A286708 -id:A286708 - cp's OEIS frontend

A365786 a(n) = squarefree kernel of A286708(n).

6, 6, 10, 6, 6, 14, 10, 6, 15, 6, 6, 14, 10, 6, 21, 22, 10, 6, 6, 15, 26, 14, 10, 6, 30, 22, 6, 10, 33, 15, 6, 34, 35, 6, 21, 26, 14, 38, 39, 14, 10, 6, 42, 30, 22, 6, 10, 15, 46, 6, 34, 10, 6, 51, 30, 26, 14, 38, 6, 55, 21, 14, 10, 57, 33, 58, 15, 6, 42, 30, 62

Offset: 1

Michael De Vlieger, Sep 19 2023

nonn

Terms are squarefree and composite, i.e., in A120944.

			Let b(n) = A286708(n) and let squarefree kernel rad(n) = A007947(n).
a(1) = 6 = rad(b(1)) = rad(36).
a(2) = 6 = rad(b(2)) = rad(72).
a(3) = 10 = rad(b(3)) = rad(100), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A007947, A084371, A120944, A126706, A286708.

With[{nn = 2^12}, Map[Times @@ FactorInteger[#][[All, 1]] &, Rest@ Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Not @* PrimePowerQ]] ]

apply(x->factorback(factorint(x)[, 1]), select(x->((x>1) && ispowerful(x) && !isprimepower(x)), [1..5000])) \\ Michel Marcus, Sep 20 2023

a(n) = A007947(A286708(n)).

A365792 a(n) = number of k <= b(n) such that rad(k) | b(n), where rad(n) = A007947(n) and b(n) = A286708(n).

Original entry on oeis.org

14, 18, 15, 21, 23, 16, 19, 26, 13, 29, 30, 20, 23, 32, 14, 18, 24, 35, 36, 18, 19, 24, 28, 39, 83, 21, 40, 29, 15, 20, 42, 21, 13, 43, 18, 22, 27, 21, 15, 28, 33, 46, 91, 104, 25, 47, 34, 23, 22, 50, 24, 36, 51, 16, 120, 26, 32, 24, 52, 13, 22, 33, 39, 16, 19

Offset: 1

Michael De Vlieger, Sep 22 2023

nonn

Alternatively, position of A286708(n) in the list R(rad(n)) of k such that rad(k) | n, where rad(n) = A007947(n).

			a(1) = 14 since rad(b(1)) = rad(36) = 6, and in the sequence R(6) = A003586 = {1, 2, 3, 4, 6, 8, 9, ..., 36, ...}, 36 is the 14th term.
a(2) = 18 since rad(b(2)) = rad(72) = 6, and 72 is the 18th term in R(6).
a(3) = 15 since rad(b(3)) = rad(100) = 10, and in the sequence R(10) = A003592 = {1, 2, 4, 5, 8, 10, ..., 100, ...}, 100 is the 15th term, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A010846, A286708, A365786, A365793.

nn = 3300; f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
t = Select[
  Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &],
  AllTrue[FactorInteger[#][[All, -1]], # > 1 &] &];
s = Map[f, t];
Map[Function[k, Set[r[k], Select[Range[nn], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]]][[1]] &, Length[t]]

A358173 First differences of A286708.

Original entry on oeis.org

36, 28, 8, 36, 52, 4, 16, 9, 63, 36, 68, 8, 32, 9, 43, 16, 76, 72, 27, 1, 108, 16, 64, 36, 68, 4, 28, 89, 36, 27, 4, 69, 71, 27, 29, 20, 72, 77, 47, 32, 128, 36, 36, 136, 8, 56, 25, 91, 188, 8, 188, 92, 9, 99, 4, 40, 144, 28, 109, 62, 49, 64, 49, 18, 97, 11, 81

Offset: 1

Michael De Vlieger, Nov 01 2022

nonn

Consider the sequence of powerful numbers A001694, superset of A246547, the sequence of composite prime powers. Let s = A001694(k) such that omega(s) > 1 be followed by t = A001694(k+1) such that omega(t) = 1.
Since A286708 = A001694 \ A246547, prime powers t are missing in A286708. We consider s = A286708(j) and note that the difference A286708(j+1) - A286708(j) > A001694(k+1) - A001694(k).
Therefore we see a subset S containing s in A286708 that plots "out of place" with respect to the complementary subset R = A286708 \ S; some of this subset S exceeds the maxima of R in the scatterplot of this sequence. The plot of the R resembles the scatterplot of A001694.

			The number 36 is the smallest powerful number that is not a prime power; the next powerful number that is not a prime power is 72, and their difference is 36, hence a(1) = 36.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n), n = 1..2^16, highlighting in red terms s that in A001694 are succeeded by a prime power t.

Cf. A001694, A053707, A076446, A246547, A286708.

With[{nn = 2^25}, Differences@ Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], ! PrimePowerQ[#] &]]

from math import isqrt
from sympy import integer_nthroot, primepi, mobius
def A358173(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length())), 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        return c
    return -(a:=bisection(f,n,n))+bisection(lambda x:f(x)+1,a,a) # Chai Wah Wu, Sep 10 2024

A365787 a(n) = A286708(n) divided by its squarefree kernel.

Original entry on oeis.org

6, 12, 10, 18, 24, 14, 20, 36, 15, 48, 54, 28, 40, 72, 21, 22, 50, 96, 108, 45, 26, 56, 80, 144, 30, 44, 162, 100, 33, 75, 192, 34, 35, 216, 63, 52, 98, 38, 39, 112, 160, 288, 42, 60, 88, 324, 200, 135, 46, 384, 68, 250, 432, 51, 90, 104, 196, 76, 486, 55, 147

Offset: 1

Michael De Vlieger, Sep 19 2023

nonn

Permutation of numbers that are not prime powers A024619.

			a(1) = 2 since b(1)/rad(b(1)) = 36/6 = 6.
a(2) = 3 since b(2)/rad(b(2)) = 72/6 = 12.
a(3) = 2 since b(3)/rad(b(3)) = 100/10 = 10.
a(4) = 4 since b(4)/rad(b(4)) = 108/6 = 18.
a(5) = 2 since b(5)/rad(b(5)) = 144/6 = 24.
a(6) = 6 since b(6)/rad(b(6)) = 196/14 = 14, etc

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A024619, A120944, A286708, A365786.

nn = 5000;
s = Rest@ Select[Union@ Flatten@
  Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
  Not @* PrimePowerQ];
t = Select[Range[nn/6], And[SquareFreeQ[#], CompositeQ[#]] &];
Map[FirstPosition[t, Times @@ FactorInteger[#][[All, 1]]][[1]] &, s]

a(n) = A286708(n)/A007947(A286708(n)) = A286708(n)/A365786(n).

Let b(n) = A286708(n) and let squarefree kernel rad(n) = A007947(n). a(n) >= n such that rad(a(n)) | n.

A365793 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A286708(n).

Original entry on oeis.org

5, 8, 6, 10, 11, 6, 8, 14, 5, 15, 16, 8, 11, 18, 5, 7, 12, 20, 21, 8, 7, 11, 14, 23, 18, 9, 24, 15, 6, 9, 25, 8, 5, 26, 8, 9, 13, 8, 6, 14, 18, 29, 19, 26, 11, 30, 19, 12, 8, 31, 10, 20, 32, 6, 32, 11, 16, 10, 33, 5, 10, 17, 22, 6, 8, 8, 13, 35, 28, 36, 8, 14

Offset: 1

Michael De Vlieger, Sep 22 2023

nonn

Alternatively, position of A126706(n) in the list k*{R(k)} containing m such that A007947(m) = k, where k = A007947(n).

			a(1) = 5 since rad(b(1)) = rad(36) = 6, and in the sequence k*{R(6)} = 6*{A003586} = {6, 12, 18, 24, 36, ...}, 36 is the 5th term.
a(2) = 8 since rad(b(2)) = rad(72) = 6, and 72 is the 8th term in k*{R(6)}.
a(3) = 6 since rad(b(3)) = rad(100) = 10, and in the sequence k*{R(10)} = 10*{A003592} = {10, 20, 40, 50, 80, 100, ...}, 100 is the 6th term, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A286708, A365786, A365792.

nn = 4000;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
t = Select[
  Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &],
  AllTrue[FactorInteger[#][[All, -1]], # > 1 &] &];
s = Map[f, t];
Map[Function[k, Set[r[k], k*Select[Range[nn/k], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]]][[1]] &, Length[t]]

a(n) = A008479(A286708(n)).

a(n) > 1 for all n.

A380430 Number of powerful numbers k that are not powers of primes (i.e., k is in A286708) that do not exceed the primorial number A002110(n).

Original entry on oeis.org

0, 0, 0, 0, 7, 50, 254, 1245, 5898, 29600, 163705, 925977, 5690175, 36681963, 241663896, 1662446097, 12134853382, 93406989325, 730785520398, 5990426525483, 50538885715526, 432266550168097, 3845700235189327, 35065304557027821, 334652745159828239, 3262707438761612651

Offset: 0

Michael De Vlieger, Jan 24 2025

			Let P = A002110 and let s = A286708 = A001694 \ A246547 \ {1}.
a(0..3) = 0 since the smallest number in s is 36.
a(4) = 7 since P(4) = 210 and numbers in s that are less than 210 include {36, 72, 100, 108, 144, 196, 200}, etc.

Cf. A001694, A126706, A246547, A286708, A380254, A380402.

Table[-1 + Sum[If[MoebiusMu[j] != 0, Floor[Sqrt[#/j^3]], 0], {j, #^(1/3)}] - Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor[Log2[#]]}] &[Product[Prime[i], {i, n}]], {n, 0, 12} ]

from math import isqrt
from sympy import primorial, primepi, integer_nthroot, mobius
def A380430(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    if n == 0: return 0
    m = primorial(n)
    c, l, j = int(squarefreepi(integer_nthroot(m, 3)[0])-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))-1), 0, isqrt(m)
    while j>1:
        k2 = integer_nthroot(m//j**2,3)[0]+1
        w = squarefreepi(k2-1)
        c += j*(w-l)
        l, j = w, isqrt(m//k2**3)
    return c-l # Chai Wah Wu, Feb 25 2025

a(n) = A380254(n) - A380402(n) - 1.

a(n) <= A380403(n) since A286708 is a proper subset of A126706.

A380431 Number of powerful numbers that are not powers of primes (i.e. are in A286708) that do not exceed 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 9, 17, 28, 48, 75, 115, 178, 266, 403, 590, 865, 1263, 1830, 2644, 3810, 5466, 7838, 11210, 16011, 22841, 32530, 46315, 65886, 93658, 133060, 188952, 268204, 380564, 539823, 765481, 1085224, 1538194, 2179816, 3088481, 4375308, 6197420, 8777222

Offset: 0

Michael De Vlieger, Jan 24 2025

			Let s = A286708 = A001694 \ A246547 \ {1}.
a(0..5) = 0 since the smallest number in s is 36.
a(6) = 1 since only s(1) = 36 is smaller than 2^6 = 64.
a(7) = 4 since s(1..4) = {36, 72, 100, 108} are smaller than 2^7 = 128.
a(8) = 9 since s(1..9) = {36, 72, 100, 108, 144, 196, 200, 216, 225} are smaller than 2^8 = 256, etc.

Amiram Eldar, Table of n, a(n) for n = 0..90

Cf. A001694, A036386, A062762, A246547, A286708, A372403.

Table[-1 + Sum[If[MoebiusMu[j] != 0, Floor[Sqrt[(2^n)/j^3]], 0], {j, 2^(n/3)}] - Sum[PrimePi@ Floor[2^(n/k)], {k, 2, n}], {n, 0, 45} ]

from math import isqrt
from sympy import mobius, integer_nthroot, primepi
def A380431(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    l, m = 0, 1<1:
        k2 = integer_nthroot(m//j**2,3)[0]+1
        w = squarefreepi(k2-1)
        c += j*(w-l)
        l, j = w, isqrt(m//k2**3)
    return c-l # Chai Wah Wu, Jan 30 2025

a(n) = A062762(n) - A036386(n) - 1.

a(n) <= A372403(n), since A286708 is a proper subset of A126706.

A378899 Number of primes between successive powerful numbers k that are not prime powers (i.e., k in A286708).

Original entry on oeis.org

11, 9, 5, 3, 6, 10, 2, 1, 1, 13, 5, 11, 1, 5, 2, 7, 3, 10, 13, 4, 0, 15, 2, 11, 4, 9, 1, 4, 13, 7, 2, 1, 9, 10, 6, 1, 2, 9, 12, 7, 4, 18, 5, 4, 17, 0, 8, 3, 13, 23, 2, 23, 10, 1, 15, 0, 7, 18, 3, 13, 7, 4, 7, 5, 4, 13, 2, 6, 10, 11, 29, 4, 2, 11, 1, 28, 2, 14

Offset: 0

Michael De Vlieger, Dec 10 2024

			Let s = A286708.
a(0) = 11 since {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31} are primes less than s(1) = 36.
a(1) = 9 since {37, 41, 43, 47, 53, 59, 61, 67, 71} are primes that exceed s(1) but not s(2) = 72.
a(2) = 5 since {73, 79, 83, 89, 97} are primes p such that s(2) < p < s(3), where s(3) = 100.
a(3) = 3 since {101, 103, 107} are primes p such that s(3) < p < s(4), where s(4) = 108, etc.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000040, A000720, A240590, A286708, A378699.

s = With[{nn = 5000},
  Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Not@*PrimePowerQ]];
{PrimePi[s[[1]]]}~Join~Differences@ Map[PrimePi, s]

a(0) = pi(36) = A000720(36) = 11.

For n > 0, a(n) = pi(A286708(n+1)) - pi(A286708(n)).

A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

Original entry on oeis.org

1, 36, 100, 144, 196, 216, 225, 324, 400, 441, 484, 576, 676, 784, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1728, 1764, 1936, 2025, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3249, 3364, 3375, 3600, 3844, 3969, 4225, 4356, 4624

Offset: 1

Daniel Forgues, May 27 2008

nonn

Although 1 is a square, is a cube, and so on..., 1 is sometimes excluded from perfect powers since it is not a well-defined power of 1 (1 = 1^k for any k in [2, 3, 4, 5, ...])
From Michael De Vlieger, Aug 11 2025: (Start)
This sequence is A001597 \ A246547, i.e., perfect powers without proper prime powers.
Union of {1} with the intersection of A001597 and A126706, where A126706 is the sequence of numbers that are neither prime powers nor squarefree.
Union of {1} and A286708 \ A052486, i.e., powerful numbers that are not prime powers, without Achilles numbers, but including the empty product. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (Terms a(n), n = 1..1323 from Klaus Brockhaus, and n = 1324..8649 from Daniel Forgues.)

Cf. A000961, A001597, A024619, A025475, A052486, A072102, A126706, A136141, A246547, A286708.

With[{nn = 2^20}, {1}~Join~Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], And[Length[#2] > 1, GCD @@ #2 > 1] & @@ {#, FactorInteger[#][[;; , -1]]} &] ] (* Michael De Vlieger, Aug 11 2025 *)

isok(n) = if (n == 1, return (1), return (ispower(n, ,&np) && (! isprime(np)))); \\ Michel Marcus, Jun 12 2013

from sympy import mobius, integer_nthroot, primepi
def A131605(n):
    def f(x): return int(n-2+x+sum(mobius(k)*((a:=integer_nthroot(x,k)[0])-1)+primepi(a) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Sum_{n>=1} 1/a(n) = 1 + A072102 - A136141 = 1.10130769935514973882... . - Amiram Eldar, Aug 15 2025

A360519 Let C consist of 1 together with all numbers with at least two distinct prime factors; this is the lexicographically earliest infinite sequence {a(n)} of distinct elements of C such that, for n>2, a(n) has a common factor with a(n-1) but not with a(n-2).

Original entry on oeis.org

1, 6, 10, 35, 21, 12, 20, 55, 33, 18, 14, 77, 99, 15, 40, 22, 143, 39, 24, 28, 91, 65, 30, 34, 119, 63, 36, 26, 221, 51, 42, 38, 95, 45, 48, 44, 187, 85, 50, 46, 69, 57, 76, 52, 117, 75, 70, 58, 87, 93, 62, 56, 105, 111, 74, 68, 153, 123, 82, 80, 115, 161, 84, 60, 145, 203, 98, 54, 129, 215, 100, 66, 141

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 01 2023

nonn

In other words, C contains all positive numbers except powers of primes p^k, k>=1.
This is a modified version of the Enots Wolley sequence A336957. The modification ensures that the sequence does not contain the prime 2.
Let Ker(k), the kernel of k, denote the set of primes dividing k. Thus Ker(36) = {2,3}, Ker(1) = {}.
Theorem: a(1)=1, a(2)=6; thereafter, a(n) is the smallest number m not yet in the sequence such that
(i) Ker(m) intersect Ker(a(n-1)) is nonempty,
(ii) Ker(m) intersect Ker(a(n-2)) is empty, and
(iii) The set Ker(m) \ Ker(a(n-1)) is nonempty.
Conjecture: The sequence is a permutation of C.

Scott R. Shannon, Table of n, a(n) for n = 1..100000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing squarefree composites (in A120944) in green, numbers neither prime power nor squarefree (in A126706) in blues, highlighting powerful numbers (in A001694 and A286708) in large dark blue.
Scott R. Shannon, White on black graph of first 10^5 terms [Some aspects of the graph are more visible when black and white are swapped. The green line is a(n) = n]
Scott R. Shannon, Image of the first 1 million terms in color. The terms with a lowest prime factor of 2,3,5,7,9,11,13,17,19,>=23 are colored white, red, orange, yellow, green, blue, indigo, violet, gray respectively.
N. J. A. Sloane, Table showing a(1)-a(13), also the smallest missing number (smn, A361109 and A361110), binary vectors showing which terms are divisible by the primes 2, 3, 5, 7, 11; and phi, a decimal representation of those binary vectors (A361111).

Cf. A098550, A336957.

For a number of sequences related to this, see A361102 (the sequence C) and the following entries.

with(numtheory);
N:= 10^4: # to get a(1) to a(n) where a(n+1) is the first term > N
B:= Vector(N, datatype=integer[4]):
A[1]:=1; A[2]:=6;
for n from 3 do
  for k from 10 to N do
    if B[k] = 0 and igcd(k, A[n-1]) > 1 and igcd(k, A[n-2]) = 1 then
          if nops(factorset(k) minus factorset(A[n-1])) > 0 then
       A[n]:= k;
       B[k]:= 1;
       break;
          fi;
    fi
  od:
  if k > N then break; fi;
od:
s1:=[seq(A[i], i=1..n-1)];

nn = 2^12; c[_] = False;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
MapIndexed[
 Set[{a[First[#2]], c[#1]}, {#1, True}] &, {1, 6}];
 u = 10; i = a[1]; j = a[2];
Do[k = u;
  While[Nand[! PrimePowerQ[k], ! c[k],
    CoprimeQ[i, k], ! CoprimeQ[j, k], ! Divisible[j, f[k]]], k++];
  Set[{a[n], c[k], i, j}, {k, True, j, f[k]}];
  If[k == u, While[Or[c[u], PrimePowerQ[u]], u++]]
  , {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Mar 03 2023 *)

cp's OEIS Frontend

A365786 a(n) = squarefree kernel of A286708(n).

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A365792 a(n) = number of k <= b(n) such that rad(k) | b(n), where rad(n) = A007947(n) and b(n) = A286708(n).

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A358173 First differences of A286708.

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A365787 a(n) = A286708(n) divided by its squarefree kernel.

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A365793 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A286708(n).

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A380430 Number of powerful numbers k that are not powers of primes (i.e., k is in A286708) that do not exceed the primorial number A002110(n).

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A380431 Number of powerful numbers that are not powers of primes (i.e. are in A286708) that do not exceed 2^n.

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A378899 Number of primes between successive powerful numbers k that are not prime powers (i.e., k in A286708).

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