A027564 -id:A027564 - cp's OEIS frontend

A025052 Numbers not of form ab + bc + ca for 1<=a<=b<=c (probably the list is complete).

1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462

Offset: 1

According to Borwein and Choi, if the Generalized Riemann Hypothesis is true, then this sequence has no larger terms, otherwise there may be one term greater than 10^11. - T. D. Noe, Apr 08 2004
Note that n+1 must be prime for all n in this sequence. - T. D. Noe, Apr 28 2004
Borwein and Choi prove (Theorem 6.2) that the equation N=xy+xz+yz has an integer solution x,y,z>0 if N contains a square factor and N is not 4 or 18. In the following simple proof explicit solutions are given. Let N=mn^2, m,n integer, m>0, n>1. If n3: x=6, y=n-3, z=n^2-4n+6. If n>m+1: if n=0 (mod m+1): x=m+1, y=m(m+1), z=m(n^2/(m+1)^2-1), if n=k (mod m+1), 0

J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.
Maohua Le, A note on positive integer solutions of the equation xy+yz+zx=n, Publ. Math. Debrecen 52 (1998) 159-165; Math. Rev. 98j:11016.
M. Peters, The Diophantine Equation xy + yz + zx = n and Indecomposable Binary Quadratic Forms, Experiment. Math., Volume 13, Issue 3 (2004), 273-274.

Subsequence of A000926 (numbers not of the form ab+ac+bc, 0A006093.
Cf. A027563, A027564, A027565, A027566, A055745, A034168, A094379, A094380, A094381.
Cf. A093669 (numbers having a unique representation as ab+ac+bc, 0A093670 (numbers having a unique representation as ab+ac+bc, 0<=a<=b<=c).

n=500; lim=Ceiling[(n-1)/2]; lst={}; Do[m=a*b+a*c+b*c; If[m<=n, lst=Union[lst, {m}]], {a, lim}, {b, lim}, {c, lim}]; Complement[Range[n], lst]

Corrected by R. H. Hardin

A027563 Numbers not of form abc + abd + acd + bcd for 1<=a<=b<=c<=d.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 21, 23, 26, 29, 30, 33, 35, 41, 48, 51, 53, 63, 65, 74, 75, 86, 89, 90, 98, 111, 113, 119, 125, 131, 141, 155, 158, 173, 179, 191, 209, 210, 233, 239, 251, 254, 273, 285, 293, 321, 323, 326, 338, 341, 345, 363, 419

Offset: 1

R. H. Hardin

This list is conjecturally complete, but this has not been proved. It may be complete as a consequence of the Generalized Riemann Hypothesis; see comments for A025052. - Harry Richman, Jan 09 2025

T. D. Noe, Table of n, a(n) for n=1..126 (complete sequence)

Cf. A025052 (3 variables), A027564 (5 variables), A027565, A027566.

A027565 Largest number not of form k_1 k_2 .. k_n (1/k_1 + .. + 1/k_n), k_i integers >= 1.

Original entry on oeis.org

1, 462, 52061, 33952307

Offset: 2

R. H. Hardin

			For n=2, the largest number not of the form k_1+k_2 with k_1>=1, k_2>=1 is 1! So a(2) = 1. For n=3 the largest number not of the form ab+bc+ca (a,b,c >= 1) is believed to be 462 (see A025052).

See A025052 for numbers of form ab+bc+ca (which is the case n=3), A027563 for n=4, A027564 for n=5. See also A027566.

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A025052 Numbers not of form ab + bc + ca for 1<=a<=b<=c (probably the list is complete).

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A027563 Numbers not of form abc + abd + acd + bcd for 1<=a<=b<=c<=d.

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A027565 Largest number not of form k_1 k_2 .. k_n (1/k_1 + .. + 1/k_n), k_i integers >= 1.

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