A274141 -id:A274141 - cp's OEIS frontend

A274188 Number n such that there is a smaller positive number j == n (mod 5) such that sqrt(j*n) is an integer.

9, 16, 18, 20, 27, 32, 36, 40, 45, 48, 49, 54, 60, 63, 64, 72, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 120, 121, 125, 126, 128, 135, 140, 144, 147, 153, 160, 162, 169, 171, 176, 180, 189, 192, 196, 198, 200, 207, 208, 216, 220, 224, 225, 234, 240, 242, 243, 245, 250

Offset: 1

Vladimir Shevelev, Jun 12 2016

Or numbers n>=9 having a divisor t^2 > 1, where t=k/m, 1<= m < k, such that n == n/t^2 (mod 5).

Or positive numbers n such that if n == 0 (mod 5), then it divisible by 5^3 or by the square of some other prime; otherwise n divisible by k^2, such that there is a k_1, 0< k_1

A generalization see in our seqfan list from Jun 13 (correction Jun 14) 2016.

			9 is member, since 4 == 9 (mod 5) and 4*9 is a square;
32 is member, since 2 == 32 (mod 5) and 2*32 is a square;
45 is member, since 5 == 45 (mod 5) and 5*45 is a square.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A046790, A274141.

Select[Range@ 250, Function[n, Count[Sqrt[# n] & /@ Select[Range[n - 1], Mod[#, 5] == Mod[n, 5] &], k_ /; IntegerQ@ k] > 0]] (* Michael De Vlieger, Jun 14 2016 *)

isok(n)=j = n-5; while (j >0, if (issquare(j*n), return (1)); j -= 5;); 0; \\ Michel Marcus, Jun 14 2016

is(n)=!issquarefree(n/if(n%5, if(n%4, 1, 4), 5)) \\ Charles R Greathouse IV, Jun 15 2016

Let A(x) be the number of a(n)<=x. Then A(x)/x ~ 1 - 149/(20*Pi^2) = 1 - 7.45/Pi^2 = 0.24515718... as x goes to infinity. - Vladimir Shevelev, Jun 15 2016; corrected by Charles R Greathouse IV, Jun 15 2016

Corrected and extended by Michel Marcus, Jun 14 2016

A274240 Numbers n such that there is a smaller positive number j == n (mod 7) such that j*n is a square.

Original entry on oeis.org

16, 25, 28, 32, 36, 48, 50, 56, 63, 64, 72, 75, 80, 81, 84, 96, 100, 108, 112, 121, 125, 126, 128, 140, 144, 150, 160, 162, 168, 169, 175, 176, 180, 189, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 250, 252, 256, 272, 275, 280, 288, 289, 300, 304, 308

Offset: 1

Vladimir Shevelev, Jun 15 2016

nonn

Or numbers n >= 16 having a divisor t^2 > 1, where t=k/m, 1 <= m < k, such that n == n/t^2 (mod 7).
Or positive numbers n such that if n == 0 (mod 7), then n is divisible by 7^3 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 such that k_1^2 == k^2 (mod 7) (or, according to the comment in A130290, n is divisible by some k^2 >= 16).
For a generalization, see the seqfan list from Jun 13 (correction Jun 14) 2016.

			25 is a member, since 4 == 25 (mod 7) and 4*25 is a square.
32 is a member, since 18 == 32 (mod 7) and 18*32 is a square.

Cf. A046790, A274141, A274188.

is(n) = for(j=1, n-1, if(Mod(j, 7)==n && issquare(j*n), return(1))); return(0) \\ Felix Fröhlich, Jun 15 2016

More terms from Felix Fröhlich, Jun 15 2016

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

Original entry on oeis.org

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

Vladimir Shevelev, Jun 15 2016

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)

			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.

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

Cf. A046790, A274141, A274188, A274240.

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 *)

is(n) = for(j=1, n-1, if(Mod(j, 11)==n && issquare(j*n), return(1))); return(0) \\ Felix Fröhlich, Jun 15 2016

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

More terms from Felix Fröhlich, Jun 15 2016

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A274188 Number n such that there is a smaller positive number j == n (mod 5) such that sqrt(j*n) is an integer.

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A274240 Numbers n such that there is a smaller positive number j == n (mod 7) such that j*n is a square.

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions