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

36, 44, 49, 64, 72, 81, 88, 98, 99, 100, 108, 128, 132, 144, 147, 162, 169, 176, 180, 192, 196, 198, 200, 216, 220, 225, 243, 245, 252, 256, 264, 275, 288, 289, 294, 297, 300, 308, 320, 324, 338, 343, 352, 360, 361, 384, 392, 396, 400, 405, 432, 440, 441, 448

Offset: 1

Author

Vladimir Shevelev, Jun 15 2016

Keywords

• nonn
• easy

Comments

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).

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).

For a generalization, see the Sequence Fans mailing list for Jun 13 2016 (correction Jun 14 2016).

From David A. Corneth, Jun 26 2016: (Start)

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.

Summarizing, k is a term iff

- k is of the form k^2 for floor(11/2) < k except k = 11.

- k is of the form 11 * p^2 for p < floor(11/2)

- of the form k * t for k of one of the forms above and integer t > 0. (End)

Examples

			49 is member, since 16 == 49 (mod 11) and 16*49 is a square.
108 is member, since 75 == 108 (mod 11) and 75*108 is a square.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• David A. Corneth, n, a(n) and j as described in name for n = 1..10000

Crossrefs

Cf. A046790, A274141, A274188, A274240.

Programs

Mathematica

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 *)

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

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

Extensions

More terms from Felix Fröhlich, Jun 15 2016