A355582 -id:A355582 - cp's OEIS frontend

A379006 Ordinal transform of A355582, where A355582 is the largest 5-smooth divisor of n.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 4, 2, 1, 1, 5, 1, 6, 1, 2, 3, 7, 1, 1, 4, 1, 2, 8, 1, 9, 1, 3, 5, 2, 1, 10, 6, 4, 1, 11, 2, 12, 3, 1, 7, 13, 1, 14, 1, 5, 4, 15, 1, 3, 2, 6, 8, 16, 1, 17, 9, 2, 1, 4, 3, 18, 5, 7, 2, 19, 1, 20, 10, 1, 6, 21, 4, 22, 1, 1, 11, 23, 2, 5, 12, 8, 3, 24, 1, 25, 7, 9, 13, 6, 1, 26, 14, 3, 1, 27, 5, 28, 4, 2

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A355582.
Cf. A379005 (ordinal transform).
Cf. also A003602, A126760.

up_to = 20000;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
is_A051037(n) = (n<7||vecmax(factor(n, 6)[, 1])<7); \\ From A051037
A355582(n) = fordiv(n,d,if(is_A051037(n/d),return(n/d)));
v379006 = ordinal_transform(vector(up_to, n, A355582(n)));
A379006(n) = v379006[n];

A132741 Largest divisor of n having the form 2^i*5^j.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 1, 2, 5, 16, 1, 2, 1, 20, 1, 2, 1, 8, 25, 2, 1, 4, 1, 10, 1, 32, 1, 2, 5, 4, 1, 2, 1, 40, 1, 2, 1, 4, 5, 2, 1, 16, 1, 50, 1, 4, 1, 2, 5, 8, 1, 2, 1, 20, 1, 2, 1, 64, 5, 2, 1, 4, 1, 10, 1, 8, 1, 2, 25, 4, 1, 2, 1, 80, 1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 1, 2, 5, 32, 1, 2

Offset: 1

Reinhard Zumkeller, Aug 27 2007

The range of this sequence, { a(n); n>=0 }, is equal to A003592. - M. F. Hasler, Dec 28 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003592, A006519, A007732, A007814, A051626, A060904, A112765, A132740.
Cf. A001620, A082633.
Cf. A379003 (ordinal transform), A379004 (rgs-transform).
Cf. also A355582.

a132741 = f 2 1 where
   f p y x | r == 0    = f p (y * p) x'
           | otherwise = if p == 2 then f 5 y x else y
           where (x', r) = divMod x p
-- Reinhard Zumkeller, Nov 19 2015

A132741 := proc(n) local f,a; f := ifactors(n)[2] ; a := 1; for f in ifactors(n)[2] do if op(1,f) =2 then a := a*2^op(2,f) ; elif op(1,f) =5 then a := a*5^op(2,f) ; end if; end do;a; end proc: # R. J. Mathar, Sep 06 2011

a[n_] := SelectFirst[Reverse[Divisors[n]], MatchQ[FactorInteger[#], {{1, 1}} | {{2, }} | {{5, }} | {{2, }, {5, }}]&]; Array[a, 100] (* Jean-François Alcover, Feb 02 2018 *)
a[n_] := Times @@ ({2, 5}^IntegerExponent[n, {2, 5}]); Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

A132741(n)=5^valuation(n,5)<M. F. Hasler, Dec 28 2015

a(n) = n / A132740(n).
a(A003592(n)) = A003592(n).
A051626(a(n)) = 0.
A007732(a(n)) = 1.
From R. J. Mathar, Sep 06 2011: (Start)
Multiplicative with a(2^e)=2^e, a(5^e)=5^e and a(p^e)=1 for p=3 or p>=7.
Dirichlet g.f. zeta(s)*(2^s-1)*(5^s-1)/((2^s-2)*(5^s-5)). (End)
a(n) = A006519(n)*A060904(n) = 2^A007814(n)*5^A112765(n). - M. F. Hasler, Dec 28 2015
Sum_{k=1..n} a(k) ~ n*(12*log(n)^2 + (24*gamma + 36*log(2) - 24)*log(n) + 24 - 24*gamma - 36*log(2) + 36*gamma*log(2) + 2*log(2)^2 - 18*log(5) + 18*gamma*log(5) + 27*log(2)*log(5) + 2*log(5)^2 + 18*log(5)*log(n) - 24*gamma_1)/(60*log(2)*log(5)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Jan 26 2023

A355583 a(n) is the number of the 5-smooth divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 1, 4, 3, 4, 1, 6, 1, 2, 4, 5, 1, 6, 1, 6, 2, 2, 1, 8, 3, 2, 4, 3, 1, 8, 1, 6, 2, 2, 2, 9, 1, 2, 2, 8, 1, 4, 1, 3, 6, 2, 1, 10, 1, 6, 2, 3, 1, 8, 2, 4, 2, 2, 1, 12, 1, 2, 3, 7, 2, 4, 1, 3, 2, 4, 1, 12, 1, 2, 6, 3, 1, 4, 1, 10, 5, 2, 1, 6, 2, 2

Offset: 1

Amiram Eldar, Jul 08 2022

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

Cf. A000005, A007814, A007949, A051037, A072078, A112765, A355582, A355584.

a[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); Array[a, 100]

a(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);

from sympy import multiplicity as v
def a(n): return (v(2, n)+1)*(v(3, n)+1)*(v(5, n)+1)
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Jul 08 2022

Multiplicative with a(p^e) = e+1 if p <= 5 and 1 otherwise.
a(n) = (A007814(n) + 1)*(A007949(n) + 1)*(A112765(n) + 1).
a(n) = A000005(A355582(n)).
a(n) <= A000005(n), with equality if and only if n is in A051037.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15/4.
Dirichlet g.f.: zeta(s)/((1-1/2^s)*(1-1/3^s)*(1-1/5^s)). - Amiram Eldar, Dec 25 2022

A379005 Lexicographically earliest infinite sequence such that a(i) = a(j) => v_2(i) = v_2(j), v_3(i) = v_3(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814), v_3 (A007949) and v_5 (A112765) give the 2-, 3- and 5-adic valuations of n respectively.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 7, 8, 9, 1, 10, 1, 2, 11, 12, 1, 13, 1, 14, 3, 2, 1, 15, 16, 2, 17, 4, 1, 18, 1, 19, 3, 2, 5, 20, 1, 2, 3, 21, 1, 6, 1, 4, 22, 2, 1, 23, 1, 24, 3, 4, 1, 25, 5, 7, 3, 2, 1, 26, 1, 2, 8, 27, 5, 6, 1, 4, 3, 9, 1, 28, 1, 2, 29, 4, 1, 6, 1, 30, 31, 2, 1, 10, 5, 2, 3, 7, 1, 32, 1, 4, 3, 2, 5, 33, 1, 2, 8, 34, 1, 6, 1, 7, 11

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

Restricted growth sequence transform of A355582.
For all i, j:
  A379001(i) = A379001(j) => a(i) = a(j),
  a(i) = a(j) => A322026(i) = A322026(j),
  a(i) = a(j) => A379004(i) = A379004(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A007814, A007949, A112765, A355582, A379006 (ordinal transform).

Cf. A322026, A379001, A379004.

up_to = 100000;
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; };
v379005 = rgs_transform(vector(up_to, n, [valuation(n,2), valuation(n,3), valuation(n,5)]));
A379005(n) = v379005[n];

A343431 Part of n composed of prime factors of the form 6k-1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 11, 1, 1, 1, 5, 1, 17, 1, 1, 5, 1, 11, 23, 1, 25, 1, 1, 1, 29, 5, 1, 1, 11, 17, 5, 1, 1, 1, 1, 5, 41, 1, 1, 11, 5, 23, 47, 1, 1, 25, 17, 1, 53, 1, 55, 1, 1, 29, 59, 5, 1, 1, 1, 1, 5, 11, 1, 17, 23, 5, 71, 1, 1, 1, 25, 1, 11, 1, 1, 5, 1, 41, 83, 1, 85, 1, 29, 11, 89, 5

Offset: 1

Peter Munn, Apr 15 2021

Completely multiplicative with a(p) = p if p is of the form 6k-1 and a(p) = 1 otherwise.

Largest term of A259548 that divides n.

Equivalent sequence for distinct prime factors: A170825.
Equivalent sequences for prime factors of other forms: A000265 (2k+1), A343430 (3k-1), A170818 (4k+1), A097706 (4k-1), A248909 (6k+1), A065330 (6k+/-1), A065331 (<= 3), A355582 (<= 5).
Range of terms: A259548.

f[p_, e_] := If[Mod[p, 6] == 5, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* after Amiram Eldar at A248909 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 1] + 1) % 6, f[i, 1] = 1); ); factorback(f); } \\ after Michel Marcus at A248909

from math import prod
from sympy import factorint
def A343431(n): return prod(p**e for p, e in factorint(n).items() if not (p+1)%6) # Chai Wah Wu, Dec 26 2022

a(n) = n / A065331(n) / A248909(n) = A065330(n) / A248909(n).

A355584 a(n) is the sum of the 5-smooth divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 1, 15, 13, 18, 1, 28, 1, 3, 24, 31, 1, 39, 1, 42, 4, 3, 1, 60, 31, 3, 40, 7, 1, 72, 1, 63, 4, 3, 6, 91, 1, 3, 4, 90, 1, 12, 1, 7, 78, 3, 1, 124, 1, 93, 4, 7, 1, 120, 6, 15, 4, 3, 1, 168, 1, 3, 13, 127, 6, 12, 1, 7, 4, 18, 1, 195, 1, 3, 124, 7

Offset: 1

Amiram Eldar, Jul 08 2022

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

Sum of the p-smooth divisors of n: A038712 (2), A072079 (3), this sequence (5).

Cf. A000203, A007814, A007949, A051037, A112765, A355582, A355583.

a[n_] := (Times @@ ({2, 3, 5}^(IntegerExponent[n, {2, 3, 5}] + 1) - 1))/8; Array[a, 100]

a(n) = (2^(valuation(n, 2) + 1) - 1) * (3^(valuation(n, 3) + 1) - 1) * (5^(valuation(n, 5) + 1) - 1) / 8;

from sympy import multiplicity as v
def a(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1) * (5**(v(5, n)+1)-1) // 8
print([a(n) for n in range(1, 77)]) # Michael S. Branicky, Jul 08 2022

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 5, and 1 otherwise.
a(n) = (2^(A007814(n)+1)-1)*(3^(A007949(n)+1)-1)*(5^(A112765(n)+1)-1)/8.
a(n) = A000203(A355582(n)).
a(n) <= A000203(n), with equality if and only if n is in A051037.
Dirichlet g.f.: zeta(s)*(2^s/(2^s-2))*(3^s/(3^s-3))*(5^s/(5^s-5)). - Amiram Eldar, Dec 25 2022

A165725 Largest divisor of n coprime to 30. I.e., a(n) = max { k | gcd(n, k) = k and gcd(k, 30) = 1 }.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 13, 7, 1, 1, 17, 1, 19, 1, 7, 11, 23, 1, 1, 13, 1, 7, 29, 1, 31, 1, 11, 17, 7, 1, 37, 19, 13, 1, 41, 7, 43, 11, 1, 23, 47, 1, 49, 1, 17, 13, 53, 1, 11, 7, 19, 29, 59, 1, 61, 31, 7, 1, 13, 11, 67, 17, 23, 7, 71, 1, 73, 37, 1, 19, 77, 13, 79, 1, 1, 41, 83, 7

Offset: 1

Barry Wells (wells.barry(AT)gmail.com), Sep 25 2009

This is the sequence of the largest divisor of n which is coprime to 30. The product of the first 3 prime numbers is 2*3*5=30. This sequence gives the largest factor of n which does not include 2, 3 or 5 in its prime factorization.

			The largest factor of 1, 2, 3, 4, 5 and 6 not including the primes 2, 3 and 5 is 1. 7 is prime and therefore its sequence value is 7. For p > 5, p prime, gives a(p) = p. As 14 = 2*7, a(14)= 7. As 98 = 2*7*7, a(98)= 49.

Barry Wells, Table of n, a(n) for n = 1..1024 [a(392) and a(704) corrected by Sean A. Irvine]

A051037 gives the smooth five numbers, numbers whose prime divisor only include 2, 3 and 5. A132740 gives the largest divisor of n coprime to 10. A065330 gives a(n) = max { k | gcd(n, k) = k and gcd(k, 6) = 1 }.
Largest divisor of n coprime to a prime factor of 30: A000265 (2), A038502 (3), A132739 (5).
Cf. A355582.

a[n_] := n / Times @@ ({2, 3, 5}^IntegerExponent[n, {2, 3, 5}]); Array[a, 100] (* Amiram Eldar, Jul 10 2022 *)

a(n)=n>>valuation(n,2)/3^valuation(n,3)/5^valuation(n,5) \\ Charles R Greathouse IV, Jul 16 2017

From Amiram Eldar, Jul 10 2022: (Start)
Multiplicative with a(p^e) = p^e if p >= 7 and 1 otherwise.
a(n) = n/A355582(n). (End)
Sum_{k=1..n} a(k) ~ (5/24) * n^2. - Amiram Eldar, Nov 28 2022
Dirichlet g.f.: zeta(s-1)*(2^s-2)*(3^s-3)*(5^s-5)/((2^s-1)*(3^s-1)*(5^s-1)). - Amiram Eldar, Jan 04 2023

A178146 a(n) is the number of distinct prime factors <= 5 of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2

Offset: 1

Vladimir Shevelev, May 21 2010

The sequence is periodic with period {0 1 1 1 1 2 0 1 1 2 0 2 0 1 2 1 0 2 0 2 1 1 0 2 1 1 1 1 0 3} of length 30. There are 26 coincidences on the interval [1,30] with A156542.

Vladimir Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1,0,1,2,2,1).

Cf. A000005, A001221, A156542, A171182, A178143, A178144, A178142.
Cf. A059841, A079978, A079998, A355582, A356006.
Number of distinct prime factors <= p: A171182 (p=3), this sequence (p=5), A210679 (p=7).

Rest@ CoefficientList[Series[-x^2*(3*x^6 + 6*x^5 + 7*x^4 + 6*x^3 + 5*x^2 + 3*x + 1)/((x - 1)*(x + 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* G. C. Greubel, May 16 2017 *)
LinearRecurrence[{-2,-2,-1,0,1,2,2,1},{0,1,1,1,1,2,0,1},120] (* Harvey P. Dale, Sep 29 2021 *)
a[n_] := PrimeNu[GCD[n, 30]]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

my(x='x+O('x^50)); concat([0], Vec(-x^2*(3*x^6+6*x^5+7*x^4+6*x^3+5*x^2+3*x+1)/((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)))) \\ G. C. Greubel, May 16 2017

a(n) = omega(gcd(n, 30)); \\ Amiram Eldar, Sep 16 2023

a(n) = a(n-2) + a(n-3) - a(n-7) - a(n-8) + a(n-10), a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 2, a(7) = 0, a(8) = 1, a(9) = 1, a(10) = 2.
G.f.: -x^2*(3*x^6+6*x^5+7*x^4+6*x^3+5*x^2+3*x+1) / ((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)). - Colin Barker, Mar 13 2013
From Amiram Eldar, Sep 16 2023: (Start)
Additive with a(p^e) = 1 if p <= 5, and 0 otherwise.
a(n) = A059841(n) + A079978(n) + A079998(n).
a(n) = A001221(gcd(n, 30)).
a(n) = A001221(A355582(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 31/30. (End)

Name edited by Amiram Eldar, Sep 16 2023

A356006 The number of prime divisors of n that are not greater than 5, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 0, 3, 2, 2, 0, 3, 0, 1, 2, 4, 0, 3, 0, 3, 1, 1, 0, 4, 2, 1, 3, 2, 0, 3, 0, 5, 1, 1, 1, 4, 0, 1, 1, 4, 0, 2, 0, 2, 3, 1, 0, 5, 0, 3, 1, 2, 0, 4, 1, 3, 1, 1, 0, 4, 0, 1, 2, 6, 1, 2, 0, 2, 1, 2, 0, 5, 0, 1, 3, 2, 0, 2, 0, 5, 4, 1, 0, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Jul 23 2022

Equivalently, the number of prime divisors, counted with multiplicity, of the largest 5-smooth divisor of n.

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

Cf. A007814, A007949, A112765.

Cf. A001222, A112759, A169611, A355582, A355583, A355584.

a[n_] := Plus @@ IntegerExponent[n, {2, 3, 5}]; Array[a, 100]

a(n) = valuation(n, 2) + valuation(n, 3) + valuation(n, 5);

from sympy import multiplicity as v
def a(n): return v(2, n) + v(3, n) + v(5, n)
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jul 25 2022

Totally additive with a(p) = 1 if p <= 5, and 0 otherwise.
a(n) = A007814(n) + A007949(n) + A112765(n).
a(n) = A001222(A355582(n)).
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 7/4.

A375536 The maximum exponent in the prime factorization of the largest 5-smooth divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 0, 3, 2, 1, 0, 2, 0, 1, 1, 4, 0, 2, 0, 2, 1, 1, 0, 3, 2, 1, 3, 2, 0, 1, 0, 5, 1, 1, 1, 2, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 2, 0, 1, 2, 6, 1, 1, 0, 2, 1, 1, 0, 3, 0, 1, 2, 2, 0, 1, 0, 4, 4, 1, 0, 2, 1, 1, 1, 3, 0, 2, 0, 2, 1, 1, 1, 5, 0, 1, 2, 2, 0, 1, 0, 3, 1

Offset: 1

Amiram Eldar, Aug 19 2024

Cf. A007775, A051903, A244417 (3-smooth analog), A355582, A375537, A375538, A375539.

Cf. A007814, A007949, A112765.

a[n_] := Max[IntegerExponent[n, {2, 3, 5}]]; Array[a, 100]

a(n) = max(max(valuation(n, 2), valuation(n, 3)), valuation(n, 5));

a(n) = A051903(A355582(n)).
a(n) = max(A007814(n), A007949(n), A112765(n)).
a(n) = 0 if and only if n is a 7-rough number (A007775).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A375538(3)/A375539(3) = 51227/36540 = 1.401943076...

cp's OEIS Frontend

A379006 Ordinal transform of A355582, where A355582 is the largest 5-smooth divisor of n.

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A132741 Largest divisor of n having the form 2^i*5^j.

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A355583 a(n) is the number of the 5-smooth divisors of n.

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A379005 Lexicographically earliest infinite sequence such that a(i) = a(j) => v_2(i) = v_2(j), v_3(i) = v_3(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814), v_3 (A007949) and v_5 (A112765) give the 2-, 3- and 5-adic valuations of n respectively.

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A343431 Part of n composed of prime factors of the form 6k-1.

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A355584 a(n) is the sum of the 5-smooth divisors of n.

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A165725 Largest divisor of n coprime to 30. I.e., a(n) = max { k | gcd(n, k) = k and gcd(k, 30) = 1 }.

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A178146 a(n) is the number of distinct prime factors <= 5 of n.

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