A008833 -id:A008833 - cp's OEIS frontend

A056647 a(n) = A056623(A001405(n)).

1, 1, 1, 1, 1, 4, 1, 1, 9, 36, 1, 4, 4, 1, 9, 9, 1, 4, 1, 4, 4, 1, 1, 4, 100, 25, 100, 25, 9, 144, 9, 9, 1, 4, 25, 100, 100, 25, 9, 36, 4, 1, 4, 1, 25, 400, 225, 900, 1764, 441, 196, 49, 49, 784, 4, 1, 1, 16, 1, 16, 16, 4, 441, 441, 49, 196, 49, 196, 36, 9, 1, 4, 4, 1, 100, 25, 1225

Offset: 1

Labos Elemer, Aug 09 2000

nonn

Previous name "Largest unitary square divisor of central binomial coefficient" was incorrect. See A376553 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

			a(28) = A056623(binomial(28,14)) = A056623(40116600) = 25.

Cf. A001405, A008833, A055229, A056056, A056057, A056059, A056623, A376553.

a(n) = A008833(A001405(n))/A055229(A001405(n))^2 = A056057(n)/A056059(n)^2.

Incorrect name replaced with a formula by Amiram Eldar, Sep 28 2024

A083730 Greatest prime^2 factor of n, or a(n)=1 for squarefree n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 25, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 49, 9

Offset: 1

Reinhard Zumkeller, Jun 14 2003

Not multiplicative, for example a(4)*a(9) <> a(36). - R. J. Mathar, Oct 31 2011

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A006530, A001248, A005117, A013929, A008833, A046028, A249740.

a[n_] := If[(pos = Position[(f = FactorInteger[n])[[;; , 2]], ?(# >= 2 &)]) == {}, 1, f[[pos[[-1, 1]], 1]]^2];  Array[a, 100] (* _Amiram Eldar, Nov 14 2020 *)

a(n)=my(f=factor(n)); forstep(i=#f~,1,-1, if(f[i,2]>1, return(f[i,1]^2))); 1 \\ Charles R Greathouse IV, Jul 23 2017

a(n) = A249740(n)^2. - Amiram Eldar, Feb 11 2021

A282344 The smallest square referenced in A086982 (Numbers n such that 10^n+1 is not squarefree).

Original entry on oeis.org

121, 49, 121, 169, 121, 49, 121, 121, 49, 169, 121, 289, 121, 49, 121, 361, 121, 49, 169, 10201, 121, 49, 237169, 121, 49, 121, 5329, 121, 49, 121, 121, 169, 49, 121, 121, 49, 841, 121, 289, 121, 49, 121, 121, 49, 121, 169, 361, 121, 49, 121, 18769, 121, 49

Offset: 1

Robert Price, Feb 12 2017

nonn

Cf. A086982, A282345.

a(n) = A008833( 1+10^A086982(n) ). - R. J. Mathar, Feb 13 2017

A306300 Discriminant D of real quadratic number field Q(sqrt(D)) associated with fundamental discriminant d = A003658(n).

Original entry on oeis.org

1, 5, 2, 3, 13, 17, 21, 6, 7, 29, 33, 37, 10, 41, 11, 53, 14, 57, 15, 61, 65, 69, 73, 19, 77, 85, 22, 89, 23, 93, 97, 101, 26, 105, 109, 113, 30, 31, 129, 133, 34, 137, 35, 141, 145, 149, 38, 39, 157, 161, 165, 42, 43, 173, 177, 181, 46, 185, 47, 193, 197

Offset: 1

N. J. A. Sloane, Apr 02 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Britta Habdank-Eichelsbacher, Unimodulare Gitter über Reell-Quadratischen Zahlkörpern, Ergänzungsreihe 95-005, Univ. Bielefeld, 1995. See Section 4.2.

Cf. A003658, A007913, A008833.

a(n) = A007913(A003658(n)) = A003658(n) / (A008833(A003658(n))).

A358272 Multiplicative sequence with a(p^e) = (-1)^e * p^(2*floor(e/2)) for prime p and e >= 0.

Original entry on oeis.org

1, -1, -1, 4, -1, 1, -1, -4, 9, 1, -1, -4, -1, 1, 1, 16, -1, -9, -1, -4, 1, 1, -1, 4, 25, 1, -9, -4, -1, -1, -1, -16, 1, 1, 1, 36, -1, 1, 1, 4, -1, -1, -1, -4, -9, 1, -1, -16, 49, -25, 1, -4, -1, 9, 1, 4, 1, 1, -1, 4, -1, 1, -9, 64, 1, -1, -1, -4, 1, -1, -1, -36, -1, 1, -25, -4, 1, -1, -1, -16

Offset: 1

Werner Schulte, Nov 07 2022

Signed version of A008833.

Cf. A000010, A008833, A008836, A034444, A061019.

A358272 := proc(n)
    local a,pe,e,p ;
    a := 1;
    for pe in ifactors(n)[2] do
        e := op(2,pe) ;
        p := op(1,pe) ;
        a := a*(-1)^e*p^(2*floor(e/2)) ;
    end do:
    a ;
end proc:
seq(A358272(n),n=1..80) ; # R. J. Mathar, Jan 17 2023

f[p_, e_] := (-1)^e * p^(2*Floor[e/2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 07 2022 *)

from math import prod
from sympy import factorint
def A358272(n): return prod(-p**(e&-2) if e&1 else p**(e&-2) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 17 2023

a(n) = lambda(n) * A008833(n) for n > 0 where lambda(n) = A008836(n).
Dirichlet g.f.: zeta(2*s-2) / zeta(s).
Dirichlet inverse b(n), n > 0, is multiplicative with b(p) = 1 and b(p^e) = 1 - p^2 for prime p and e > 1.
Dirichlet convolution with A034444 equals A008833.
Equals Dirichlet convolution of A000010 and A061019.
Conjecture: a(n) = Sum_{k=1..n} gcd(k, n) * lambda(gcd(k, n)) for n > 0.
a(n) = Sum_{d|n} lambda(d)*d*phi(n/d), where lambda(n) = A008836(n). - Ridouane Oudra, May 11 2025

A365334 The sum of exponentially odd divisors of the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 11, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 11, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 11, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 43, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 11, 31, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Sep 01 2023

The number of these divisors is A365333(n).

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

Cf. A008833, A005117, A033634, A365333.

f[p_, e_] := (p^(e + 1 - Mod[e, 2]) - p)/(p^2 - 1) + 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] + 1 - f[i,2]%2) - f[i,1])/(f[i,1]^2 - 1) + 1);}

a(n) = A033634(A008833(n)).
a(n) = 1 if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = 1 + (p^(e + 1 - (e mod 2)) - 1)/(p^2 - 1).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s-1)).

A365403 The sum of the unitary divisors of the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 17, 1, 1, 1, 50, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 17, 82

Offset: 1

Amiram Eldar, Sep 03 2023

The number of these divisors is A323308(n).

The sum of the unitary divisors of the square root of the largest square dividing n is A365404(n).

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

Cf. A000290, A005117, A008833, A034448, A323308, A365404.

Cf. A013662, A078434.

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

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

a(n) = A034448(A008833(n)).
a(n) <= A034448(n) with equality if and only if n is a square (A000290).
a(n) >= 1 with equality if and only if n is squarefree (A005117).
Multiplicative with a(p) = 1 and a(p^e) = p^(2*floor(e/2)) + 1 for e >= 2.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) / zeta(4*s-2).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(4)) = 30*zeta(3/2)/Pi^4 = 0.804557969165... .

A366523 Largest square divisor of n which is <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4

Offset: 1

Ilya Gutkovskiy, Oct 11 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A008833, A033676, A366524.

Table[Last[Select[Divisors[n], # <= Sqrt[n] && IntegerQ[Sqrt[#]] &]], {n, 100}]

a(n) = {my(m=1); fordiv(sqrtint(n/core(n)), d, if(d^4 <= n, m=max(m,d))); m^2} \\ Andrew Howroyd, Oct 11 2023

A379663 a(n) is the number of integer-sided triangles whose sides are in geometric progression with smallest side n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 4, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 5, 4, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 4, 2, 1, 1, 1, 3, 6, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Felix Huber, Jan 07 2025

nonn

The integer sides of the triangles are n, n*r, n*r^2 with rational r >= 1. From the triangle inequality n + n*r >= n*r^2 follows r <= (1 + sqrt(5))/2 (golden ratio). Therefore 1 <= r = c/d < (1 + sqrt(5))/2, where c and d are coprimes and d^2 divides n.

			The a(18) = 2 integer-sided triangles whose sides form a geometric sequence are [18, 18, 18] with r = 1, [18, 24, 32] with r = 4/3.
The a(25) = 4 integer-sided triangles whose sides form a geometric sequence are [25, 25, 25] with r = 1, [25, 30, 36] with r = 6/5, [25, 35, 49] with r = 7/5, [25, 40, 64] with r = 8/5.
The a(36) = 4 integer-sided triangles whose sides form a geometric sequence are [36, 36, 36] with r = 1, [36, 54, 81] with r = 3/2, [36, 48, 64] with r = 4/3, [36, 42, 49] with r = 7/6.
See also the linked Maple program "Triangles for a given n".

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Triangles for a given n
Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Geometric Progression
Wikipedia, Triangle Inequality

Cf. A000188, A000201, A008833, A027750, A046951, A057918, A060143, A366398, A370599, A376900, A379033, A379340, A379341, A379705.

A379663:=n->floor(2*expand(NumberTheory:-LargestNthPower(n,2))/(1+sqrt(5)))+1;
seq(A379663(n),n=1..88);

a(n) = A060143(A000188(n)) + 1.

A379705 a(n) is the least integer k > n such that integers p, q exist for which n, p, k are in arithmetic and n, q, k are in geometric progression.

Original entry on oeis.org

9, 8, 27, 16, 45, 24, 63, 18, 25, 40, 99, 48, 117, 56, 135, 36, 153, 32, 171, 80, 189, 88, 207, 54, 49, 104, 75, 112, 261, 120, 279, 50, 297, 136, 315, 64, 333, 152, 351, 90, 369, 168, 387, 176, 125, 184, 423, 108, 81, 72, 459, 208, 477, 96, 495, 126, 513, 232

Offset: 1

Felix Huber, Jan 07 2025

nonn

			a(9) = 25 because 9, 17, 25 are in arithmetic progression (common difference = 8) and 9, +-15, 25 are in geometric progression (common ratio = +-5/3) and there is no other integer k with 9 < k < 25 such that integers p and q exist for which 9, p, k are in arithmetic and 9, q, k are in geometric progression.

Felix Huber, Table of n, a(n) for n = 1..10000
Wikipedia, Arithmetic Progression
Wikipedia, Geometric Progression

Cf. A000188, A008833, A057918, A072905, A379663.

A379705:=proc(n)
   local d;
   d:=expand(NumberTheory:-LargestNthPower(n,2));
   if is(n*(1+(d+1)^2/d^2),even) then
      n*(d+1)^2/d^2
   else
      n*(d+2)^2/d^2
   fi;
end proc;
seq(A379705(n),n=1..58);

a(n) = n/A008833(n)*(A000188(n) + k)^2, where k = 1 if n*(1+(A000188(n)+1)^2/A008833(n)) is even or k = 2 else.

a(n) = A072905(n) if n*(1+(A000188(n)+1)^2/A008833(n)) is even.

cp's OEIS Frontend

A056647 a(n) = A056623(A001405(n)).

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A083730 Greatest prime^2 factor of n, or a(n)=1 for squarefree n.

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A282344 The smallest square referenced in A086982 (Numbers n such that 10^n+1 is not squarefree).

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A306300 Discriminant D of real quadratic number field Q(sqrt(D)) associated with fundamental discriminant d = A003658(n).

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A358272 Multiplicative sequence with a(p^e) = (-1)^e * p^(2*floor(e/2)) for prime p and e >= 0.

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A365334 The sum of exponentially odd divisors of the largest square dividing n.

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A365403 The sum of the unitary divisors of the largest square dividing n.

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A366523 Largest square divisor of n which is <= sqrt(n).

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A379663 a(n) is the number of integer-sided triangles whose sides are in geometric progression with smallest side n.

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