A019554 -id:A019554 - cp's OEIS frontend

A365347 The sum of divisors of the smallest number whose square is divisible by n.

1, 3, 4, 3, 6, 12, 8, 7, 4, 18, 12, 12, 14, 24, 24, 7, 18, 12, 20, 18, 32, 36, 24, 28, 6, 42, 13, 24, 30, 72, 32, 15, 48, 54, 48, 12, 38, 60, 56, 42, 42, 96, 44, 36, 24, 72, 48, 28, 8, 18, 72, 42, 54, 39, 72, 56, 80, 90, 60, 72, 62, 96, 32, 15, 84, 144, 68, 54

Offset: 1

Amiram Eldar, Sep 02 2023

The number of divisors of the smallest number whose square is divisible by n is A322483(n).

The sum of divisors of the smallest square divisible by n is A365346(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A019554, A322483, A365346.

Cf. A002117, A065465.

f[p_, e_] := (p^((e + Mod[e, 2])/2 + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^((f[i,2] + f[i,2]%2)/2 + 1) - 1)/(f[i,1] - 1));}

a(n) = sigma(n/core(n, 1)[2]); \\ Michel Marcus, Sep 02 2023

a(n) = A000203(A019554(n)).
Multiplicative with a(p^e) = (p^(e + 1 + (e mod 2)) - 1)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/2) * A002117 * A065465 = 0.529814898136... .

A365481 The sum of unitary divisors of the smallest number whose square is divisible by n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 5, 4, 18, 12, 12, 14, 24, 24, 5, 18, 12, 20, 18, 32, 36, 24, 20, 6, 42, 10, 24, 30, 72, 32, 9, 48, 54, 48, 12, 38, 60, 56, 30, 42, 96, 44, 36, 24, 72, 48, 20, 8, 18, 72, 42, 54, 30, 72, 40, 80, 90, 60, 72, 62, 96, 32, 9, 84, 144, 68, 54, 96

Offset: 1

Amiram Eldar, Sep 05 2023

The number of unitary divisors of the smallest number whose square is divisible by n is the same as the number of unitary divisors of n, A034444(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002117, A013661, A019554, A034444, A034448, A365347, A365479, A365480.

f[p_, e_] := p^Ceiling[e/2] + 1; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i,1]^ceil(f[i,2]/2) + 1);}

from math import prod
from sympy import factorint
def A365481(n): return prod(p**((e>>1)+(e&1))+1 for p,e in factorint(n).items()) # Chai Wah Wu, Sep 05 2023

a(n) = A034448(A019554(n)).
Multiplicative with a(p^e) = p^(ceiling(e/2)) + 1.
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-1) - 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 - 1/p^5 + 1/p^6) = 0.515959523197... .

A055491 Smallest square divisible by n divided by largest square which divides n.

Original entry on oeis.org

1, 4, 9, 1, 25, 36, 49, 4, 1, 100, 121, 9, 169, 196, 225, 1, 289, 4, 361, 25, 441, 484, 529, 36, 1, 676, 9, 49, 841, 900, 961, 4, 1089, 1156, 1225, 1, 1369, 1444, 1521, 100, 1681, 1764, 1849, 121, 25, 2116, 2209, 9, 1, 4, 2601, 169, 2809, 36, 3025, 196, 3249, 3364

Offset: 1

Henry Bottomley, Jun 28 2000

			a(12) = 36/4 = 9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.

Cf. A056551, A056552.
Cf. A000188, A000290, A007913, A008833, A019554, A053143.
Cf. A013661, A013664.

a055491 = (^ 2) . a007913  -- Reinhard Zumkeller, Jul 23 2014

With[{sqs=Range[100]^2},Table[SelectFirst[sqs,Divisible[#,n]&]/ SelectFirst[ Reverse[sqs],Divisible[n,#]&],{n,60}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 18 2018 *)
f[p_, e_] := p^(2 * Mod[e, 2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2*(f[i,2]%2)));} \\ Amiram Eldar, Oct 27 2022

If n is written as Product(Pj^Ej) then a(n) = Product(Pj^(2*(Ej mod 2))).
a(n) = A053143(n)/A008833(n) = A007913(n)^2 = (A019554(n)/A000188(n))^2 = A000290(n)/A008833(n)^2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(6)/(3*zeta(2))) = 2*Pi^4/945 = 0.206156... . - Amiram Eldar, Oct 27 2022
Dirichlet g.f.: zeta(s-2) * zeta(2*s) / zeta(2*s-4). - Amiram Eldar, Sep 16 2023

A369430 Smallest number whose square is divisible by the n-th powerful number.

Original entry on oeis.org

1, 2, 4, 3, 4, 5, 9, 8, 6, 7, 8, 12, 9, 10, 18, 11, 25, 16, 12, 13, 14, 20, 36, 15, 27, 16, 24, 17, 18, 49, 19, 28, 20, 36, 21, 22, 50, 32, 23, 24, 25, 36, 45, 26, 27, 28, 40, 29, 72, 30, 31, 44, 54, 100, 32, 33, 75, 48, 34, 35, 36, 63, 121, 52, 37, 98, 38, 39

Offset: 1

Amiram Eldar, Jan 23 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001694, A019554, A369429.

Cf. A001620, A002117, A078434.

f[p_, e_] := p^Ceiling[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[2000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]
(* or *)
f[p_, e_] := p^Ceiling[e/2]; f[p_, 1] := 0; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 2000], # > 0 &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 0, f[i,1]^ceil(f[i,2]/2)));}
lista(kmax) = {my(s1); for(k = 1, kmax, s1 = s(k); if(s1 > 0, print1(s1, ", ")));}

a(n) = A019554(A001694(n)).
a(n) > 1 for n >= 2.
Sum_{k=1..n} a(k) ~ (c*n/Pi)^2 * (2*log(n) + 2*log(c) + 5*gamma - 1 - 6*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620), and c = zeta(3)/zeta(3/2).

A375359 The maximum exponent in the prime factorization of the smallest number whose square is divisible by n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Amiram Eldar, Aug 13 2024

Differs from A050361 at n = 1, 64, 128, 192, ... . Differs from A366902 at n = 1, 64, 192, 216, ... . Differs from A325837 at n = 1, 216, 432, 648, ... .

Cf. A019554, A051903, A372602, A375360.

Cf. A050361, A325837, A366902.

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[(If[EvenQ[#], #, # + 1]) & /@ e]/2]; a[1] = 0; Array[a, 100]

a(n) = if(n == 1, 0, vecmax(apply(x -> if(x % 2, x+1, x), factor(n)[,2]))/2);

a(n) = A051903(A019554(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum{k>=1} (1 - 1/zeta(2*k+1)) = 1.21464720975357037829... .

A056134 Smallest positive integer which is the geometric mean of n and an integer other than n.

Original entry on oeis.org

2, 4, 6, 2, 10, 12, 14, 4, 3, 20, 22, 6, 26, 28, 30, 4, 34, 6, 38, 10, 42, 44, 46, 12, 5, 52, 9, 14, 58, 60, 62, 8, 66, 68, 70, 6, 74, 76, 78, 20, 82, 84, 86, 22, 15, 92, 94, 12, 7, 10, 102, 26, 106, 18, 110, 28, 114, 116, 118, 30, 122, 124, 21, 8, 130, 132, 134, 34, 138, 140

Offset: 1

Henry Bottomley, Jun 13 2000

a(n)=sqrt(n*A056135(n)). If n is squarefree then a(n)=2n, otherwise a(n)=A019554(n)

A085767 Smallest m such that n divides the pentagonal number A000326(m).

Original entry on oeis.org

1, 3, 3, 3, 2, 3, 5, 11, 9, 7, 4, 3, 9, 7, 12, 11, 6, 27, 13, 27, 12, 4, 8, 27, 17, 35, 27, 19, 10, 12, 21, 43, 15, 23, 5, 27, 25, 19, 9, 27, 14, 12, 29, 11, 27, 8, 16, 75, 33, 67, 6, 35, 18, 27, 15, 75, 51, 39, 20, 27, 41, 31, 54, 43, 22, 15, 45, 40, 54, 7, 24, 27, 49, 99, 42, 19, 26, 39

Offset: 1

Jon Perry, Jul 22 2003

nonn

			Let pe(m)=m*(3m-1)/2. The pe(1)=1, pe(2)=5, pe(3)=12. As pe(3) is the first divisible by 6, a(6)=3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A011772 (triangular numbers), A019554 (squares).

smn[n_]:=Module[{m=1,c},c=(m(3m-1))/2;While[!Divisible[c,n],m++;c=(m(3m-1))/2];m]; Array[smn, 80] (* Harvey P. Dale, Feb 03 2015 *)

pe(n)=n*(3*n-1)/2 for (n=1,50,c=1; while (pe(c)%n!=0,c++); print1(c","))

More terms from David Wasserman, Feb 10 2005

A072391 D2(n,n) = Sum_{1<=k<=n} (d_n(k^2)), where d_a(k^2)=card{d: d|k and 1<=d<=a} for real a.

Original entry on oeis.org

1, 3, 5, 9, 11, 16, 18, 23, 28, 33, 35, 44, 46, 51, 56, 64, 66, 76, 78, 87, 92, 97, 99, 111, 118, 123, 129, 138, 140, 154, 156, 165, 170, 175, 180, 198, 200, 205, 210, 222, 224, 238, 240, 249, 259, 264, 266, 283, 292, 304, 309, 318, 320, 333, 338, 350, 355, 360

Offset: 1

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 20 2002

nonn

Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, vol. 101, (2002), pp. 105-114.

Cf. A019554.

a(n)=Sum_{k<=n} (floor(n/A019554(k))) Asymptotic expression: a(n)=(n*log(n)^2/(4*zeta(2)))+(n*log(n)/zeta(2))*((3*gamma/2)-(zeta'(2)/zeta(2))), gamma = A001620.

Asymptotic expression (includes error term): a(n)=(n*log(n)^2/(4*zeta(2)))+(n*log(n)/zeta(2))*((3*gamma/2)-(zeta'(2)/zeta(2)))+O(n), gamma = A001620.

A085766 Smallest m such that n divides the tetrahedral number A000292(m+1).

Original entry on oeis.org

1, 1, 6, 1, 2, 6, 4, 5, 24, 2, 8, 6, 10, 5, 7, 13, 14, 25, 16, 3, 6, 9, 20, 7, 22, 10, 78, 5, 26, 7, 28, 29, 8, 14, 4, 25, 34, 17, 24, 7, 38, 6, 40, 9, 24, 21, 44, 15, 46, 22, 15, 11, 50, 78, 8, 5, 16, 26, 56, 7, 58, 29, 25, 61, 12, 42, 64, 14, 43, 13, 68, 53, 70, 34, 24, 17, 19, 25, 76, 13

Offset: 1

Jon Perry, Jul 22 2003

nonn

			Let te(m)=(m+1)(m+2)(m+3)/6. Then te(1)=4, te(2)=10, te(3)=20, te(4)=35, te(5)=56 and te(6)=84. te(6) is the first tetrahedral number divisible by 3, hence a(3)=6.

Cf. A011772 (triangular numbers), A019554 (squares).

te(n)=(n+1)*(n+2)*(n+3)/6 for (n=1,50,c=1; while (te(c)%n!=0,c++); print1(c","))

first(n) = {my(res = vector(n), todo = n); res[1] = 1; todo--; for(i = 1, oo, t = binomial(i + 2, 3); d = divisors(t); for(j = 1, #d, if(d[j] <= n && res[d[j]] == 0, res[d[j]] = i - 1; todo--; if(todo <= 0, return(res); ) ) ) ) } \\ David A. Corneth, Mar 22 2021

More terms from David Wasserman, Feb 10 2005

Definition corrected by David A. Corneth, Mar 22 2021

cp's OEIS Frontend

A365347 The sum of divisors of the smallest number whose square is divisible by n.

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A365481 The sum of unitary divisors of the smallest number whose square is divisible by n.

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A055491 Smallest square divisible by n divided by largest square which divides n.

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A369430 Smallest number whose square is divisible by the n-th powerful number.

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A375359 The maximum exponent in the prime factorization of the smallest number whose square is divisible by n.

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A056134 Smallest positive integer which is the geometric mean of n and an integer other than n.

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A085767 Smallest m such that n divides the pentagonal number A000326(m).

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A072391 D2(n,n) = Sum_{1<=k<=n} (d_n(k^2)), where d_a(k^2)=card{d: d|k and 1<=d<=a} for real a.

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