cp's OEIS Frontend

A274241 Numbers n such that there is a smaller positive number j == n (mod 11) such that sqrt(j*n) is an integer.

%I A274241 #32 Jan 24 2019 11:45:55
%S A274241 36,44,49,64,72,81,88,98,99,100,108,128,132,144,147,162,169,176,180,
%T A274241 192,196,198,200,216,220,225,243,245,252,256,264,275,288,289,294,297,
%U A274241 300,308,320,324,338,343,352,360,361,384,392,396,400,405,432,440,441,448
%N A274241 Numbers n such that there is a smaller positive number j == n (mod 11) such that sqrt(j*n) is an integer.
%C A274241 Or numbers n >= 36 having a divisor t^2 > 1, where t=k/m, 1 <= m < k, such that n == n/t^2 (mod 11).
%C A274241 Or positive numbers n such that if n == 0 (mod 11), then n is divisible by 11^3 or by the square of some other prime; otherwise n is divisible by k^2, such that there is a k_1, 0 < k_1 < k with k_1^2 == k^2 (mod 11) (or, according to the comment in A130290, n is divisible by some k^2 >= 36).
%C A274241 For a generalization, see the Sequence Fans mailing list for Jun 13 2016 (correction Jun 14 2016).
%C A274241 From _David A. Corneth_, Jun 26 2016: (Start)
%C A274241 If k is a term then m * k is a term for m > 0. Hence closed under multiplication. For k > 11, k^2 is in the sequence. So k^t is as well for t > 2.
%C A274241 Summarizing, k is a term iff
%C A274241 - k is of the form k^2 for floor(11/2) < k except k = 11.
%C A274241 - k is of the form 11 * p^2 for p < floor(11/2)
%C A274241 - of the form k * t for k of one of the forms above and integer t > 0. (End)
%H A274241 David A. Corneth, <a href="/A274241/b274241.txt">Table of n, a(n) for n = 1..10000</a>
%H A274241 David A. Corneth, <a href="/A274241/a274241.gp.txt">n, a(n) and j as described in name for n = 1..10000</a>
%e A274241 49 is member, since 16 == 49 (mod 11) and 16*49 is a square.
%e A274241 108 is member, since 75 == 108 (mod 11) and 75*108 is a square.
%t A274241 Select[Range@500, Function[n, AnyTrue[Range[n - 1], And[Mod[#, 11] == Mod[n, 11], IntegerQ@ Sqrt[# n]] &]]] (* _Michael De Vlieger_, Jun 23 2016, Version 10 *)
%o A274241 (PARI) is(n) = for(j=1, n-1, if(Mod(j, 11)==n && issquare(j*n), return(1))); return(0) \\ _Felix Fröhlich_, Jun 15 2016
%o A274241 (PARI) is(n)=my(f=factor(n)); f[,2]=f[,2]%2; t=prod(i=1,matsize(f)[1], f[i,1] ^ f[i,2]); for(i=1,sqrtint((n-1)\t), if(Mod(t*i^2, 11)==n,return(1))); 0 \\ _David A. Corneth_, Jun 26 2016
%Y A274241 Cf. A046790, A274141, A274188, A274240.
%K A274241 nonn,easy
%O A274241 1,1
%A A274241 _Vladimir Shevelev_, Jun 15 2016
%E A274241 More terms from _Felix Fröhlich_, Jun 15 2016