A130284 -id:A130284 - cp's OEIS frontend

A130280 a(n) = smallest integer k>1 such that n(k^2-1)+1 is a perfect square, or 0 if no such number exists.

2, 5, 3, 0, 2, 3, 5, 2, 0, 3, 7, 5, 4, 11, 3, 2, 4, 13, 9, 7, 2, 5, 19, 4, 0, 5, 21, 3, 11, 9, 11, 14, 2, 29, 5, 3, 6, 31, 21, 2, 13, 11, 13, 169, 3, 7, 41, 6, 0, 7, 5, 11, 22, 419, 3, 2, 5, 23, 461, 27, 8, 55, 7, 4, 2, 3, 49, 29

Offset: 1

M. F. Hasler, May 20 2007, May 25 2007

A084702(n) = a(n)^2-1, resp. a(n) = sqrt(A084702(n)+1). See A130283 for values where A130280(n)=0.

			a( (2k)^2 ) <= k since (2k)^2(k^2-1)+1 = (2k^2-1)^2 (but k=1 is excluded since with k^2-1=0 this would be a trivial solution for any n).

M. F. Hasler, Table of n, a(n) for n = 1..1000

Cf. A084702, A094357, A130281, A130282, A130283, A130284.

See also A306767.

A130280:=proc(n) local x,y,z; if n=1 then return 2 fi; isolve(n*(x^2-1)+1=y^2,z); select(has,`union`(%),x); map(rhs,%); simplify(eval(%,z=1) union eval(%,z=0)) minus {-1,1}; if %={} then 0 else (min@op@map)(abs,%) fi end;

$MaxExtraPrecision = 100;
r[n_, c_] := Reduce[k > 1 && j > 1 && n*(k^2 - 1) + 1 == j^2, {j, k}, Integers] /. C[1] -> c // Simplify;
a[n_] := If[rn = r[n,0] || r[n,1] || r[n,2]; rn === False, 0, k /. {ToRules[rn]} // Min];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 800}] (* Jean-François Alcover, May 12 2017 *)

{A130280(n,L=10^15)=if(issquare(n),L=2+sqrtint(n>>2)); for( k=2, L, if( issquare(n*(k^2-1)+1),return(k)))}

If n=(2k)^2, then A130280(n) <= k, since (2k)^2(k^2-1)+1 = (2k^2-1)^2. See A130281 for the cases where equality does not hold. If n=k^2-1, then A130280(n) <= k-1 since (k^2-1)((k-1)^2-1)+1 = (k^2-k-1)^2. See A130282 for the cases where equality does not hold.

A130283 Integers n > 0 for which A130280(n) = 0, i.e., such that there is no integer m > 1 for which n(m^2 - 1) + 1 is a square.

Original entry on oeis.org

4, 9, 25, 49, 81, 121, 169, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569, 7921, 8281, 8649, 9025, 9409

Offset: 1

M. F. Hasler, May 24 2007

nonn

No term > 4 in this sequence is an even square (see formula in A130280).

A001248(k) is a term for any k. - Jinyuan Wang, Apr 14 2019

			a(1)=4 since 1(2^2-1)+1=2^2, 2(5^2-1)+1=7^2, 3(3^2-1)+1=5^2 but 4(m^2-1)+1 = 4m^2-3 can't be a square because the largest square < 4m^2 is (2m-1)^2 = 4m^2-4m+1 < 4m^2-3 for m>1.
a(2)=9 since for n=5,6,7,8 one has m=2,3,5,2, but 9(m^2-1)+1 = 9m^2-8 > 9m^2-11 >= 9m^2-6m+1 = (3m-1)^2 and therefore can't be a square.

Cf. A001248, A084702, A130280, A130284, A130288.

$MaxExtraPrecision = 200;
r[n_, c_] := Reduce[k > 1 && j > 1 && n*(k^2 - 1) + 1 == j^2, {j, k}, Integers] /. C[1] -> c // Simplify;
A130280[n_] := If[rn = r[n, 0] || r[n, 1] || r[n, 2]; rn === False, 0, k /. {ToRules[rn]} // Min];
Reap[For[n=1, n <= 2000, n++, If[A130280[n]==0, Print[n]; Sow[n]]]][[2,1]] (* Jean-François Alcover, May 12 2017 *)

f(n) = for(k=2, n+1, if( issquare(n*(k^2-1)+1), return(k)))
is(n) = issquare(n) && f(n) == 0; \\ Jinyuan Wang, Apr 14 2019

More terms from Jean-François Alcover, May 12 2017

More terms from Jinyuan Wang, Apr 14 2019

A130288 Record indices of A130280: integers n>0 for which min{ m>1 | (2n+1)^2(m^2-1)+1 is a square} < oo but bigger than for all preceding n.

Original entry on oeis.org

1, 2, 11, 14, 18, 23, 27, 34, 38, 44, 54, 59, 74, 158, 179, 284, 524

Offset: 1

M. F. Hasler, May 24 2007

nonn

Most elements of this sequence seem to be 1,2 or 4 times a prime.

Corresponding values of A130280 are given in A130289. - M. F. Hasler, May 24 2007

Cf. A084702, A130280, A130283, A130284, A130289.

A130288(L=999,S=1)={local(R,T);for(n=S,L, if(issquare(n) || R>=T=A130280(n),next); print1(n", ");R=T)}

A130289 Record values in A130280: minima of { m>1 | (2n+1)^2(m^2-1)+1 is a square} bigger than for all preceding n.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 21, 29, 31, 169, 419, 461, 3269, 7127, 3877019, 22783559, 215308729

Offset: 1

M. F. Hasler, May 24 2007

nonn

Most elements of this sequence seem to be 1,2 or 4 times a prime.

Cf. A084702, A130280, A130283, A130284, A130288.

A130289( L=999, S=1 )={ local( R, T ); for( n=S, L, if( issquare(n) || R >= T = A130280(n), next ); print1( T ", " ); R=T )}

cp's OEIS Frontend

A130280 a(n) = smallest integer k>1 such that n(k^2-1)+1 is a perfect square, or 0 if no such number exists.

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A130283 Integers n > 0 for which A130280(n) = 0, i.e., such that there is no integer m > 1 for which n(m^2 - 1) + 1 is a square.

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A130288 Record indices of A130280: integers n>0 for which min{ m>1 | (2n+1)^2(m^2-1)+1 is a square} < oo but bigger than for all preceding n.

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A130289 Record values in A130280: minima of { m>1 | (2n+1)^2(m^2-1)+1 is a square} bigger than for all preceding n.

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