A006446 -id:A006446 - cp's OEIS frontend

A079647 Integer part of the cube root of n and integer part of the square root of n both divide n.

1, 2, 3, 4, 6, 8, 12, 16, 20, 24, 30, 36, 42, 48, 63, 64, 72, 80, 100, 120, 195, 210, 240, 288, 306, 324, 342, 399, 420, 441, 462, 483, 528, 552, 576, 600, 624, 728, 729, 756, 783, 900, 1190, 1260, 1764, 1848, 1980, 2600, 2652, 2704, 3024, 3080, 3136, 3192

Offset: 1

Benoit Cloitre, Jan 31 2003

nonn

			floor(20^(1/2)) = 4 and floor(20^(1/3)) = 2, hence 20 is in the sequence.

Cf. A006446.

Select[Range[3192], Mod[#, Floor[Sqrt[#]]] == 0 && Mod[#, Floor[#^(1/3)]] == 0 &] (* T. D. Noe, Dec 04 2013 *)

isok(n) = !(n % sqrtint(n)) && !(n % sqrtnint(n, 3)); \\ Michel Marcus, Dec 02 2013

Terms corrected by Michel Marcus, Dec 02 2013

A161416 Partial sums of A056737.

Original entry on oeis.org

0, 1, 3, 3, 7, 8, 14, 16, 16, 19, 29, 30, 42, 47, 49, 49, 65, 68, 86, 87, 91, 100, 122, 124, 124, 135, 141, 144, 172, 173, 203, 207, 215, 230, 232, 232, 268, 285, 295, 298, 338, 339, 381, 388, 392, 413, 459, 461, 461, 466, 480, 489, 541, 544, 550, 551, 567, 594

Offset: 1

Omar E. Pol, Jun 21 2009

nonn

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Divisors and pi(x)

Cf. A056737, A018253, A160811, A160812, A161205, A161344, A161345, A161424, A006446, A161828, A161835, A219729, A219730.

a(n) = A219730(n) - A219729(n). - Tamas Sandor Nagy, Jan 20 2024

Extended beyond a(16) by R. J. Mathar, Aug 01 2009

A162192 Triangle read by rows in which row n lists the divisors of n, prime(n), the consecutive composites that are greater than prime(n), and prime (n+1), but row 0 is formed by 1 and 2.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, 3, 5, 6, 7, 1, 2, 4, 7, 8, 9, 10, 11, 1, 5, 11, 12, 13, 1, 2, 3, 6, 13, 14, 15, 16, 17, 1, 7, 17, 18, 19, 1, 2, 4, 8, 19, 20, 21, 22, 23, 1, 3, 9, 23, 24, 25, 26, 27, 28, 29, 1, 2, 5, 10, 29, 30, 31

Offset: 0

Omar E. Pol, Jun 30 2009

See also A162190, a sequence with a similar structure.

			Triangle begins:
1,(2);
1,(2),(3);
1,.2.,(3),4,(5);
1,.....3,...(5),6,(7);
1,.2,.....4,......(7),8,.9,10,(11);
1,...........5,...............(11),12,(13);
1,.2,..3,.......6,....................(13),14,15,16,(17);
1,.................7,...............................(17),18,(19);
1,.2,.....4,..........8,....................................(19),20,21,22,(23);

Omar E. Pol, Illustration: Divisors and pi(x)
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos

Cf. A000005, A000040, A000720, A027750, A018253, A160811, A160812, A161205, A161344, A161345, A161424, A006446, A161827, A161828, A161835, A162190.

A204330 a(n) is the number of k satisfying 1 <= k <= n and such that floor(sqrt(k)) divides k.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 21, 22, 22, 22, 22, 22, 22, 22

Offset: 1

Benoit Cloitre, Jan 14 2012

nonn

a(n) = floor(2*sqrt(n)) + floor(sqrt(n-1)) - 1 if n belongs to A135106 otherwise a(n) = floor(2*sqrt(n)) + floor(sqrt(n-1)) - 2.

B. Cloitre, Some divisibility sequences

Cf. A006446, A079631.

Accumulate[Boole[Table[IntegerQ[n/Floor[n^(1/2)]], {n, 1, 70}]]]  (* Geoffrey Critzer, May 25 2013 *)

PARI
```
a(n)=sum(k=1,n,if(k%sqrtint(k),0,1));
```

a(n) = card{j>=1, A006446(j)<=n}.

Corrected by Geoffrey Critzer, May 25 2013

A339473 Numbers k such that floor(sqrt(k)) divides k^2, but does not divide k.

Original entry on oeis.org

18, 22, 68, 76, 84, 87, 93, 96, 150, 162, 260, 264, 268, 276, 280, 284, 330, 336, 348, 354, 410, 430, 588, 612, 630, 635, 640, 645, 655, 660, 665, 670, 738, 747, 765, 774, 798, 826, 1032, 1040, 1048, 1064, 1072, 1080, 1302, 1308, 1314, 1320, 1326, 1338, 1344, 1350

Offset: 1

Wesley Ivan Hurt, Apr 24 2021

nonn

			18 is in the sequence since floor(sqrt(18)) = 4, which does not divide 18, but it does divide 18^2 = 324.

Cf. A006446.

Flatten[Table[If[(1 - Ceiling[n^2/Floor[Sqrt[n]]] + Floor[n^2/Floor[Sqrt[n]]]) (Ceiling[n/Floor[Sqrt[n]]] - Floor[n/Floor[Sqrt[n]]]) == 1, n, {}], {n, 2000}]]

isok(k) = (k % sqrtint(k)) && !(k^2 % sqrtint(k)); \\ Michel Marcus, Apr 24 2021

from math import isqrt
def ok(k): r = isqrt(k); return k % r != 0 and k**2 % r == 0
print(list(filter(ok, range(1, 1351)))) # Michael S. Branicky, Apr 24 2021

cp's OEIS Frontend

A079647 Integer part of the cube root of n and integer part of the square root of n both divide n.

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A161416 Partial sums of A056737.

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A162192 Triangle read by rows in which row n lists the divisors of n, prime(n), the consecutive composites that are greater than prime(n), and prime (n+1), but row 0 is formed by 1 and 2.

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A204330 a(n) is the number of k satisfying 1 <= k <= n and such that floor(sqrt(k)) divides k.

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Mathematica

PARI

Formula

Extensions

A339473 Numbers k such that floor(sqrt(k)) divides k^2, but does not divide k.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Python