A195085 -id:A195085 - cp's OEIS frontend

A019554 Smallest number whose square is divisible by n.

1, 2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, 19, 10, 21, 22, 23, 12, 5, 26, 9, 14, 29, 30, 31, 8, 33, 34, 35, 6, 37, 38, 39, 20, 41, 42, 43, 22, 15, 46, 47, 12, 7, 10, 51, 26, 53, 18, 55, 28, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70, 71, 12, 73, 74, 15, 38, 77

Offset: 1

R. Muller

A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), and b*c = A019554(n) = "outer square root" of n.
Instead of the terms "inner square root" and "outer square root", we may use the terms "lower square root" and "upper square root", respectively. Upper k-th roots have been studied by Broughan (2002, 2003, 2006). - Petros Hadjicostas, Sep 15 2019
The number of times each number k appears in this sequence is A034444(k). The first time k appears is at position A102631(k). - N. J. A. Sloane, Jul 28 2021

T. D. Noe, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105-114. See also here for another copy.
Kevin A. Broughan, Relationship between the integer conductor and k-th root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.
Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121-136.
Florentin Smarandache, Collected Papers, Vol. II, Tempus Publ. Hse, Bucharest, 1996.
Eric Weisstein's World of Mathematics, Smarandache Ceil Function.

Cf. A000188 (inner square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A053166 (outer 4th root), A015052 (outer 5th root), A015053 (outer 6th root).

Cf. also A007913, A007947, A008833, A015049, A034444, A053143, A102631, A346602, A005117, A133466, A195085.

a019554 n = product $ zipWith (^)
            (a027748_row n) (map ((`div` 2) . (+ 1)) $ a124010_row n)
-- Reinhard Zumkeller, Apr 13 2013
(Python 3.8+)
from math import prod
from sympy import factorint
def A019554(n): return n//prod(p**(q//2) for p, q in factorint(n).items()) # Chai Wah Wu, Aug 18 2021

with(numtheory):A019554 := proc(n) local i: RETURN(op(mul(i,i=map(x->x[1]^ceil(x[2]/2),ifactors(n)[2])))); end;

Flatten[Table[Select[Range[n],Divisible[#^2,n]&,1],{n,100}]] (* Harvey P. Dale, Oct 17 2011 *)
f[p_, e_] := p^Ceiling[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

a(n)=n/core(n,1)[2] \\ Charles R Greathouse IV, Feb 24 2011

Replace any square factors in n by their square roots.
Multiplicative with a(p^e) = p^ceiling(e/2).
Dirichlet series:
   Sum_{n>=1} a(n)/n^s = zeta(2*s-1)*zeta(s-1)/zeta(2*s-2), (Re(s) > 2);
   Sum_{n>=1} (1/a(n))/n^s = zeta(2*s+1)*zeta(s+1)/zeta(2*s+2), (Re(s) > 0).
a(n) = n/A000188(n).
a(n) = denominator of n/n^(3/2). - Arkadiusz Wesolowski, Dec 04 2011
a(n) = Product_{k=1..A001221(n)} A027748(n,k)^ceiling(A124010(n,k)/2). - Reinhard Zumkeller, Apr 13 2013
Sum_{k=1..n} a(k) ~ 3*zeta(3)*n^2 / Pi^2. - Vaclav Kotesovec, Sep 18 2020
Sum_{k=1..n} 1/a(k) ~ 3*log(n)^2/(2*Pi^2) + (9*gamma/Pi^2 - 36*zeta'(2)/Pi^4)*log(n) + 6*gamma^2/Pi^2 - 108*gamma*zeta'(2)/Pi^4 + 432*zeta'(2)^2/Pi^6 - 36*zeta''(2)/Pi^4 - 15*sg1/Pi^2, where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Jul 27 2021
a(n) = sqrt(n*A007913(n)). - Jianing Song, May 08 2022
a(n) = sqrt(A053143(n)). - Amiram Eldar, Sep 02 2023
From Mia Boudreau, Jul 17 2025: (Start)
a(n^2) = n.
a(A005117(n)) = A005117(n).
a(A133466(n)) = A133466(n)/2.
a(A195085(n)) = A195085(n)/3. (End)

A057918 Number of pairs of numbers (r,s) each less than n such that (r,s,n) is in geometric progression.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 3, 0, 2, 0, 1, 0, 0, 0, 1, 4, 0, 2, 1, 0, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 3, 6, 4, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 2, 7, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 4, 1, 0, 0, 0, 3, 8, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 3, 0, 6, 2, 9, 0, 0, 0, 1, 0

Offset: 1

Henry Bottomley, Nov 22 2000

Also, the number of integers k in {1,2,...,n-1} such that k*n is square. - John W. Layman, Sep 08 2011

			a(72)=5 since (2,12,72), (8,24,72), (18,36,72), (32,48,72), (50,60,72) are the possible three term geometric progressions.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000188, A005117, A133466.

Cf. A132345 (partial sums).

a057918 n = sum $ map ((0 ^) . (`mod` n) . (^ 2)) [1..n-1]
-- Reinhard Zumkeller, Mar 27 2012

a(n) = A000188(n) - 1.

a(A005117(n)) = 0; a(A013929(n)) > 0; A008966(n) = A000007(a(n)); a(A133466(n)) = 1; a(A195085(n)) = 2. - Reinhard Zumkeller, Mar 27 2012

A133466 Positive integers k for which there is exactly one integer i in {1,2,3,...,k-1} such that i*k is a square.

Original entry on oeis.org

4, 8, 12, 20, 24, 28, 40, 44, 52, 56, 60, 68, 76, 84, 88, 92, 104, 116, 120, 124, 132, 136, 140, 148, 152, 156, 164, 168, 172, 184, 188, 204, 212, 220, 228, 232, 236, 244, 248, 260, 264, 268, 276, 280, 284, 292, 296, 308, 312, 316, 328, 332, 340, 344, 348, 356

Offset: 1

John W. Layman, Nov 28 2007

nonn

It appears that all terms of this sequence are exactly four times those of the squarefree integers (A005117).
The observed behavior is true for all n. All positive integers n are written uniquely as k*m^2 where k is squarefree, k >=1, m >= 1. The square multiples of n are j^2*k*n, j >= 1. We seek n with exactly 1 multiple that is square and less than n^2. If m = 1, there are no such multiples as we have k = n, so the least square multiple is n^2. If m >= 2, k*n is square and less than n^2. However, 4*k*n also qualifies as square and less than n^2 if m > 2. So the qualifying values of n are those with m=2. - Peter Munn, Nov 28 2019
The asymptotic density of this sequence is 3/(2*Pi^2). - Amiram Eldar, Mar 08 2021

			4 is in the sequence because among the products 1*4,2*4,3*4 = 4,8,12 there is exactly one square.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A005117, A007283, A225546.

Cf. A057918, A195085.

a133466 n = a133466_list !! (n-1)
a133466_list = map (+ 1) $ elemIndices 1 a057918_list
-- Reinhard Zumkeller, Mar 27 2012

[k:k in [1..350]|#[m:m in [1..k-1]| IsSquare(m*k)] eq 1]; // Marius A. Burtea, Dec 03 2019

eoiQ[n_]:=Count[n*Range[n-1],?(IntegerQ[Sqrt[#]]&)]==1; Select[Range[ 400],eoiQ] (* _Harvey P. Dale, Mar 14 2015 *)

isok(n) = sum(k=1, n-1, issquare(k*n)) == 1; \\ Michel Marcus, Nov 29 2019

from math import isqrt
from sympy import mobius
def A133466(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m)<<2 # Chai Wah Wu, Aug 15 2024

A057918(a(n)) = 1. - Reinhard Zumkeller, Mar 27 2012
From Peter Munn, Nov 28 2019: (Start)
a(n) = 4 * A005117(n).
{a(n)} = {A225546(A007283(n)) : n >= 0}, where {a(n)} denotes the set of integers in the sequence.
(End)
Sum_{n>=1} 1/a(n)^s = zeta(s)/(4^s*zeta(2*s)), s>1. - Amiram Eldar, Sep 26 2023

A335437 Numbers k with a partition into two distinct parts (s,t) such that k | s*t.

Original entry on oeis.org

9, 16, 18, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 121, 125, 126, 128, 135, 144, 147, 150, 153, 160, 162, 169, 171, 175, 176, 180, 189, 192, 196, 198, 200, 207, 208, 216, 224, 225, 234, 240, 242, 243, 245, 250, 252, 256

Offset: 1

Wesley Ivan Hurt, Jun 10 2020

nonn

All values of this sequence are nonsquarefree (A013929).
From Peter Munn, Nov 23 2020: (Start)
Numbers whose square part is greater than 4. [Proof follows from s and t having to be multiples of A019554(k), the smallest number whose square is divisible by k.]
Compare with A116451, numbers whose odd part is greater than 3. The self-inverse function A225546(.) maps the members of either one of these sets 1:1 onto the other set.
Compare with A028983, numbers whose squarefree part is greater than 2.
(End)
The asymptotic density of this sequence is 1 - 15/(2*Pi^2). - Amiram Eldar, Mar 08 2021
From Bernard Schott, Jan 09 2022: (Start)
Numbers of the form u*m^2, for u >= 1 and m >= 3 (union of first 2 comments).
A geometric application: in trapezoid ABCD, with AB // CD, the diagonals intersect at E. If the area of triangle ABE is u and the area of triangle CDE is v, with u>v, then the area of trapezoid ABCD is w = u + v + 2*sqrt(u*v); in this case, u, v, w are integer solutions iff (u,v,w) = (k*s^2,k*t^2,k*(s+t)^2), with s>t and k positives; hence, w is a term of this sequence (see IMTS link). (End)

			16 is in the sequence since it has a partition into two distinct parts (12,4), such that 16 | 12*4 = 48.

S. Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 of International Mathematical Talent Search, round 7, page 285.

Amiram Eldar, Table of n, a(n) for n = 1..10000.
International Mathematical Talent Search, Problem 1/7, Round 7.
Eric Weisstein's World of Mathematics, Square part.
Index to sequences related to Olympiads and other Mathematical Competitions.
Index entries for sequences related to partitions.

Complement of A133466 within A013929.
Cf. A019554, A028983, A335234, A335438.
A038838, A046101, A062312\{1}, A195085 are subsequences.
Related to A116451 via A225546.

Table[If[Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[(n - 1)/2]}] > 0, n, {}], {n, 300}] // Flatten
f[p_, e_] := p^(2*Floor[e/2]); sqpart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[256], sqpart[#] > 4 &] (* Amiram Eldar, Mar 08 2021 *)

A328285 Smallest positive number k >= 2 for which there exist exactly n >= 1 integers m in M = {1, 2, 3, ..., k-1} such that k*m is a perfect power.

Original entry on oeis.org

12, 4, 8, 48, 16, 32, 49, 640, 108, 162, 64, 121, 243, 144, 196, 225, 867, 289, 324, 361, 256, 400, 484, 529, 512, 1250, 676, 625, 576, 1682, 784, 900, 961, 1458, 729, 1156, 1225, 2312, 1369, 1024, 1521, 2048, 1681, 1600, 1849, 1936, 6348, 2025, 2209, 4232

Offset: 1

Marius A. Burtea, Nov 29 2019

nonn

			For n = 1 and k = 12 the set M = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} and we obtain only 12 * 3 = 36 = 6^2, so a(1) = 12.
For n = 2 and k = 4 the set M = {1, 2, 3} and we obtain 4 * 1 = 4 = 2^2 and 4 * 2 = 8 = 2^3 so a(2) = 4.
For n = 3 and k = 8 the set M = {1, 2, 3, 4, 5, 6, 7}. The powers 8 * 1 = 2^3, 8 * 2 = 16 = 2^4 and 8 * 4 = 32 = 2^5 are obtained, so a(3) = 8.

Chai Wah Wu, Table of n, a(n) for n = 1..763

Cf. A001597, A133466, A195085.

a:=[]; for n in [1..40] do k:=1; while #[m:m in [1..k-1]| IsPower(m*k)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

ppQ[n_] := 1 < GCD @@ FactorInteger[n][[All, 2]]; cnt[k_] := cnt[k] = Length[ Select[ Range[k-1], ppQ[k #] &]]; a[n_] := Block[{k = n + 1}, While[ cnt[k] != n, k++]; k]; Array[a, 40] (* Giovanni Resta, Dec 05 2019 *)

a(n) = {my(k=2); while (sum(m=1, k-1, ispower(m*k) != 0) != n, k++); k;} \\ Michel Marcus, Dec 05 2019

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A019554 Smallest number whose square is divisible by n.

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A057918 Number of pairs of numbers (r,s) each less than n such that (r,s,n) is in geometric progression.

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A133466 Positive integers k for which there is exactly one integer i in {1,2,3,...,k-1} such that i*k is a square.

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A335437 Numbers k with a partition into two distinct parts (s,t) such that k | s*t.

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A328285 Smallest positive number k >= 2 for which there exist exactly n >= 1 integers m in M = {1, 2, 3, ..., k-1} such that k*m is a perfect power.

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