A062822 -id:A062822 - cp's OEIS frontend

A265668 Table read by rows: prime factors of squarefree numbers; a(1) = 1 by convention.

1, 2, 3, 5, 2, 3, 7, 2, 5, 11, 13, 2, 7, 3, 5, 17, 19, 3, 7, 2, 11, 23, 2, 13, 29, 2, 3, 5, 31, 3, 11, 2, 17, 5, 7, 37, 2, 19, 3, 13, 41, 2, 3, 7, 43, 2, 23, 47, 3, 17, 53, 5, 11, 3, 19, 2, 29, 59, 61, 2, 31, 5, 13, 2, 3, 11, 67, 3, 23, 2, 5, 7, 71, 73, 2

Offset: 1

Reinhard Zumkeller, Dec 13 2015

For n > 1: A072047(n) = length of row n;
T(n,1) = A073481(n); T(n,A001221(n)) = A073482(n);
for n > 1: A111060(n) = sum of row n;
A005117(n) = product of row n.

			.   n | T(n,*)  A5117(n)    n | T(n,*)  A5117(n)    n | T(n,*)   A5117(n)
. ----+---------+------   ----+---------+------   ----+----------+------
.   1 | [1]     |  1       21 | [3,11]  | 33       41 | [2,3,11] | 66
.   2 | [2]     |  2       22 | [2,17]  | 34       42 | [67]     | 67
.   3 | [3]     |  3       23 | [5,7]   | 35       43 | [3,23]   | 69
.   4 | [5]     |  5       24 | [37]    | 37       44 | [2,5,7]  | 70
.   5 | [2,3]   |  6       25 | [2,19]  | 38       45 | [71]     | 71
.   6 | [7]     |  7       26 | [3,13]  | 39       46 | [73]     | 73
.   7 | [2,5]   | 10       27 | [41]    | 41       47 | [2,37]   | 74
.   8 | [11]    | 11       28 | [2,3,7] | 42       48 | [7,11]   | 77
.   9 | [13]    | 13       29 | [43]    | 43       49 | [2,3,13] | 78
.  10 | [2,7]   | 14       30 | [2,23]  | 46       50 | [79]     | 79
.  11 | [3,5]   | 15       31 | [47]    | 47       51 | [2,41]   | 82
.  12 | [17]    | 17       32 | [3,17]  | 51       52 | [83]     | 83
.  13 | [19]    | 19       33 | [53]    | 53       53 | [5,17]   | 85
.  14 | [3,7]   | 21       34 | [5,11]  | 55       54 | [2,43]   | 86
.  15 | [2,11]  | 22       35 | [3,19]  | 57       55 | [3,29]   | 87
.  16 | [23]    | 23       36 | [2,29]  | 58       56 | [89]     | 89
.  17 | [2,13]  | 26       37 | [59]    | 59       57 | [7,13]   | 91
.  18 | [29]    | 29       38 | [61]    | 61       58 | [3,31]   | 93
.  19 | [2,3,5] | 30       39 | [2,31]  | 62       59 | [2,47]   | 94
.  20 | [31]    | 31       40 | [5,13]  | 65       60 | [5,19]   | 95  .

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Cf. A005117, A027748, A124010, A072047 (row lengths), A073481, A073482, A001221, A111060, A049200, A062822.

import Math.NumberTheory.Primes.Factorisation (factorise)
import Data.Maybe (mapMaybe)
a265668 n k = a265668_tabf !! (n-1) !! (k-1)
a265668_row n = a265668_tabf !! (n-1)
a265668_tabf = [1] : mapMaybe f [2..] where
   f x = if all (== 1) es then Just ps else Nothing
         where (ps, es) = unzip $ factorise x

FactorInteger[#][[All,1]]&/@Select[Range[100],SquareFreeQ]//Flatten (* Harvey P. Dale, Apr 27 2018 *)

A366439 The sum of divisors of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 15, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 60, 42, 40, 30, 72, 32, 63, 48, 54, 48, 38, 60, 56, 90, 42, 96, 44, 72, 48, 72, 54, 120, 72, 120, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 180, 90

Offset: 1

Amiram Eldar, Oct 10 2023

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

Cf. A000203, A065463, A268335, A366438.

Similar sequences: A062822, A065764, A180114, A362986, A366440.

f[p_, e_] := (p^(e+1)-1)/(p-1); s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;;, 2]], OddQ], Times @@ f @@@ fct, Nothing]]; s[1] = 1; Array[s, 100]

lista(max) = for(k = 1, max, my(f = factor(k), isexpodd = 1); for(i = 1, #f~, if(!(f[i, 2] % 2), isexpodd = 0; break)); if(isexpodd, print1(sigma(f), ", ")));

from math import prod
from itertools import count, islice
from sympy import factorint
def A366439_gen(): # generator of terms
    for n in count(1):
        f = factorint(n)
        if all(e&1 for e in f.values()):
            yield prod((p**(e+1)-1)//(p-1) for p,e in f.items())
A366439_list = list(islice(A366439_gen(),30)) # Chai Wah Wu, Oct 11 2023

a(n) = A000203(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/(2*d^2)) * Product_{p prime} (1 + 1/(p^5-p)) = 1.045911669131479732932..., where d = 0.7044422... (A065463) is the asymptotic density of the exponentially odd numbers.
The asymptotic mean of the abundancy index of the exponentially odd numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A268335(k) = (1/d) * Product_{p prime} (1 + 1/(p^5-p)) = 2 * c * d = 1.4735686365073812503199... .

A072048 Number of divisors of the squarefree numbers: tau(A005117(n)).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 8, 2, 4, 4, 4, 2, 4, 4, 2, 8, 2, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 2, 2, 8, 2, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 4, 4, 2, 4, 8, 2, 4, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 2, 8, 4, 2

Offset: 1

Reinhard Zumkeller, Jun 09 2002

nonn

Also the number of cubefree numbers with the same squarefree kernel as the n-th squarefree number, see A073245.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
B. Gordon and K. Rogers, Sums of the divisor function, Canadian Journal of Mathematics, Vol. 16 (1964), pp. 151-158.

Cf. A000005, A001620, A062822, A065473.

Cf. A000079, A001221, A005117, A072047.

a072048 = (2 ^) . a072047  -- Reinhard Zumkeller, Dec 13 2015

A072048:=n->`if`(numtheory[issqrfree](n) = true, numtheory[tau](n), NULL); seq(A072048(k), k=1..100); # Wesley Ivan Hurt, Oct 13 2013

DivisorSigma[0, Select[Range[200], SquareFreeQ]] (* Amiram Eldar, Oct 29 2022 *)

from math import isqrt
from sympy import mobius, divisor_count
def A072048(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 divisor_count(m) # Chai Wah Wu, Aug 12 2024

a(n) = A000005(A005117(n)).
a(n) = 2^A072047(n) = 2^A001221(A005117(n)).
Sum_{k=1..n} a(k) ~ A * n * log(n) + B * n + O(n^(1/2+eps)), where  A = A065473, B = A * ((2*gamma-1) + 6 * Sum_{p prime} (p-1)*log(p)/(p^2*(p+2)) = 0.236184..., and gamma = A001620 (Gordon and Rogers, 1964). - Amiram Eldar, Oct 29 2022

A366440 The sum of divisors of the cubefree numbers (A004709).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 13, 18, 12, 28, 14, 24, 24, 18, 39, 20, 42, 32, 36, 24, 31, 42, 56, 30, 72, 32, 48, 54, 48, 91, 38, 60, 56, 42, 96, 44, 84, 78, 72, 48, 57, 93, 72, 98, 54, 72, 80, 90, 60, 168, 62, 96, 104, 84, 144, 68, 126, 96, 144, 72, 74, 114, 124, 140

Offset: 1

Amiram Eldar, Oct 10 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634; p. 619, eq. (2).

Cf. A000203, A004709, A358040.
Cf. A002117, A082020.
Similar sequences: A062822, A065764, A180114, A362986, A366439.

f[p_, e_] := (p^(e+1)-1)/(p-1); s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;;, 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; s[1] = 1; Array[s, 100]

lista(max) = for(k = 1, max, my(f = factor(k), iscubefree = 1); for(i = 1, #f~, if(f[i, 2] > 2, iscubefree = 0; break)); if(iscubefree, print1(sigma(f), ", ")));

from sympy import mobius, integer_nthroot, divisor_sigma
def A366440(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return divisor_sigma(m) # Chai Wah Wu, Aug 06 2024

a(n) = A000203(A004709(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 15*zeta(3)/(2*Pi^2) = A082020 * A002117 / 2 = 0.913453711751... .
The asymptotic mean of the abundancy index of the cubefree numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A004709(k) = 15/Pi^2 = 1.519817... (A082020).

A366537 The sum of unitary divisors of the cubefree numbers (A004709).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 10, 18, 12, 20, 14, 24, 24, 18, 30, 20, 30, 32, 36, 24, 26, 42, 40, 30, 72, 32, 48, 54, 48, 50, 38, 60, 56, 42, 96, 44, 60, 60, 72, 48, 50, 78, 72, 70, 54, 72, 80, 90, 60, 120, 62, 96, 80, 84, 144, 68, 90, 96, 144, 72, 74, 114, 104, 100, 96

Offset: 1

Amiram Eldar, Oct 12 2023

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

Cf. A002117, A004709, A005117, A034448, A062822, A077610, A358040, A366440, A366536.

Similar sequences: A034676, A366535, A366539.

s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, # < 3 &], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]

lista(max) = for(k = 1, max, my(f = factor(k), e = f[, 2], iscubefree = 1); for(i = 1, #e, if(e[i] > 2, iscubefree = 0; break)); if(iscubefree, print1(prod(i = 1, #e, 1 + f[i, 1]^e[i]), ", ")));

from sympy.ntheory.factor_ import udivisor_sigma
from sympy import mobius, integer_nthroot
def A366537(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return udivisor_sigma(m) # Chai Wah Wu, Aug 05 2024

a(n) = A034448(A004709(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(3)^2 * Product_{p prime} (1 + 1/p^2 - 2/p^3 + 1/p^4 - 1/p^5) = 1.665430860774244601005... .
The asymptotic mean of the unitary abundancy index of the cubefree numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A004709(k) = c / zeta(3) = 1.38548421160152785073... .

A369889 The sum of squarefree divisors of the cubefree numbers.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 4, 18, 12, 12, 14, 24, 24, 18, 12, 20, 18, 32, 36, 24, 6, 42, 24, 30, 72, 32, 48, 54, 48, 12, 38, 60, 56, 42, 96, 44, 36, 24, 72, 48, 8, 18, 72, 42, 54, 72, 80, 90, 60, 72, 62, 96, 32, 84, 144, 68, 54, 96, 144, 72, 74, 114, 24, 60, 96, 168

Offset: 1

Amiram Eldar, Feb 15 2024

The number of squarefree divisors of the n-th cubefree number is A366536(n).

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

Cf. A004709, A048250, A062822, A366440, A366536, A366537.

Cf. A002117, A013663.

f[p_, e_] := p + 1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; cubefreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 3; s /@ Select[Range[100], cubefreeQ]
(* or *)
f[p_, e_] := If[e > 2, 0, p + 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &]

lista(kmax) = {my(f, s, p, e); for(k = 1, kmax, f = factor(k); s = prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(e < 3, p + 1, 0)); if(s > 0, print1(s, ", ")));}

from math import prod
from sympy import mobius, integer_nthroot, primefactors
def A369889(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return prod(p+1 for p in primefactors(m)) # Chai Wah Wu, Aug 12 2024

a(n) = A048250(A004709(n)).
Sum_{j=1..n} a(j) ~ c * n^2, where c = zeta(3)^2/(2*zeta(5)) = 0.6967413068... .
In general, the formula holds for the sum of squarefree divisors of the k-free numbers with c = zeta(k)^2/(2*zeta(2*k-1))..., for k >= 2.

A073245 Sum of all cubefree numbers with the same squarefree kernel as the n-th squarefree number.

Original entry on oeis.org

1, 6, 12, 30, 72, 56, 180, 132, 182, 336, 360, 306, 380, 672, 792, 552, 1092, 870, 2160, 992, 1584, 1836, 1680, 1406, 2280, 2184, 1722, 4032, 1892, 3312, 2256, 3672, 2862, 3960, 4560, 5220, 3540, 3782, 5952, 5460, 9504, 4556, 6624, 10080, 5112, 5402

Offset: 1

Reinhard Zumkeller, Jul 21 2002

nonn

			14 is the 10th squarefree number: A005117(10)=14=2*7, the cubefree numbers with squarefree kernel =14 are 14, 28=2*2*7, 98=2*7*7 and 196=2*2*7*7; therefore a(10)=14+28+98+196=336; a(10)=A062822(10)*A005117(10)=24*14=336.

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

Cf. A004709, A005117, A007947, A062822, A064987, A072048, A306633.

Cf. A002117, A013661.

Map[# * DivisorSigma[1, #] &, Select[Range[200], SquareFreeQ]] (* Amiram Eldar, Oct 14 2020 *)

apply(x->(x*sigma(x)), select(issquarefree, [1..100])) \\ Michel Marcus, Oct 18 2020

from math import isqrt
from sympy import mobius, divisor_sigma
def A073245(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 m*divisor_sigma(m) # Chai Wah Wu, Aug 12 2024

a(n) = A062822(n)*A005117(n).
Sum_{n>=1} 1/a(n) = A306633. - Amiram Eldar, Oct 14 2020
a(n) = A064987(A005117(n)). - Michel Marcus, Oct 18 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)^3/(3*zeta(3)) = 1.23423882415851340020... . - Amiram Eldar, Oct 09 2023

A366442 The sum of divisors of the 5-rough numbers (A007310).

Original entry on oeis.org

1, 6, 8, 12, 14, 18, 20, 24, 31, 30, 32, 48, 38, 42, 44, 48, 57, 54, 72, 60, 62, 84, 68, 72, 74, 96, 80, 84, 108, 90, 112, 120, 98, 102, 104, 108, 110, 114, 144, 144, 133, 156, 128, 132, 160, 138, 140, 168, 180, 150, 152, 192, 158, 192, 164, 168, 183, 174, 248

Offset: 1

Amiram Eldar, Oct 10 2023

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

Cf. A000203, A013661, A007310, A100044, A366441.

Similar sequences: A008438, A062822, A065764, A180114, A362986, A366439, A366440.

a[n_] := DivisorSigma[1, 2*Floor[3*n/2] - 1]; Array[a, 100]

PARI
```
a(n) = sigma((3*n)\2 << 1 - 1)
```

from sympy import divisor_sigma
def A366442(n): return divisor_sigma((n+(n>>1)<<1)-1) # Chai Wah Wu, Oct 10 2023

a(n) = A000203(A007310(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2) = 1.644934... (A013661).
The asymptotic mean of the abundancy index of the 5-rough numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A007310(k) = Pi^2/9 = 1.0966227... (A100044).
In general, the asymptotic mean of the abundancy index of the prime(k)-rough numbers is zeta(2) * Product_{i=1..k-1} (1 - 1/prime(i)^2).

A291109 Numbers that are not the sum of the squarefree divisors of some natural number.

Original entry on oeis.org

2, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 111

Offset: 1

Ilya Gutkovskiy, Aug 17 2017

nonn

Impossible values for A048250 (numbers k in increasing order such that A048250(m) = k has no solution).
Numbers that are not of the form Product (p_i + 1), p is a prime, so all odd numbers (except 1 and 3) are in this sequence.
Also numbers that are not the sum of the divisors of some squarefree number.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors

Cf. A005114, A005117, A007369, A048250, A048995, A062822, A063948, A064000, A203444, A206778, A274793.

sort(convert({$1..1000} minus map(numtheory:-sigma, select(numtheory:-issqrfree, {$1..1000})),list)); # Robert Israel, Jun 26 2018

TakeWhile[Complement[Range@ #, Union@ Table[Total@ Select[Divisors@ n, SquareFreeQ], {n, 2 #}]], Function[k, k <= #]] &@ 111

cp's OEIS Frontend

A265668 Table read by rows: prime factors of squarefree numbers; a(1) = 1 by convention.

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A366439 The sum of divisors of the exponentially odd numbers (A268335).

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A072048 Number of divisors of the squarefree numbers: tau(A005117(n)).

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A366440 The sum of divisors of the cubefree numbers (A004709).

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A366537 The sum of unitary divisors of the cubefree numbers (A004709).

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A369889 The sum of squarefree divisors of the cubefree numbers.

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A073245 Sum of all cubefree numbers with the same squarefree kernel as the n-th squarefree number.

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