A349162 -id:A349162 - cp's OEIS frontend

A369260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A349162(i) = A349162(j), for all i, j >= 1.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 15, 23, 16, 24, 25, 26, 27, 28, 21, 29, 30, 31, 30, 32, 33, 34, 26, 35, 36, 37, 38, 39, 40, 28, 30, 41, 42, 43, 44, 45, 46, 47, 44, 48, 49, 50, 51, 52, 53, 37, 54, 55, 56, 57, 58, 59, 60, 57, 44, 61, 62, 63, 41, 64, 60, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 65, 79, 57

Offset: 1

Antti Karttunen, Jan 25 2024

nonn

Restricted growth sequence transform of the ordered pair [A342671(n), A349162(n)], or equally, of the pair [A000203(n), A342671(n)], or equally, of the pair [A000203(n), A349162(n)].
For all i, j >= 1:
  A369259(i) = A369259(j) => a(i) = a(j) => A286603(i) = A286603(j).

Cf. A000203, A003961, A342671, A348993, A349162.

Cf. also A286603, A291751, A369259, A369261.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
Aux369260(n) = { my(u=A342671(n)); [u, sigma(n)/u]; };
v369260 = rgs_transform(vector(up_to, n, Aux369260(n)));
A369260(n) = v369260[n];

A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 21, 1, 3, 1, 15, 1, 3, 5, 1, 1, 3, 1, 9, 1, 3, 1, 1, 1, 3, 1, 9, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 3, 5, 3, 1, 21, 1, 3, 1, 1, 7, 3, 1, 9, 1, 3, 1, 15, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 5, 9, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 13, 7, 1, 3, 1, 3, 1

Offset: 1

Antti Karttunen, Mar 20 2021

nonn

Cf. A000203, A003961, A161942, A286385, A341529, A342672, A342673, A348992, A349161, A349162, A349163, A349164, A349165 (positions of 1's), A349166 (of terms > 1), A349167, A349756, A350071 [= a(n^2)], A355828 (Dirichlet inverse).
Cf. A349169, A349745, A355833, A355924 (applied onto prime shift array A246278).
Cf. also A336850, A354827/A354828, A355932.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));

a(n) = gcd(A000203(n), A003961(n)).
a(n) = gcd(A000203(n), A286385(n)) = gcd(A003961(n), A286385(n)).
a(n) = A341529(n) / A342672(n).
From Antti Karttunen, Jul 21 2022: (Start)
a(n) = A003961(n) / A349161(n).
a(n) = A000203(n) / A349162(n).
a(n) = A161942(n) / A348992(n).
a(n) = A003961(A349163(n)) = A003961(n/A349164(n)).
(End)

A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 15, 105, 3003, 3465, 13923, 45045, 264537, 459459, 745875, 1541475, 5221125, 8729721, 10790325, 14171625, 29288025, 34563375, 57034575, 71430975, 99201375, 109643625, 144729585, 205016175, 255835125, 295708875, 356080725, 399242025, 419159475, 449323875, 928602675, 939495375, 1083656925, 1941623775, 1962350685, 2083228875

Offset: 1

Antti Karttunen, Nov 10 2021

Numbers k such that A348990(k) [= k/gcd(k, A003961(k))] is equal to A348992(k), which is the odd part of A349162(k), thus all terms must be odd, as A348990 preserves the parity of its argument.
Equally, numbers k for which gcd(A064987(k), A191002(k)) is equal to A000265(gcd(A064987(k), A341529(k))).
Also odd numbers k for which A348993(k) = A319627(k).
Odd terms of A336702 are given by the intersection of this sequence and A349174.
Conjectures:
  (1) After 1, all terms are multiples of 3. (Why?)
  (2) After 1, all terms are in A104210, in other words, for all n > 1, gcd(a(n), A003961(a(n))) > 1. Note that if we encountered a term k with gcd(k, A003961(k)) = 1, then we would have discovered an odd multiperfect number.
  (3) Apart from 1, 15, 105, 3003, 13923, 264537, all other terms are abundant. [These apparently are also the only terms that are not Zumkeller, A083207. Note added Dec 05 2024]
  (4) After 1, all terms are in A248150. (Cf. also A386430).
  (5) After 1, all terms are in A348748.
  (6) Apart from 1, there are no common terms with A349753.
Note: If any of the last four conjectures could be proved, it would refute the existence of odd perfect numbers at once. Note that it seems that gcd(sigma(k), A003961(k)) < k, for all k except these four: 1, 2, 20, 160.
Questions:
  (1) For any term x here, can 2*x be in A349745? (Partial answer: at least x should be in A191218 and should not be a multiple of 3). Would this then imply that x is an odd perfect number? (Which could explain the points (1) and (4) in above, assuming the nonexistence of opn's).

Cf. A000203, A003961, A007949, A064989, A083207, A104210, A161942, A191002, A191218, A228058, A319627, A322361, A326134, A329963, A342671, A348748, A348990, A348992, A348993, A349162, A349164, A349174.
Cf. also A349745, A349746, A349753.
Subsequence of A349749, and of A386430.

Select[Range[10^6], #1/GCD[#1, #3] == #2/(2^IntegerExponent[#2, 2]*GCD[#2, #3]) & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A000265(n) = (n >> valuation(n, 2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349169(n) = { my(s=sigma(n),u=A003961(n)); (n*gcd(s,u) == A000265(s)*gcd(n,u)); }; \\ (Program simplified Nov 30 2021)

For all n >= 1, A007949(A000203(a(n))) = A007949(a(n)). [sigma preserves the 3-adic valuation of the terms of this sequence] - Antti Karttunen, Nov 29 2021

Name changed and comment section rewritten by Antti Karttunen, Nov 29 2021

A349161 a(n) = A003961(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 5, 9, 7, 5, 11, 9, 25, 7, 13, 45, 17, 11, 35, 81, 19, 25, 23, 3, 55, 13, 29, 9, 49, 17, 25, 99, 31, 35, 37, 27, 65, 19, 77, 225, 41, 23, 85, 21, 43, 55, 47, 39, 175, 29, 53, 405, 121, 49, 95, 153, 59, 25, 91, 99, 23, 31, 61, 15, 67, 37, 275, 729, 17, 65, 71, 19, 145, 77, 73, 45, 79, 41, 245, 207, 143, 85, 83

Offset: 1

Antti Karttunen, Nov 09 2021

Numerator of ratio A003961(n) / A000203(n). Sequence A349162 gives the denominators.
Numerator of ratio A003961(n) / A161942(n). Sequence A348992 gives the denominators.
Both ratios are multiplicative because the constituent sequences are.
No 1's occur as terms after a(2), because for n > 2, sigma(n) < A003961(n). (See A286385).

Cf. A000203, A003961, A161942, A286385, A319626, A319627, A326042, A336702, A342671, A349163, A349164, A349627, A349628.

Cf. A348992, A349162 (denominators).

Array[#2/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 79] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349161(n) = { my(u=A003961(n)); (u/gcd(u,sigma(n))); };

from math import prod, gcd
from sympy import nextprime, factorint
def A349161(n):
    f = factorint(n).items()
    a = prod(nextprime(p)**e for p, e in f)
    b = prod((p**(e+1)-1)//(p-1) for p, e in f)
    return a//gcd(a,b) # Chai Wah Wu, Mar 17 2023

a(n) = A003961(n) / A342671(n) = A003961(n) / gcd(A000203(n), A003961(n)).

a(n) = A003961(A349164(n)).

A349164 a(n) = A064989(A003961(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 4, 9, 5, 11, 12, 13, 7, 15, 16, 17, 9, 19, 2, 21, 11, 23, 4, 25, 13, 9, 28, 29, 15, 31, 8, 33, 17, 35, 36, 37, 19, 39, 10, 41, 21, 43, 22, 45, 23, 47, 48, 49, 25, 51, 52, 53, 9, 55, 28, 19, 29, 59, 6, 61, 31, 63, 64, 13, 33, 67, 17, 69, 35, 71, 12, 73, 37, 75, 76, 77, 39, 79, 40, 81, 41, 83, 84

Offset: 1

Antti Karttunen, Nov 09 2021

nonn

Cf. A000203, A003961, A319630, A326042, A336702, A342671, A348993, A349161, A349162, A349169, A349174.

Cf. A349144 and A349168 [positions where a(n) is / is not relatively prime with A349163(n) = n/a(n)].

Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#2/GCD[##]]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 84] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A349164(n) = { my(u=A003961(n)); A064989(u/gcd(u,sigma(n))); };

a(n) = A064989(A349161(n)).

a(n) = n / A349163(n).

A351546 a(n) is the largest unitary divisor of sigma(n) coprime with A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 4, 7, 6, 4, 8, 5, 13, 2, 12, 28, 14, 8, 24, 31, 18, 13, 20, 2, 32, 4, 24, 4, 31, 14, 8, 56, 30, 8, 32, 7, 48, 2, 48, 91, 38, 20, 56, 10, 42, 32, 44, 28, 78, 8, 48, 124, 57, 31, 72, 98, 54, 8, 72, 40, 16, 10, 60, 8, 62, 32, 104, 127, 12, 16, 68, 14, 96, 16, 72, 13, 74, 38, 124, 140, 96, 56, 80, 62, 121, 14, 84

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

			For n = 672 = 2^5 * 3^1 * 7^1, and the largest unitary divisor of the sigma(672) [= 2^5 * 3^2 * 7^1] coprime with A003961(672) = 13365 = 3^5 * 5^1 * 11^1 is 2^5 * 7^1 = 224, therefore a(672) = 224.

Cf. A000203, A003961, A351544, A351547, A351551, A351552, A353666 [gcd(a(n),n)], A353667, A353668, A353633, A354997.

Cf. A342671, A349161, A349162.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351546(n) = { my(f=factor(sigma(n)),u=A003961(n)); prod(k=1,#f~,f[k,1]^((0!=(u%f[k,1]))*f[k,2])); };

a(n) = Product_{p^e || A000203(n)} p^(e*[p does not divide A003961(n)]), where [ ] is the Iverson bracket, returning 0 if p is a divisor of A003961(n), and 1 otherwise. Here p^e is the largest power of each prime p dividing sigma(n).
a(n) = A000203(n) / A351544(n).
a(n) = A353666(n) * A353668(n) = A351547(n) / A354997(n). - Antti Karttunen, Jul 09 2022

A349163 a(n) = A064989(gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 10, 1, 2, 1, 6, 1, 2, 3, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 10, 1, 2, 1, 1, 5, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 11, 5, 1, 2, 1, 2, 1

Offset: 1

Antti Karttunen, Nov 09 2021

nonn

Cf. A000203, A003961, A342671, A349161, A349162, A349165 (positions of 1's), A349166 (of terms > 1).

Cf. A349144 and A349168 [positions where a(n) is / is not relatively prime with A349164(n) = n/a(n)].

Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger@ GCD[##]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 105] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A349163(n) = A064989(gcd(sigma(n),A003961(n)));

a(n) = A064989(A342671(n)).

a(n) = n / A349164(n).

A348992 a(n) = A000265(sigma(n)) / gcd(sigma(n), A003961(n)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 7, 3, 1, 1, 5, 13, 3, 3, 7, 7, 1, 3, 31, 9, 13, 5, 1, 1, 3, 3, 1, 31, 7, 1, 7, 15, 3, 1, 7, 3, 9, 3, 91, 19, 5, 7, 5, 21, 1, 11, 7, 39, 3, 3, 31, 57, 31, 9, 49, 27, 1, 9, 5, 1, 15, 15, 1, 31, 1, 13, 127, 3, 3, 17, 7, 3, 3, 9, 13, 37, 19, 31, 35, 3, 7, 5, 31, 121, 21, 21, 7, 27, 11, 3, 5, 45, 39, 7, 7, 1, 3

Offset: 1

Antti Karttunen, Nov 10 2021

Denominator of ratio A003961(n) / A161942(n).

Odd part of A349162.
Cf. A000203, A000265, A003961, A161942, A342671, A348993, A349169 (where equal to A348990).
Cf. A349161 (numerators).

Array[#1/(2^IntegerExponent[#1, 2]*GCD[##]) & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 94] (* Michael De Vlieger, Nov 11 2021 *)

A000265(n) = (n >> valuation(n, 2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A348992(n) = { my(s=sigma(n)); (A000265(s)/gcd(s,A003961(n))); };

a(n) = A161942(n) / A342671(n) = A000265(A349162(n)).

a(n) = A003961(A348993(n)).

A348993 a(n) = A064989(sigma(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 1, 3, 11, 2, 2, 5, 5, 1, 2, 29, 4, 11, 3, 1, 1, 2, 2, 1, 29, 5, 1, 5, 6, 2, 1, 5, 2, 4, 2, 55, 17, 3, 5, 3, 10, 1, 7, 5, 22, 2, 2, 29, 34, 29, 4, 25, 8, 1, 4, 3, 1, 6, 6, 1, 29, 1, 11, 113, 2, 2, 13, 5, 2, 2, 4, 11, 31, 17, 29, 15, 2, 5, 3, 29, 49, 10, 10, 5, 8, 7, 2, 3, 12, 22, 5, 5, 1, 2, 6, 5

Offset: 1

Antti Karttunen, Nov 10 2021

Cf. A000203, A000265, A003961, A064989, A161942, A342671, A348992, A349162, A349169 (gives odd k for which a(k) = A319627(k)).

Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#1/GCD[##]]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 96] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };
A348993(n) = A064989(A349162(n));

a(n) = A064989(A349162(n)) = A064989(A348992(n)).

A349627 Numerators of the Möbius transform of ratio A003961(n)/sigma(n).

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 3, 18, 35, 0, 1, 1, 3, 0, 1, 126, 1, 0, 3, 1, 3, 0, 5, 9, 77, 0, 125, 3, 1, 0, 5, 270, 1, 0, 1, 5, 3, 0, 3, 3, 1, 0, 3, 1, 35, 0, 5, 63, 341, 0, 1, 3, 5, 0, 1, 27, 3, 0, 1, 1, 5, 0, 105, 1674, 1, 0, 3, 1, 5, 0, 1, 9, 5, 0, 77, 3, 1, 0, 3, 21, 1975, 0, 5, 3, 1, 0, 1, 3, 7, 0, 9, 5, 5, 0, 1, 135, 3

Offset: 1

Antti Karttunen, Nov 26 2021

Because the ratio A003961(n)/A000203(n) is multiplicative, so is also its Möbius transform. This sequence gives the numerator of that ratio when presented in its lowest terms, while A349628 gives the denominators. See the examples.

			The ratio a(n)/A349628(n) for n = 1..15: 1/1, 0/1, 1/4, 2/7, 1/6, 0/1, 3/8, 18/35, 35/52, 0/1, 1/12, 1/14, 3/14, 0/1, 1/24.

Cf. A000203, A003961.
Cf. A349628 (denominators).
Cf. also A342671, A349161, A349162, A349633.

f[p_, e_] := NextPrime[p]^e*(p - 1)/(p^(e + 1) - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Numerator @ DivisorSum[n, s[#] * MoebiusMu[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A349627(n) = numerator(sumdiv(n,d,moebius(n/d)*(A003961(d)/sigma(d))));

cp's OEIS Frontend

A369260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A349162(i) = A349162(j), for all i, j >= 1.

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A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

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A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

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A349161 a(n) = A003961(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

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A349164 a(n) = A064989(A003961(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

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A351546 a(n) is the largest unitary divisor of sigma(n) coprime with A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A349163 a(n) = A064989(gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

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A348992 a(n) = A000265(sigma(n)) / gcd(sigma(n), A003961(n)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

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