A321491 -id:A321491 - cp's OEIS frontend

A321499 Numbers of the form (x - y)(x^2 - y^2) with x > y > 0.

3, 5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 96, 97, 99, 101, 103, 104, 105, 107, 109, 111, 112, 113, 115

Offset: 1

M. F. Hasler, Nov 22 2018

Equivalently, numbers of the form (x - y)^2*(x + y) or d^2*(2m + d), for (x, y) = (m+d, m). This shows that this consists of all squares d^2 > 0 times all numbers of the same parity and larger than d. In particular, for d=1, all odd numbers > 1, and for d=2, 4*(even numbers > 2) = 8*(any number > 1). Larger d can't yield additional terms, neither odd nor even: The sequence consists exactly of all odd numbers > 2 and multiples of 8 larger than 8.

			a(1) = 3 = 1*3 = (2 - 1)*(2^2 - 1^2). Similarly any larger odd number 2k+1 = (k+1 - k)((k+1)^2 - k^2) is in this sequence.
a(8) = 16 = 2*8 = (3 - 1)*(3^2 - 1^2). Similarly, any larger multiple of 8, 8*(1 + k) = 2*(4k + 4) = (k+2 - k)((k+2)^2 - k^2) is in this sequence.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

See A321491 for numbers of the form (x+y)(x^2+y^2).
Cf. A321501 (complement).
See A321498 for numbers that have two representations of the form (x-y)(x^2-y^2).
Cf. A106505 (conjectured to be the sequence without the 3).

is(n)={n&&fordiv(n,d, d^2*(d+2)>n && break; n%d^2&&next; bittest(n\d^2-d,0)||return(1))} \\ This uses the definition. More efficient variant below.

select( is_A321499(n)=if(bittest(n,0),n>1,n%8,0,n>8), [0..99]) \\ Defines the function is_A321499(). The select() command is just an illustration and check.

A321499_list(M)=setunion(vector(M\2-1,k,2*k+1),[2..M\8]*8) \\ list all terms up to M; more efficient than select() above.

apply( A321499(n)=if(n<8, 2*n+1, n%5!=3, (n+2)*4\5*2+1, n\5*8+8), [1..30]) \\ Defines A321499(n). The apply() command provides a check & illustration.

def A321499(n): return (n<<1)+1 if n<4 else (((n+2)<<2)//5<<1)+(n%5!=3) # Chai Wah Wu, Feb 26 2025

Asymptotic density is 5/8. Complement is A321501.
a(5k-2) = 8k for all k > 1, a(n) = floor((n+2)*4/5)*2 + 1 for all other n > 3.
a(n + 5) = a(n) + 8 for n > 3. - David A. Corneth, Nov 23 2018
O.g.f. 3*x+5*x^2+7*x^3 -x^4*(-9-2*x-2*x^2-2*x^3-x^4+8*x^5) / ( (x^4+x^3+x^2+x+1) *(x-1)^2 ). - R. J. Mathar, Nov 29 2018

A321490 Triangular table T[n,k] = (n+k)(n^2+k^2), 1 <= k <= n = 1, 2, 3, ...; read by rows.

Original entry on oeis.org

4, 15, 32, 40, 65, 108, 85, 120, 175, 256, 156, 203, 272, 369, 500, 259, 320, 405, 520, 671, 864, 400, 477, 580, 715, 888, 1105, 1372, 585, 680, 803, 960, 1157, 1400, 1695, 2048, 820, 935, 1080, 1261, 1484, 1755, 2080, 2465, 2916, 1111, 1248, 1417, 1624, 1875, 2176, 2533, 2952, 3439, 4000, 1464, 1625, 1820, 2055, 2336

Offset: 1

M. F. Hasler, Nov 22 2018

			The table starts:
Row 1:    4;
Row 2:   15,  32;
Row 3:   40,  65, 108;
Row 4:   85, 120, 175, 256;
Row 5:  156, 203, 272, 369,  500;
Row 6:  259, 320, 405, 520,  671,  864;
Row 7:  400, 477, 580, 715,  888, 1105, 1372;
Row 8:  585, 680, 803, 960, 1157, 1400, 1695, 2048;
etc.

Cf. A321491 (numbers of the form T(n,k) with n > k > 0).
Cf. A321492 (numbers which can be written at least twice in this form).
Cf. A033430 (diagonal), A053698 (column 1).
Cf. A198063 (read as a square array equals T(n,k) for all n, k >= 0).
Cf. A321500 (variant of this table with additional row 0 and column 0).

t[n_, k_] := (n + k) (n^2 + k^2); Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 22 2018 *)

A321490(n,k)=(n+k)*(n^2+k^2)
A321490_row(n)=vector(n,k,(n+k)*(n^2+k^2))
A321490_list(N=12)=concat(apply(A321490_row,[1..N]))

Diagonal: T(n,n) = 4*n^3 = A033430(n).

Column 1: T(n,1) = (n + 1)(n^2 + 1) = A053698(n) = (n^4-1)/(n-1) for n > 1.

A321492 Numbers that can be written as (x + y)(x^2 + y^2), x > y > 0, in at least two ways.

Original entry on oeis.org

12325, 98600, 117720, 146705, 206312, 263840, 332775, 378505, 400945, 500200, 651456, 687245, 734400, 741845, 773800, 788800, 799240, 941760, 1173640, 1327360, 1533195, 1540625, 1650496, 1735105, 1836680, 1943240, 2048320, 2050880, 2110720, 2217280, 2662200, 2704360, 2965685

Offset: 1

Geoffrey B. Campbell and M. F. Hasler, Nov 22 2018

nonn

See A321491 for numbers of the form (x+y)(x^2+y^2) = A321490(x,y) with x > y > 0.

			12325 = (13+16)(13^2+16^2) = (3+22)(3^2+22^2).
98600 = (26+32)(26^2+32^2) = (6+44)(6^2+44^2).
117720 = (21+39)(21^2+39^2) = (8+46)(8^2+46^2).
146705 = (24+41)(24^2+41^2) = (14+47)(14^2+47^2).
206312 = (15+53)(15^2+53^2) = (32+42)(32^2+42^2).
263840 = (6+62)(6^2+62^2) = (33+47)(33^2+47^2).

Geoffrey B. Campbell, (m+n)(m²+n²) in two different ways, LinkedIn Number Theory Group, Aug. 2018

Cf. A321490, A321491.

A321492_list(L=1e6)={my(S=[],T=List(),t);for(m=2,sqrtn(L,3),while(#S&&S[1]<=m^3, S=S[^1]); for(n=1,m-1,if(L

A321501 Numbers not of the form (x - y)(x^2 - y^2) with x > y > 0; complement of A321499.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 10, 12, 14, 18, 20, 22, 26, 28, 30, 34, 36, 38, 42, 44, 46, 50, 52, 54, 58, 60, 62, 66, 68, 70, 74, 76, 78, 82, 84, 86, 90, 92, 94, 98, 100, 102, 106, 108, 110, 114, 116, 118, 122, 124, 126, 130, 132, 134, 138, 140, 142, 146, 148, 150, 154, 156, 158, 162, 164, 166, 170, 172, 174, 178

Offset: 1

M. F. Hasler, Nov 22 2018

Equivalently, numbers not of the form (x - y)^2*(x + y) or d^2*(2m + d), for (x, y) = (m+d, m). This shows that excluded are all squares d^2 > 0 times any number of the same parity and larger than d. In particular, for d=1, all odd numbers > 1, and for d=2, 4*(even numbers > 4) = 8*(odd numbers > 2). For larger d, no further (neither odd nor even) numbers are excluded.

So apart from 0, 1 and 8, this consists of even numbers not multiple of 8. All these numbers occur, since for larger (odd or even) d, no additional term is excluded.

			a(1) = 0, a(2) = 1 and a(3) = 2 obviously can't be of the form (x - y)(x^2 - y^2) with x > y > 0, which is necessarily greater than 1*3 = 3.
See A321499 for examples of the terms that are not in the sequence.

See A321499 for the complement: numbers of the form (x-y)(x^2-y^2).

See A321491 for numbers of the form (x+y)(x^2+y^2).

is(n)={!n||!fordiv(n,d, d^2*(d+2)>n && break; n%d^2&&next; bittest(n\d^2-d,0)||return)} \\ Uses the initial definition. More efficient variant below:

select( is_A321501(n)=!bittest(n,0)&&(n%8||n<9)||n<3, [0..99]) \\ Defines the function is_A321501(). The select() command is an illustration and a check.

A321501_list(M)={setunion([1],setminus([0..M\2]*2,[2..M\8]*8))} \\ Return all terms up to M; more efficient than to use select(...,[0..M]) as above.

A321501(n)=if(n>6,(n-2)*9\/8*2,n>3,n*2-4,n-1)

Asymptotic density is 3/8.

a(n) = round((n-2)*9/8)*2 for all n > 6.

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A321499 Numbers of the form (x - y)(x^2 - y^2) with x > y > 0.

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A321490 Triangular table T[n,k] = (n+k)(n^2+k^2), 1 <= k <= n = 1, 2, 3, ...; read by rows.

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A321492 Numbers that can be written as (x + y)(x^2 + y^2), x > y > 0, in at least two ways.

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A321501 Numbers not of the form (x - y)(x^2 - y^2) with x > y > 0; complement of A321499.

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