A049200 -id:A049200 - 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 *)

A374456 The Euler phi function values of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 4, 4, 10, 12, 6, 8, 16, 18, 12, 10, 22, 8, 12, 18, 28, 8, 30, 16, 20, 16, 24, 36, 18, 24, 16, 40, 12, 42, 22, 46, 32, 52, 18, 40, 24, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 40, 88, 72, 60, 46, 72, 32, 96

Offset: 1

Amiram Eldar, Jul 09 2024

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

Cf. A000010, A065463, A268335, A307868.
Similar sequences related to phi: A002618, A049200, A323333, A358039.
Similar sequences related to exponentially odd numbers: A366438, A366439, A366534, A366535, A367417, A368711, A374457.

f[p_, e_] := If[OddQ[e], (p-1) * p^(e-1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &]

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

a(n) = A000010(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A307868 / A065463^2 = 0.95051132596733153581... .

A379715 The second Jordan totient function applied to the squarefree numbers.

Original entry on oeis.org

1, 3, 8, 24, 24, 48, 72, 120, 168, 144, 192, 288, 360, 384, 360, 528, 504, 840, 576, 960, 960, 864, 1152, 1368, 1080, 1344, 1680, 1152, 1848, 1584, 2208, 2304, 2808, 2880, 2880, 2520, 3480, 3720, 2880, 4032, 2880, 4488, 4224, 3456, 5040, 5328, 4104, 5760, 4032

Offset: 1

Amiram Eldar, Dec 30 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mohammadreza Esfandiari, On the Means of Jordan's Totient Function, Bull. Iran. Math. Soc., Vol. 46 (2020), pp. 1753-1765.
R. Sitaramachandrarao, On an error term of Landau - II, Rocky Mountain J. Math., Vol. 15, No. 2 (1985), pp. 579-588. See p. 581.

Cf. A005117, A007434, A013661, A049200 (analogous with J_1 = phi), A330523, A379716, A379717, A379718.

f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; j2 /@ Select[Range[100], SquareFreeQ]

j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
list(lim) = apply(j2, select(issquarefree, vector(lim, i, i)));

a(n) = A007434(A005117(n)).
Sum_{n>=1} 1/a(n) = zeta(2) (A013661) (Sitaramachandrarao, 1985).
In general, Sum_{m squarefree} 1/J_k(m) = zeta(k), for k >= 2, where J_k is the k-th Jordan totient function.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2)^3 * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) =  A013661^3 * A330523 = 2.38520727393117206135... . - Amiram Eldar, Jan 03 2025

A049225 Values of totient function applied to squarefree numbers; or numbers of form Product (p_i-1) where p_i are distinct primes.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 56, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 92, 96, 100, 102, 104, 106, 108, 112, 116, 120, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 156, 160, 162, 164, 166, 168, 172, 176

Offset: 1

Labos Elemer

nonn

Numbers m such that m = phi(k) and |mu(k)| = 1.

			8, 120, 48 are terms as totients of the squarefree numbers 15, 143, 210.
54, 110 are not terms since there are no squarefree numbers k such that phi(k) = 54, 110.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Terms of A049200 after ordering and omitting multiple occurrences.

Cf. A000010, A005117, A008683, A013929, A002174, A002202, A049200.

isok(n) = {my(v = invphi(n)); (#v) && (#select(x->issquarefree(x), v));} \\ Michel Marcus, Feb 25 2019

A358039 a(n) is the Euler totient function phi applied to the n-th cubefree number.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 6, 4, 10, 4, 12, 6, 8, 16, 6, 18, 8, 12, 10, 22, 20, 12, 12, 28, 8, 30, 20, 16, 24, 12, 36, 18, 24, 40, 12, 42, 20, 24, 22, 46, 42, 20, 32, 24, 52, 40, 36, 28, 58, 16, 60, 30, 36, 48, 20, 66, 32, 44, 24, 70, 72, 36, 40, 36, 60, 24, 78, 40

Offset: 1

Amiram Eldar, Oct 29 2022

nonn

The analogous sequence with squarefree numbers is A049200.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Zhu Weiyi, On the cube free number sequences, Smarandache Notions J., Vol. 14 (2004), pp. 199-202.

Cf. A000010, A002117, A004709, A049200, A358040.

EulerPhi[Select[Range[100], Max[FactorInteger[#][[;; , 2]]] < 3 &]]

from sympy import mobius, integer_nthroot, totient
def A358039(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 totient(m) # Chai Wah Wu, Aug 06 2024

a(n) = A000010(A004709(n)).
Sum_{k=1..n} a(k) = (c/(2*zeta(3)))*n^2 + O(n^(3/2+eps)), where c = Product_{p prime} (1 - (p+1)/(p^3+p^2+1)) = 0.62583324412633345811... (Weiyi, 2004).
From Amiram Eldar, Oct 09 2023: (Start)
Sum_{n>=1} 1/(A004709(n)*a(n)) = Product_{p prime} (1 + (p^2+1)/((p-1)*p^3)) = 2.14437852780769816048... .
Sum_{n>=1} 1/a(n)^2 = Product_{p prime} (1 + (p^2+1)/((p-1)^2*p^2)) = 3.26032746607943673536... . (End)

cp's OEIS Frontend

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

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A374456 The Euler phi function values of the exponentially odd numbers (A268335).

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A379715 The second Jordan totient function applied to the squarefree numbers.

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A049225 Values of totient function applied to squarefree numbers; or numbers of form Product (p_i-1) where p_i are distinct primes.

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PARI

A358039 a(n) is the Euler totient function phi applied to the n-th cubefree number.

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Python

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