A321499 -id:A321499 - cp's OEIS frontend

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

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.

A106505 Ordered and uniqued length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.

Original entry on oeis.org

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

Offset: 1

Lekraj Beedassy, May 04 2005

nonn

The terms are proposed without repetition. For example, there exist two such triangles with a length of side = 33. They correspond respectively to s^2 - r^2 = 33 (see formula) with (r, s) = (4, 7) and sides (33, 28, 16), and the other triangle with (r, s) = (16, 17) and sides (33, 272, 256). Lengths = 39, 51, 57, 69, 75, ... correspond to two distinct triangles ... The lengths of these sides are proposed with repetition in A343064. - Bernard Schott, Apr 22 2021

Cf. A106499-A106506, A106410, A106420, A106430, A321499.

Cf. A343063, A343064, A343065, A343066, A343067.

Values s^2 - r^2, where r

Conjecture: for n>2, a(n+5) = a(n) + 8. - Ralf Stephan, Nov 16 2010.

Empirical g.f.: -x*(x^7+x^6+3*x^5-2*x^4-2*x^3-2*x^2-2*x-5) / ((x-1)^2*(x^4+x^3+x^2+x+1)). - Colin Barker, Oct 05 2013

Extended by Ray Chandler, May 09 2005

A321498 Numbers which can be written in at least two ways in the form (x-y)*(x^2-y^2) with x > y > 0.

Original entry on oeis.org

45, 63, 81, 96, 99, 117, 128, 135, 153, 160, 171, 175, 189, 192, 207, 224, 225, 243, 256, 261, 275, 279, 288, 297, 315, 320, 325, 333, 351, 352, 360, 369, 375, 384, 387, 405, 416, 423, 425, 432, 441, 448, 459, 475, 477, 480, 495, 504, 512, 513, 525, 531, 539, 544, 549, 567, 575, 576, 585

Offset: 1

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

nonn

An equivalent form is (x - y)^2*(x + y), or d^2*(d + 2y), where d = x - y > 0 and y > 0. See also A321499.

			   45 = (4 - 1)*(4^2 - 1^2) = (23 - 22)*(23^2 - 22^2),
   63 = (5 - 2)*(5^2 - 2^2) = (32 - 31)*(32^2 - 31^2),
   81 = (6 - 3)*(6^2 - 3^2) = (41 - 40)*(41^2 - 40^2),
   96 = (5 - 1)*(5^2 - 1^2) = (13 - 11)*(13^2 - 11^2),
   99 = (7 - 4)*(7^2 - 4^2) = (50 - 49)*(50^2 - 49^2),
  117 = (8 - 5)*(8^2 - 5^2) = (59 - 58)*(59^2 - 58^2).

David A. Corneth, Table of n, a(n) for n = 1..13937 (terms < 10^5)
Geoffrey B. Campbell, (m-n)(m^2-n^2) in two different ways, LinkedIn Number Theory Group, Aug. 2018.

Cf. A321499.

aQ[n_] := Length[Solve[(x-y)*(x^2-y^2) ==n && x > y && y > 0, {x,y}, Integers]] > 1; Select[Range[600], aQ] (* Amiram Eldar, Dec 06 2018 *)

select( is_A321498(n,c=2)={n&&!issquarefree(n)&&fordiv(n, d, d^2*(d+2)>n && break; n%d^2&&next; bittest(n\d^2-d, 0)||c--||return(1))}, [0..999]) \\ Define the function is_A321498(). \\ ~30% speed up by David A. Corneth, Nov 23 2018

is(n) = {if(issquarefree(n), return(0)); if(n % 2 == 0, if(n % 8 == 0, n\=8, return(0))); f = factor(n); e = select(x -> x > 1, f[, 2], 1); if(#e == 0 || n == 1, return(0), k = e[1]); n > f[k, 1]^3} \\ David A. Corneth, Dec 01 2018

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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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A106505 Ordered and uniqued length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.

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A321498 Numbers which can be written in at least two ways in the form (x-y)*(x^2-y^2) with x > y > 0.

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