A324177 -id:A324177 - cp's OEIS frontend

A324174 Integers k such that 2*floor(sqrt(k)) divides k.

2, 4, 8, 12, 16, 24, 30, 36, 48, 56, 64, 80, 90, 100, 120, 132, 144, 168, 182, 196, 224, 240, 256, 288, 306, 324, 360, 380, 400, 440, 462, 484, 528, 552, 576, 624, 650, 676, 728, 756, 784, 840, 870, 900, 960, 992, 1024, 1088, 1122, 1156, 1224, 1260, 1296

Offset: 1

Jinyuan Wang, Mar 09 2019

nonn

Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A006446, A324175, A324176, A324177, A324178.

Select[ Range[ 1000 ], Mod[ #, 2*Floor[ Sqrt[ # ]//N ] ]==0& ]
LinearRecurrence[{1,0,2,-2,0,-1,1},{2,4,8,12,16,24,30},70] (* Harvey P. Dale, Dec 11 2022 *)

PARI
```
is(n) = n%(2*sqrtint(n)) == 0;
```

For k >= 1, a(3k-2) = 4k^2 - 2k, a(3k-1) = 4k^2 and a(3k) = 4k^2 + 4k.

A324178 Integers k such that floor(sqrt(k)) + floor(sqrt(k/5)) divides k.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 24, 28, 35, 40, 45, 50, 60, 66, 77, 91, 112, 128, 153, 190, 200, 220, 231, 276, 312, 338, 378, 406, 435, 450, 480, 496, 512, 561, 578, 648, 703, 722, 741, 780, 800, 840, 882, 903, 946, 968, 990, 1058, 1152, 1176, 1250, 1300, 1352, 1378

Offset: 1

Jinyuan Wang, Mar 09 2019

nonn

This sequence is infinite for the same reason that A324175 is: if x > y satisfies x^2 - 5*y^2 = -1 (x=A075796(j), y=A007805(j-1), j>0), then x < 5*y. Let k = 5*y^2 + m. By the pigeonhole principle there exists a number m belonging to [0, 2*x - 1] such that x + y | 5*y^2 + m, so such a k is a term.

Cf. A007805, A075796, A324174, A324175, A324176, A324177.

Select[Range[1378], Mod[#, Floor@ Sqrt@ # + Floor@ Sqrt[#/5]] == 0 &] (* Giovanni Resta, Apr 05 2019 *)

is(n) = n%(floor(sqrt(n)) + floor(sqrt(n/5))) == 0;

cp's OEIS Frontend

A324174 Integers k such that 2*floor(sqrt(k)) divides k.

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