 2022

Numbers B > 0 such that the centered cube number B^3 + (B+1)^3 is equal to the difference of two positive cubes, i.e., A = B^3 + (B+1)^3 = C^3 - D^3 and such that C - D = 2n - 1, with C > D > B > 0, and A > 0, A = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1, and where A = A352755(n), B = a(n) (this sequence), C = A352757(n) and D = A352758(n).
There are infinitely many such numbers a(n) = B in this sequence.
Subsequence of A352134 and of A352221.

			a(1) = 3 is a term because 3^3 + 4^3 = 6^3 - 5^3 and 6 - 5 = 1 = 2*1 - 1.
a(2) = 46 is a term because 46^3 + 47^3 = 151^3 - 148^3 and 151 - 148 = 3 = 2*2 - 1.
a(3) = ((2*3 - 1)*(3*(2*3 - 1)^2 + 4) - 1)/2 = 197.
a(4) = 3*197 - 3*46 + 3 + 72 = 528.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352222, A352223, A352224, A352225, A352755, A352757, A352758, A352759, A352760, A352761, A352762.

restart; for n to 20 do (1/2)* ((2*n - 1)*(3*(2*n - 1)^2 + 4) - 1);  end do;

a(n)^3 + (a(n)+1)^3 = A352757(n)^3 - A352758(n)^3 and A352757(n) - A352758(n) = 2*n - 1.
a(n) = ((2*n - 1)*(3*(2*n - 1)^2 + 4) - 1)/2.
For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 72, with a(1) = 3, a(2) = 46 and a(3) = 197.
a(n) can be extended for negative n such that a(-n) = -a(n+1) - 1.
G.f.: x*(3 + 34*x + 31*x^2 + 4*x^3)/(1 - x)^4. - Stefano Spezia, Apr 08 2022

A352755 Positive centered cube numbers that can be written as the difference of two positive cubes: a(n) = t(3t^2 + 4)(t^2(3t^2 + 4)^2 + 3)/4 with t = 2n-1 and n > 0.

Original entry on oeis.org

91, 201159, 15407765, 295233841, 2746367559, 16448122691, 73287987409, 264133278045, 811598515091, 2202365761759, 5410166901741, 12249942682409, 25914353312575, 51755729480091, 98389720844009, 179211321358741, 314429627203659, 533744613620855, 879807401606341, 1412624924155809

Offset: 1

Vladimir Pletser, Apr 02 2022

Numbers A > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 and such that C - D = 2n - 1, with C > D > B > 0, and A = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1, and where A = a(n) (this sequence), B = A352756(n), C = A352757(n) and D = A352758(n).
There are infinitely many such numbers a(n) = A in this sequence.
Subsequence of A005898, of A352133 and of A352220.

			a(1) = 91 belongs to the sequence because 91 = 3^3 + 4^3 = 6^3 - 5^3 and 6 - 5 = 1 = 2*1 - 1.
a(2) = 201159 belongs to the sequence because 201159 = 46^3 + 47^3 = 151^3 - 148^3 and 151 - 148 = 3 = 2*2 - 1.
a(3) = (2*3 - 1)*(3*(2*3 - 1)^2 + 4)*((2*3 - 1)^2*(3*(2*3 - 1)^2 + 4)^2 + 3)/4 = 15407765.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352221, A352222, A352223, A352224, A352225, A352756, A352757, A352758, A352759, A355751, A355752, A355753.

restart; for n to 20 do (1/4)*(2*n-1)*(3*(2*n-1)^2+4)*((2*n-1)^2*(3*(2*n-1)^2+4)^2+3) end do;

a(n) = A352756(n)^3 + (A352756(n) + 1)^3 = A352757(n)^3 - A352758(n)^3 and A352757(n) - A352758(n) = 2n - 1.
a(n) = (2*n - 1)*(3*(2*n - 1)^2 + 4)*((2*n - 1)^2*(3*(2*n - 1)^2 + 4)^2 + 3)/4.
a(n) can be extended for negative n such that a(-n) = -a(n+1).

A352759 Centered cube numbers that are the difference of two positive cubes; a(n) = 27t^3(27t^6 + 1)/4 with t = 2n-1.

Original entry on oeis.org

189, 3587409, 355957875, 7354447191, 70607389041, 429735975669, 1932670025559, 7006302268875, 21612640524741, 58809832966521, 144757538551899, 328260072633759, 695228576765625, 1389765141771741, 2643927354266751, 4818621138983379, 8458493032498509, 14364150148238625, 23685527077994691, 38040743821584231

Offset: 1

Vladimir Pletser, Apr 02 2022

Numbers A > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 and such that C - D = 3*(2*n - 1) == 3 (mod 6), with (for n > 1) C > D > B > 0, and A = as 27*t^3*(27*t^6 + 1)/4 with t = 2*n-1, and where A = a(n) (this sequence), B = A355751(n), C = A355752(n) and D = A355753(n).
There are infinitely many such numbers a(n) = A in this sequence.
Subsequence of A005898, and of A352133.

			a(1) = 189 belongs to the sequence because 189 = 4^3 + 5^3 = 6^3 - 3^3 and 6 - 3 = 3 = 3*(2*1 - 1).
a(2) = 3587409 belongs to the sequence because 3587409 = 121^3 + 122^3 = 369^3 - 360^3 and 369 - 360 = 9 = 3*(2*2 - 1).
a(3) = 27*(2*3 - 1)^3*(27*(2*3 - 1)^6 + 1)/4 = 355957875.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352221, A352222, A352223, A352224, A352225, A352755, A352756, A352757, A352758, A355751, A355752, A355753.

restart; for n from 1 to 20 do 27*(2*n-1)^3*(27*(2*n-1)^6+1)*(1/4); end do;

a(n) = A355751(n)^3 + (A355751(n) + 1)^3 = A355752(n)^3 - A355753(n)^3 and A355752(n) - A355753(n) = 3*(2*n - 1).
a(n) = 27*(2*n - 1)^3*(27*(2*n - 1)^6 + 1)/4.
a(n) can be extended for negative n such that a(-n) = -a(n+1).

A352757 a(n) = (3(2n - 1)^2((2n - 1)^2 + 2) + 2*n + 1)/2 for n > 0.

Original entry on oeis.org

6, 151, 1016, 3753, 10090, 22331, 43356, 76621, 126158, 196575, 293056, 421361, 587826, 799363, 1063460, 1388181, 1782166, 2254631, 2815368, 3474745, 4243706, 5133771, 6157036, 7326173, 8654430, 10155631, 11844176, 13735041, 15843778, 18186515, 20779956, 23641381, 26788646, 30240183, 34015000, 38132681

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Vladimir Pletser, Apr 02 2022

Numbers C > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 such that the difference C - D is odd, C - D = 2*n - 1, and the difference between the positive cubes C^3 - D^3 is equal to a centered cube number, C^3 - D^3 = B^3 + (B+1)^3, with C > D > B > 0, and A > 0, A = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1, and where A = A352755(n), B = A352756(n), C = a(n) (this sequence), and D = A352758(n).

There are infinitely many such numbers a(n) = C in this sequence.

			a(1) = 6 belongs to the sequence as 6^3 - 5^3 = 3^3 + 4^3 = 91 and 6 - 5 = 1 = 2*1 - 1.
a(2) = 151 belongs to the sequence as 151^3 - 148^3 = 46^3 + 47^3 = 201159 and 151 - 148 = 3 = 2*2 - 1.
a(3) = (3(2*3 - 1)^2*((2*3 - 1)^2 + 2) + 2*3 + 1)/2 = 1016.
a(4) = 3*a(3) - 3*a(2) + a(1) + 576*2 = 3*1016 - 3*151 + 6 + 576*2 = 3753.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352223, A352224, A352225, A352755, A352756, A352758, A352759, A352760, A352761, A352762.

restart; for n to 20 do (1/2)*(3*(2*n - 1)^2*((2*n - 1)^2 + 2) + 2*n + 1); end do;

def A352757(n): return n*(n*(n*(24*n - 48) + 48) - 23) + 5 # Chai Wah Wu, Jul 10 2022

a(n)^3 - A352758(n)^3 = A352756(n)^3 + (A352756(n) + 1)^3 = A352755(n) and a(n) - A352758(n) = 2*n - 1.
a(n) = (3*(2*n - 1)^2*((2*n - 1)^2 + 2) + 2*n + 1)/2.
For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 576*(n - 2), with a(1) = 6, a(2) = 151 and a(3) = 1016.
a(n) can be extended for negative n such that a(-n) = a(n+1) - (2*n + 1).
G.f.: x*(6 + 121*x + 321*x^2 + 123*x^3 + 5*x^4)/(1 - x)^5. - Stefano Spezia, Apr 08 2022

cp's OEIS Frontend

User: Vladimir Pletser

A387128 First numbers A = a(n) of two numbers (A, B) such that the sums 2*A^2 + B^2 = p == 3 mod 8, where p = A007520(n) and B = A387129(n).

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A387129 Second numbers B = a(n) of two numbers (A, B) such that the sums 2*A^2 + B^2 = p == 3 mod 8, where p = A007520(n) and A = A387128(n).

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A355753 a(n) = 3*(2*n - 1)*( 3*(2*n - 1)^3 - 1) / 2 for n > 0.

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A355752 a(n) = 3*(2*n - 1)*( 3*(2*n - 1)^3 + 1) / 2.

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A355751 Positive numbers k such that the centered cube number k^3 + (k+1)^3 is equal to the difference of two positive cubes and to A352759(n).

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A352758 a(n) = (3*(2*n - 1)^2*((2*n - 1)^2 + 2) - 2*n + 3)/2 for n > 0.

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A352756 Positive numbers k such that the centered cube number k^3 + (k+1)^3 is equal to the difference of two positive cubes and to A352755(n).

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A352755 Positive centered cube numbers that can be written as the difference of two positive cubes: a(n) = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1 and n > 0.

A355753 a(n) = 3(2n - 1)( 3(2*n - 1)^3 - 1) / 2 for n > 0.

A355752 a(n) = 3(2n - 1)( 3(2*n - 1)^3 + 1) / 2.

A352758 a(n) = (3(2n - 1)^2((2n - 1)^2 + 2) - 2*n + 3)/2 for n > 0.

A352755 Positive centered cube numbers that can be written as the difference of two positive cubes: a(n) = t(3t^2 + 4)(t^2(3t^2 + 4)^2 + 3)/4 with t = 2n-1 and n > 0.

A352759 Centered cube numbers that are the difference of two positive cubes; a(n) = 27t^3(27t^6 + 1)/4 with t = 2n-1.

A352757 a(n) = (3(2n - 1)^2((2n - 1)^2 + 2) + 2*n + 1)/2 for n > 0.