A008833 -id:A008833 - cp's OEIS frontend

A336615 Numbers of the form p * m^2, where p is prime and m > 0 is not divisible by p.

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 112, 113, 116, 117, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153, 157, 162

Offset: 1

Amiram Eldar, Jul 27 2020

nonn

Numbers k such that A008833(k) is a unitary divisor of k and A007913(k) = k / A008833(k) is a prime number.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes, IX. On the set of integers representable as a product of a prime and square, Acta Arithmetica, Vol. 7 (1962), pp. 417-420.

Intersection of A229125 and A335275.
Cf. A007913, A008833, A013661.
Subsequences: A000040, A054753, A179643.

Select[Range[2, 200], Select[FactorInteger[#][[;;, 2]], OddQ] == {1} &]

from math import isqrt
from sympy import primepi, primefactors
def A336615(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(m:=x//y**2)-sum(1 for p in primefactors(y) if p<=m) for y in range(1,isqrt(x)+1))
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

The number of terms not exceeding x is (Pi^2/6) * x/log(x) + O(x/(log(x))^2) (Cohen, 1962).

A336644 a(n) = (n-rad(n)) / core(n), where rad(n) and core(n) give the squarefree kernel and squarefree part of n, respectively.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 6, 0, 0, 2, 0, 0, 0, 14, 0, 6, 0, 2, 0, 0, 0, 3, 20, 0, 8, 2, 0, 0, 0, 15, 0, 0, 0, 30, 0, 0, 0, 3, 0, 0, 0, 2, 6, 0, 0, 14, 42, 20, 0, 2, 0, 8, 0, 3, 0, 0, 0, 2, 0, 0, 6, 62, 0, 0, 0, 2, 0, 0, 0, 33, 0, 0, 20, 2, 0, 0, 0, 14, 78, 0, 0, 2, 0, 0, 0, 3, 0, 6, 0, 2, 0, 0, 0, 15, 0, 42, 6, 90, 0, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007913, A007947, A008833, A066503, A336642, A336643.

A336644(n) = ((n-factorback(factorint(n)[, 1])) / core(n));

from math import prod
from sympy.ntheory.factor_ import primefactors, core
def A336644(n): return (n-prod(primefactors(n)))//core(n) # Chai Wah Wu, Dec 30 2021

a(n) = A066503(n) / A007913(n) = (n-A007947(n)) / A007913(n).

a(n) = A008833(n) - A336643(n).

A365332 The sum of divisors of the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 7, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 7, 31, 1, 13, 7, 1, 1, 1, 31, 1, 1, 1, 91, 1, 1, 1, 7, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 13, 1, 7, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 91, 1, 1, 31, 7, 1, 1, 1, 31, 121

Offset: 1

Amiram Eldar, Sep 01 2023

All the terms are odd.
The number of these divisors is A365331(n).
The sum of divisors of the square root of the largest square dividing n is A069290(n).

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

Cf. A000203, A005117, A008833, A069290, A078434, A365331.

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

a(n) = A000203(A008833(n)).
a(n) = 1 if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = (p^(e + 1 - (e mod 2)) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * zeta(2*s-2) / zeta(4*s-2).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = 5*zeta(3/2)/Pi^2 = 1.323444812234... .

A365333 The number of exponentially odd coreful divisors of the largest square dividing n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 01 2023

First differs from A043289, A053164, A063775, A203640 and A295658 at n = 64.
The number of squares dividing the largest exponentially odd divisor of n is A325837(n).
The sum of the exponentially odd divisors of the largest square dividing n is A365334(n). [corrected, Sep 08 2023]
The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n). - Amiram Eldar, Sep 08 2023

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

Cf. A008833, A046100, A046951, A325837, A365334.

Cf. A043289, A053164, A063775, A203640, A295658.

f[p_, e_] := Max[1, Floor[e/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> max(1, x\2), factor(n)[, 2]));

a(n) = A325837(A008833(n)).
a(n) = 1 if and only if n is a biquadratefree number (A046100).
Multiplicative with a(p^e) = max(1, floor(e/2)).
Dirichlet g.f.: zeta(s) * zeta(4*s) * zeta(6*s) / zeta(12*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15015/(1382*Pi^2) = 1.100823... .

Name corrected by Amiram Eldar, Sep 08 2023

A366524 Largest square divisor of n which is < sqrt(n), for n >= 2; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 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, 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, A060775, A366523.

Join[{1}, Table[Last[Select[Divisors[n], # < Sqrt[n] && IntegerQ[Sqrt[#]] &]], {n, 2, 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

A366765 The largest divisor of n that have no exponent 2 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 48, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 64, 65, 66, 67, 34, 69

Offset: 1

Amiram Eldar, Oct 21 2023

The largest term of A337050 that divides n.

The number of these divisors is A366763(n), and their sum is A366764(n).

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

Cf. A337050, A366763, A366764.

Similar sequences: A055231, A057521, A008833, A350390.

f[p_, e_] := p^If[e < 3, 1, e]; 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] ^ if(f[i, 2] < 3, 1, f[i, 2]));}

Multiplicative with a(p^e) = p if e <= 2 and p^e otherwise.
a(n) <= n, with equality if and only if n is in A337050.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s-1) + 1/p^(3*s-3) - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 2/p^3 - 1/p^4) = 0.83234421330425224469... .

A367418 The exponentially odd numbers (A268335) divided by their squarefree kernels (A007947).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 9, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1

Offset: 1

Amiram Eldar, Nov 17 2023

Analogous to A102631, with the exponentially odd numbers instead of the square numbers (A000290).

All the terms are square numbers.

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

Cf. A000290, A008833, A102631, A268335, A367406, A367417, A367419.

s[n_] := n / Times @@ FactorInteger[n][[;; , 1]]; s /@ Select[Range[200], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1]^(f[i, 2]-1), 0)); }
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(b1, ", "))); }

a(n) = A003557(A268335(n)).
a(n) = A268335(n)/A367417(n).
a(n) = A367419(n)^2.
a(n) = A268335(n)^2/A367406(n).
a(n) = A008833(A268335(n)). - Amiram Eldar, Nov 30 2023

A055210 Sum of totients of square divisors of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 7, 1, 1, 3, 1, 1, 1, 11, 1, 7, 1, 3, 1, 1, 1, 3, 21, 1, 7, 3, 1, 1, 1, 11, 1, 1, 1, 21, 1, 1, 1, 3, 1, 1, 1, 3, 7, 1, 1, 11, 43, 21, 1, 3, 1, 7, 1, 3, 1, 1, 1, 3, 1, 1, 7, 43, 1, 1, 1, 3, 1, 1, 1, 21, 1, 1, 21, 3, 1, 1, 1, 11, 61, 1, 1, 3, 1, 1, 1, 3, 1, 7, 1, 3, 1, 1, 1, 11, 1

Offset: 1

Labos Elemer, Jun 19 2000

			n = 400: its square divisors are {1, 4, 16, 25, 100, 400}, their totients are {1, 2, 8, 20, 40, 160} and the totient-sum over these divisors is, so a(400) = 231. This value arises at special squarefree multiples of 400 (400 times 2, 3, 5, 6, 7, 10, 11, 13, 15, 17, 19, 21, 22, 23 etc).
a(400) = a(2^4*5^2) = (2^5 + 1)/3*(5^3 + 1)/6 = 231.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A010052, A013661, A078434.

[&+[EulerPhi(d):d in Divisors(n)| IsSquare(d)]: n in [1..100]]; // Marius A. Burtea, Oct 14 2019

Array[DivisorSum[#, EulerPhi, IntegerQ@ Sqrt@ # &] &, 97] (* Michael De Vlieger, Nov 18 2017 *)
f[p_, e_] := If[EvenQ[e], (p^(e + 1) + 1)/(p + 1), (p^e + 1)/(p + 1)]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 09 2020 *)

a(n) = sumdiv(n, d, eulerphi(d)*issquare(d)); \\ Michel Marcus, Dec 31 2013

a(n) = Sum_{d is square and divides n} phi(d).
Multiplicative with a(p^e) = (p^(e+1)+1)/(p+1) for even e and a(p^e) = (p^e+1)/(p+1) for odd e. - Vladeta Jovovic, Dec 01 2001
a(n) = Sum_{d|n} A010052(d)*A000010(d). - Antti Karttunen, Nov 18 2017
Conjecture: a(n) = sigma_2(n/core(n))/sigma_1(n/core(n)) = A001157(A008833(n))/A000203(A008833(n)) for all n > 0. - Velin Yanev, Oct 13 2019
G.f.: Sum_{k>=1} k * phi(k) * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Aug 20 2021
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(2)) = 0.529377... . - Amiram Eldar, Nov 13 2022

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

A056194 Characteristic cube divisor of n!: a(n) = A056191(n!).

Original entry on oeis.org

1, 1, 1, 8, 8, 1, 1, 8, 8, 1, 1, 27, 27, 216, 1000, 1000, 1000, 125, 125, 1, 9261, 74088, 74088, 343, 343, 2744, 74088, 216, 216, 125, 125, 1000, 35937000, 4492125, 12326391, 12326391, 12326391, 98611128, 8024024008, 125375375125

Offset: 1

Labos Elemer, Aug 02 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..4054

Cf. A000142, A055229, A000188, A008833, A007913, A055231, A055230, A055772, A055071, A055204, A055773 A056552, A056195.

f[p_, e_] := If[OddQ[e] && e > 1, p^3, 1]; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* Amiram Eldar, Sep 06 2020 *)

a(n) = A056191(A000142(n)). - Amiram Eldar, Sep 06 2020

cp's OEIS Frontend

A336615 Numbers of the form p * m^2, where p is prime and m > 0 is not divisible by p.

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A336644 a(n) = (n-rad(n)) / core(n), where rad(n) and core(n) give the squarefree kernel and squarefree part of n, respectively.

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A365332 The sum of divisors of the largest square dividing n.

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A365333 The number of exponentially odd coreful divisors of the largest square dividing n.

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A366524 Largest square divisor of n which is < sqrt(n), for n >= 2; a(1) = 1.

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A366765 The largest divisor of n that have no exponent 2 in their prime factorization.

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A367418 The exponentially odd numbers (A268335) divided by their squarefree kernels (A007947).

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A055210 Sum of totients of square divisors of n.

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