A006446 -id:A006446 - cp's OEIS frontend

A066377 Number of numbers m <= n such that floor(sqrt(m)) divides m.

1, 3, 6, 10, 16, 24, 33, 45, 60, 76, 96, 120, 145, 175, 210, 246, 288, 336, 385, 441, 504, 568, 640, 720, 801, 891, 990, 1090, 1200, 1320, 1441, 1573, 1716, 1860, 2016, 2184, 2353, 2535, 2730, 2926, 3136, 3360, 3585, 3825, 4080, 4336

Offset: 1

N. J. A. Sloane, Dec 23 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Partial sums of A006446.

{ n=0; a=0; for (m=1, 10^9, if (m%floor(sqrt(m)) == 0, a+=m; write("b066377.txt", n++, " ", a); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 12 2010

G.f.: -x*(x^3-x^2-x-1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Jan 12 2013

A261341 Numbers n such that round(n^(1/k)) divides n for all integers k>=1.

Original entry on oeis.org

1, 2, 4, 6, 12, 30, 36, 42, 72, 240, 420, 600, 900, 1560, 1764, 3600, 6084, 8100, 46440, 1742400, 4062240, 35814240

Offset: 1

Max Alekseyev, Aug 15 2015

There are no other terms below 10^16.

Is this a finite sequence?

Cf. A261205, A261206

Subsequence of A006446 and A261342.

isA[n_] :=
Block[{t},
  For[k = 2, (t = Floor[1/2 + n^(1/k)]) >= 2, k++,
   If[Mod[n, t] != 0, Return[False]]]; Return[True]]
Select[Range[1, 100000], isA[#] &] (* Julien Kluge, Apr 04 2016 *)

{ isA261341(n) = my(k,t); k=2; until(t<=2, t=round(sqrtn(n,k)); if(n%t,return(0)); k++); 1; }

A336068 Numbers k such that the exponent of the highest power of 3 dividing k (A007949) is a divisor of k.

Original entry on oeis.org

3, 6, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 48, 51, 54, 57, 60, 66, 69, 72, 75, 78, 84, 87, 90, 93, 96, 102, 105, 108, 111, 114, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 156, 159, 165, 168, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 204

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

All the terms are divisible by 3 by definition.

Šalát (1994) proved that the asymptotic density of this sequence is 0.287106... (A336069).

			3 is a term since A007949(3) = 1 is a divisor of 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tibor Šalát, On the function a_p, p^a_p(n) || n (n > 1), Mathematica Slovaca, Vol. 44, No. 2 (1994), pp. 143-151.

A055777 is a subsequence.
Cf. A007949, A336069.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336066.

Select[Range[200], Mod[#, 3] == 0 && Divisible[#, IntegerExponent[#, 3]] &]

isok(m) = if (!(m%3), (m % valuation(m,3)) == 0); \\ Michel Marcus, Jul 08 2020

A162190 Triangle read by rows in which row n lists the divisors of n, the n-th prime and the consecutive composites that are greater than the n-th prime, with a(0)=1.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 3, 5, 6, 1, 2, 4, 7, 8, 9, 10, 1, 5, 11, 12, 1, 2, 3, 6, 13, 14, 15, 16, 1, 7, 17, 18, 1, 2, 4, 8, 19, 20, 21, 22, 1, 3, 9, 23, 24, 25, 26, 27, 28, 1, 2, 5, 10, 29, 30, 1, 11, 31, 32, 33, 34, 35, 36, 1, 2, 3, 4, 6, 12, 37, 38, 39, 40

Offset: 0

Omar E. Pol, Jun 30 2009

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

Omar E. Pol, Illustration of initial terms: 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.

A241083 LCM of n and largest integer <= sqrt(n).

Original entry on oeis.org

1, 2, 3, 4, 10, 6, 14, 8, 9, 30, 33, 12, 39, 42, 15, 16, 68, 36, 76, 20, 84, 44, 92, 24, 25, 130, 135, 140, 145, 30, 155, 160, 165, 170, 35, 36, 222, 114, 78, 120, 246, 42, 258, 132, 90, 138, 282, 48, 49, 350, 357, 364, 371, 378, 385, 56, 399, 406, 413, 420

Offset: 1

J. Lowell, Apr 15 2014

a(n) = n iff n is in A006446. - Ivan Neretin, Apr 27 2017

			a(18) cannot be 18 because 18 is not a multiple of 4, the largest integer <= sqrt(18).

Ivan Neretin, Table of n, a(n) for n = 1..10000

A179204 is a sequence that can be defined in terms of this sequence.

Cf. A000196.

A241083:= n-> ilcm(n,floor(sqrt(n))): seq(A241083(n), n=1..50); # Wesley Ivan Hurt, Apr 15 2014

Table[LCM[n, Floor[Sqrt[n]]], {n, 50}] (* Wesley Ivan Hurt, Apr 15 2014 *)

a(n) = lcm(n, floor(sqrt(n))) = lcm(n, A000196(n)). - Wesley Ivan Hurt, Apr 15 2014

Extended by Wesley Ivan Hurt, Apr 15 2014

A284038 Lexicographically earliest sequence of distinct positive terms such that A000196(n) divides a(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 9, 15, 18, 21, 24, 27, 30, 16, 20, 28, 32, 36, 40, 44, 48, 52, 5, 25, 35, 45, 50, 55, 60, 65, 70, 75, 80, 42, 54, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 7, 14, 49, 56, 63, 77, 91, 98, 105, 112, 119, 133, 140, 147, 154, 64

Offset: 1

Rémy Sigrist, May 25 2017

nonn

This is a permutation of the natural numbers, with inverse A287433; for any n > 0, n appears among the first n^2 terms.
This sequence is similar to A075383: here we have runs of length 2*k+1, there of length k, of multiples of k.
a(p^2) = p for any prime p > 3.
All fixed points belong to A006446.
Conjecturally:
- all fixed points > 3 are squares,
- if a(n) < n, then A000196(n) belongs to A007310 \ {1},
- if k belongs to A007310 \ {1}, then a(n) < n for some n such that A000196(n) = k.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A284038
Index entries for sequences that are permutations of the natural numbers

Cf. A000196, A006446, A007310, A075383, A287433 (inverse).

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

Original entry on oeis.org

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.

A336063 Numbers divisible by the minimal exponent in their prime factorization (A051904).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

The asymptotic density of this sequence is 1 (Schinzel and Šalát, 1994).

			4 = 2^2 is a term since A051904(4) = 2 is a divisor of 4.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 331.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Schinzel and Tibor Šalát, Remarks on maximum and minimum exponents in factoring, Mathematica Slovaca, Vol. 44, No. 5 (1994), pp. 505-514.

Cf. A051904, A124184.
A005117 (except for 1) is subsequence.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336064.

h[1] = 0; h[n_] := Min[FactorInteger[n][[;; , 2]]]; Select[Range[2, 100], Divisible[#, h[#]] &]
Select[Range[2,100],Divisible[#,Min[FactorInteger[#][[All,2]]]]&] (* Harvey P. Dale, Aug 31 2020 *)

isok(m) = if (m>1, (m % vecmin(factor(m)[,2])) == 0); \\ Michel Marcus, Jul 08 2020

A079644 a(n) = (n mod sqrtint(n)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 3, 0, 1, 2, 3, 0, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 0, 1, 2, 3, 4, 5

Offset: 1

Benoit Cloitre, Jan 31 2003

Record values: given an m>=0, the first n for which a(n)=m is n = (m+1)^2+m = A028387(m). Also, for n>3, n is a square if and only if a(n)=0 and a(n-1)=0. - Stanislav Sykora, Aug 13 2014

Stanislav Sykora, Table of n, a(n) for n = 1..1023

Cf. A000290, A006446, A028387, A033638.

a:= proc(n) local r;
r:= isqrt(n);
if r^2 > n then r:= r-1 fi;
n mod r;
end proc:
seq(a(n),n=1..100); # Robert Israel, Aug 13 2014

A079644[n_]:=Mod[n,Floor[n^(1/2)]]; Array[A079644,200] (* Enrique Pérez Herrero, Oct 06 2011 *)
Table[Mod[n,Floor[Sqrt[n]]],{n,110}] (* Harvey P. Dale, Apr 10 2016 *)

PARI
```
a(n)=n%sqrtint(n)
```

a(A006446(n))=0; a(A033638(n))=1.
When n>0, a(A000290(n))=0; when n>1, a(A000290(n)-1)=0. - Stanislav Sykora, Aug 13 2014
a(n) = 0 if n or n+1 or 4*n+1 is a square, otherwise a(n) = a(n-1)+1. - Robert Israel, Aug 13 2014
G.f.: Sum_{r>=2} x^(r^2) * (x^r + 1) * ((r-1)*x^(r+1) - r*x^r + x)/(1 - x)^2. - Robert Israel, Aug 13 2014

Definition clarified by N. J. A. Sloane, Jan 11 2025

A379604 a(n) is the maximum number of items in a bucket for the bucket sort algorithm with input {0, 1, ..., n-1} and B = floor(sqrt(n)) buckets.

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 4, 4, 3, 4, 4, 4, 5, 5, 5, 4, 5, 5, 5, 5, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 9, 10, 10, 10

Offset: 1

Darío Clavijo, Dec 28 2024

With uniformly distributed input, the maximum number of items in a bucket is ceiling(n/B).

If B divides n then all buckets hold the same number of items, which is when n is in A006446.

			For n = 10 a(10) = 4 because:
Input array: [0,1,2,3,4,5,6,7,8,9] and floor(sqrt(10)) = 3.
Resulting 3 buckets of [0, 1, 2, 3], [4, 5, 6, 7], [8, 9] and the number of items in the buckets is [4,4,2], which is maximum a(10) = 4.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Bucket sort

Cf. A000196, A000196, A006446.

A379604[n_] := Ceiling[n/Floor[Sqrt[n]]]; Array[A379604, 100] (* Paolo Xausa, Feb 03 2025 *)

from sympy.core.intfunc import isqrt
a = lambda n: ((n-1) // isqrt(n)) + 1
print([a(n) for n in range(1,85)])

a(n) = ceiling(n/floor(sqrt(n))).

cp's OEIS Frontend

A066377 Number of numbers m <= n such that floor(sqrt(m)) divides m.

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A261341 Numbers n such that round(n^(1/k)) divides n for all integers k>=1.

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A336068 Numbers k such that the exponent of the highest power of 3 dividing k (A007949) is a divisor of k.

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A162190 Triangle read by rows in which row n lists the divisors of n, the n-th prime and the consecutive composites that are greater than the n-th prime, with a(0)=1.

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A241083 LCM of n and largest integer <= sqrt(n).

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A284038 Lexicographically earliest sequence of distinct positive terms such that A000196(n) divides a(n).

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

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Mathematica

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A336063 Numbers divisible by the minimal exponent in their prime factorization (A051904).

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A079644 a(n) = (n mod sqrtint(n)).