

A130282


Numbers n such that A130280(n^21) < n1, i.e., there is a k, 1 < k < n1, such that (n^21)(k^21)+1 is a perfect square.


3



11, 23, 39, 41, 59, 64, 83, 111, 134, 143, 153, 179, 181, 219, 263, 307, 311, 363, 373, 386, 419, 479, 543, 571, 584, 611, 683, 703, 759, 781, 839, 900, 923, 989, 1011, 1103, 1156, 1199, 1299, 1403, 1405, 1425, 1511, 1546, 1623, 1739, 1769, 1859, 1983, 2111
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OFFSET

1,1


COMMENTS

For any n>1, the number (n^21)(k^21)+1 is a square for k = n1 ; this sequence lists those n>1 for which there is a smaller k>1 having this property. This sequence contains the subsequence b(k) = 2k(k+1)1, k>1, for which A130280(b(k)^21) <= k < b(k)1, since (b(k)^21)(k^21)+1 = (2k^3+2k^22k1)^2. We have n=b(k) whenever 2n+3 is a square, the square root of which is then 2k+1. (See also formula.)
The only elements of this sequence not of the form P[m](k) (see formula) are seem to be nonminimal n>k+1 such that (k^21)(n^21)+1 is a square, for some k occurring earlier in this sequence (thus having A130280(n^21)=k): { 900, 1405, 19759...} with k=11; { 6161, 8322,... } with k=23, ...


LINKS



FORMULA

If 2n+3 is a square, then n = b(k)= 2k(k+1)1, k = (sqrt(n/2+3/4)1)/2 = floor(sqrt(n/2)) >= A130280(n^21). (For all k>1, b(k) is in this sequence.)
Most terms of this sequence are in the set { P[m](k), P[m](k) ; m=2,3,4..., k=2,3,4,... } with P[m] = 2 X P[m1]  P[m2], P[1]=X1, P[0]=1. Whenever a(n) = P[m](k) or a(n) = P[m](k) (m,k>1), then A130280(a(n)^21) <= k (resp. k1 for m=2) < a(n). (No case where equality does not hold is known so far.) We have P[2] = P[2](1X) and for all integers m>2,x>0: P[m](x) < (1)^m P[m](x) <= P[m+1](x) with equality iff x=2. We have P[m](1)=(1)^m (m+1), P[m](0)=(1)^(m(m+1)/2), P[m](1)=1m, P[m](x)>0 for all x >=2 ; P[m](x) ~ 2^(m1) x^m.


EXAMPLE

a(1) = 11 since n=11 is the smallest integer > 1 such that (n^21)(k^21)+1 is a square for 1 < k < n1, namely for k=2.
Values of P[2](k+1) = 2 k^2 + 2 k  1 for k=2,3,... are { 11,23,39,... } and A130280(11^21)=2, A130280(23^21)=3, A130280(39^21)=4,...
Values of P[3](k) = 4 k^3  4 k^2  3 k + 1 for k=2,3,4... are { 11,64,181,... } and A130280(64^21)=3, A130280(181^21)=4,...
Values of P[3](k) = 4 k^3 + 4 k^2  3 k  1 for k=2,3,4... are { 41,134,307,... } and A130280(134^21)=3, A130280(307^21)=4,...


PROG

(PARI) check(n) = { local( m = n^21 ); for( i=2, n2, if( issquare( m*(i^21)+1), return(i))) } t=0; A130282=vector(100, i, until(check(t++), ); t)
(PARI) P(m, x=x)=if(m>1, 2*x*P(m1, x)P(m2, x), m*(x2)+1)


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



STATUS

approved



