A079695 -id:A079695 - cp's OEIS frontend

A005277 Nontotients: even numbers k such that phi(m) = k has no solution.

14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, 158, 170, 174, 182, 186, 188, 194, 202, 206, 214, 218, 230, 234, 236, 242, 244, 246, 248, 254, 258, 266, 274, 278, 284, 286, 290, 298, 302, 304, 308, 314, 318

Offset: 1

N. J. A. Sloane

nonn

If p is prime then the following two statements are true. I. 2p is in the sequence iff 2p+1 is composite (p is not a Sophie Germain prime). II. 4p is in the sequence iff 2p+1 and 4p+1 are composite. - Farideh Firoozbakht, Dec 30 2005
Another subset of nontotients consists of the numbers j^2 + 1 such that j^2 + 2 is composite. These numbers j are given in A106571. Similarly, let b be 3 or a number such that b == 1 (mod 4). For any j > 0 such that b^j + 2 is composite, b^j + 1 is a nontotient. - T. D. Noe, Sep 13 2007
The Firoozbakht comment can be generalized: Observe that if k is a nontotient and 2k+1 is composite, then 2k is also a nontotient. See A057192 and A076336 for a connection to Sierpiński numbers. This shows that 271129*2^j is a nontotient for all j > 0. - T. D. Noe, Sep 13 2007

			There are no values of m such that phi(m)=14, so 14 is a term of the sequence.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 44 at p. 91.
R. K. Guy, Unsolved Problems in Number Theory, B36.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 91.

Robert G. Wilson v, Table of n, a(n) for n = 1..29750 (terms 1..10000 from T. D. Noe).
Lambert A'Campo, Every 7-Dimensional Abelian Variety over the p-adic Numbers has a Reducible L-adic Galois Representation, arXiv:2006.06737 [math.NT], 2020.
Matteo Caorsi and Sergio Cecotti, Geometric classification of 4d N=2 SCFTs, arXiv:1801.04542 [hep-th], 2018.
K. Ford, S. Konyagin, and C. Pomerance, Residue classes free of values of Euler's function, arXiv:2005.01078 [math.NT] (1999).
L. Havelock, A Few Observations on Totient and Cototient Valence.
Eric Weisstein's World of Mathematics, Nontotient.
Wikipedia, Nontotient.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

See A007617 for all numbers k (odd or even) such that phi(m) = k has no solution.
All even numbers not in A002202. Cf. A000010.
Cf. A005384, A006093, A014197, A057192, A076336, A079695, A106571.

a005277 n = a005277_list !! (n-1)
a005277_list = filter even a007617_list
-- Reinhard Zumkeller, Nov 22 2015

[n: n in [2..400 by 2] | #EulerPhiInverse(n) eq 0]; // Marius A. Burtea, Sep 08 2019

A005277 := n -> if type(n,even) and invphi(n)=[] then n fi: seq(A005277(i),i=1..318); # Peter Luschny, Jun 26 2011

searchMax = 320; phiAnsYldList = Table[0, {searchMax}]; Do[phiAns = EulerPhi[m]; If[phiAns <= searchMax, phiAnsYldList[[phiAns]]++ ], {m, 1, searchMax^2}]; Select[Range[searchMax], EvenQ[ # ] && (phiAnsYldList[[ # ]] == 0) &] (* Alonso del Arte, Sep 07 2004 *)
totientQ[m_] := Select[ Range[m +1, 2m*Product[(1 - 1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; (* after Jean-François Alcover, May 23 2011 in A002202 *) Select[2 Range@160, ! totientQ@# &] (* Robert G. Wilson v, Mar 20 2023 *)

is(n)=n%2==0 && !istotient(n) \\ Charles R Greathouse IV, Mar 04 2017

a(n) = 2*A079695(n). - R. J. Mathar, Sep 29 2021

{k: k even and A014197(k) = 0}. - R. J. Mathar, Sep 29 2021

More terms from Jud McCranie, Oct 13 2000

A093819 Algebraic degree of sin(2*Pi/n).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 2, 6, 4, 10, 1, 12, 6, 8, 4, 16, 6, 18, 2, 12, 10, 22, 4, 20, 12, 18, 3, 28, 8, 30, 8, 20, 16, 24, 3, 36, 18, 24, 8, 40, 12, 42, 5, 24, 22, 46, 8, 42, 20, 32, 6, 52, 18, 40, 12, 36, 28, 58, 4, 60, 30, 36, 16, 48, 20, 66, 8, 44, 24, 70, 12, 72, 36, 40, 9, 60, 24

Offset: 1

Eric W. Weisstein, Apr 16 2004

nonn

The degree formula given in the I. Niven reference on p. 37-8 (see below) appears as part of theorem 3.9 attributed to D. H. Lehmer. However, this part, concerning sin(2*Pi/n), differs from Lehmer's result, which in fact is incorrect. - Wolfdieter Lang, Jan 09 2011
This is also the algebraic degree of the area of a regular n-gon inscribed in the unit circle. - Jack W Grahl, Jan 10 2011
Every degree appears in this sequence except for the half-nontotients, A079695. - T. D. Noe, Jan 12 2011
See A181872/A181873 for the monic rational minimal polynomial of sin(2*Pi/n), and A181871 for the non-monic integer version. In A231188 the (monic and integer) minimal polynomials for 2*sin(2*Pi/n) are given. - Wolfdieter Lang, Nov 30 2013

I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

T. D. Noe, Table of n, a(n) for n = 1..1000
Sameen Ahmed Khan, Trigonometric Ratios Using Algebraic Methods, Mathematics and Statistics (2021) Vol. 9, No. 6, 899-907.
Eric Weisstein's World of Mathematics, Trigonometry Angles

Cf. A055035, A023022 (alg. degree of cos(2*Pi/n)), A183919.

Cf. A181872/A181873, A181871, A231188. - Wolfdieter Lang, Nov 30 2013

a[4]=1; a[n_] := Module[{g=GCD[n, 8], e=EulerPhi[n]}, If[g<4, e, If[g==4, e/4, e/2]]]; Array[a, 1000]
f[n_] := Exponent[ MinimalPolynomial[ Sin[ 2Pi/n]][x], x]; Array[f, 75] (* Robert G. Wilson v, Jul 28 2014 *)

a(4)=1, a(n)=phi(n) if gcd(n,8)<4; a(n)=phi(n)/4 if gcd(n,8)=4, and a(n)=phi(n)/2 if gcd(n,8)>4. Here phi(n)=A000010(n) (Euler totient). See the I. Niven reference, Theorem 3.9, p. 37-8. - Wolfdieter Lang, Jan 09 2011

a(n) = delta(c(n)/2) if c(n) = A178182(n) is even, and delta(c(n)) if c(n) is odd, with delta(n) = A055034(n), the degree of the algebraic number 2*cos(Pi/n). - Wolfdieter Lang, Nov 30 2013

A032446 Number of solutions to phi(k) = 2n.

Original entry on oeis.org

3, 4, 4, 5, 2, 6, 0, 6, 4, 5, 2, 10, 0, 2, 2, 7, 0, 8, 0, 9, 4, 3, 2, 11, 0, 2, 2, 3, 2, 9, 0, 8, 2, 0, 2, 17, 0, 0, 2, 10, 2, 6, 0, 6, 0, 3, 0, 17, 0, 4, 2, 3, 2, 9, 2, 6, 0, 3, 0, 17, 0, 0, 2, 9, 2, 7, 0, 2, 2, 3, 0, 21, 0, 2, 2, 0, 0, 7, 0, 12, 4, 3, 2, 12, 0, 2, 0, 8, 2, 10

Offset: 1

Ursula Gagelmann (gagelmann(AT)altavista.net)

By Carmichael's conjecture, a(n) <> 1 for any n. See A074987. - Thomas Ordowski, Sep 13 2017

a(n) = 0 iff n is a term of A079695. - Bernard Schott, Oct 02 2021

			If n = 8 then phi(x) = 2*8 = 16 is satisfied for only a(8) = 6 values of x, viz. 17, 32, 34, 40, 48, 60.

Albert H. Beiler, Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, Second Edition, Dover Publications, Inc., NY, 1966, page 90.

T. D. Noe, Table of n, a(n) for n = 1..5000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Matteo Caorsi and Sergio Cecotti, Geometric classification of 4d N=2 SCFTs, arXiv:1801.04542 [hep-th], 2018.
Carl Pomerance, Popular values of Euler's function, Mathematica 27 (1980), 84-89.

Bisection of A014197.
Cf. A000010, A005277, A074987, A079695, A085758.
Cf. A006511 (largest k for which A000010(k) = A002202(n)), A057635.

[#EulerPhiInverse( 2*n):n in [1..100]]; // Marius A. Burtea, Sep 08 2019

with(numtheory); [ seq(nops(invphi(2*n)), n=1..90) ];

t = Table[0, {100} ]; Do[a = EulerPhi[n]; If[a < 202, t[[a/2]]++ ], {n, 3, 10^5} ]; t

a(n) = invphiNum(2*n); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

Extended by Robin Trew (trew(AT)hcs.harvard.edu).

A002180 Values taken by the half-totient function phi(m)/2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 35, 36, 39, 40, 41, 42, 44, 46, 48, 50, 51, 52, 53, 54, 55, 56, 58, 60, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 78, 80, 81, 82, 83, 84, 86, 88, 89, 90, 92

Offset: 2

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 2..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. D. Carmichael, A table of the values of m corresponding to given values of phi(m), Amer. J. Math., 30 (1908),394-400. [Annotated scanned copy]
K. W. Wegner, Values of phi(x) = n for n from 2 through 1978, mimeographed manuscript, no date [Annotated scanned copy]

Cf. A002202, A079695 (complementary sequence).

a002180 = flip div 2 . a002202  -- Reinhard Zumkeller, Nov 22 2015

with(numtheory); t1 := [seq(nops(invphi(n)), n=1..300)]; t2 := []: for n from 2 to 300 do if t1[n] <> 0 then t2 := [op(t2), n/2]; fi; od: t2;

phiQ[m_] := Select[Range[m+1, 2 m*Product[(1-1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; Select[Range[2, 200], phiQ]/2 (* Jean-François Alcover, Jun 13 2012, after Maxim Rytin *)

list(lim)=my(v=List()); for(n=1,lim, if(istotient(2*n), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Feb 08 2017

a(n) = A002202(n)/2 for n > 1.

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001

A051445 Smallest k such that phi(k) = 2n, or 0 if there is no such k.

Original entry on oeis.org

3, 5, 7, 15, 11, 13, 0, 17, 19, 25, 23, 35, 0, 29, 31, 51, 0, 37, 0, 41, 43, 69, 47, 65, 0, 53, 81, 87, 59, 61, 0, 85, 67, 0, 71, 73, 0, 0, 79, 123, 83, 129, 0, 89, 0, 141, 0, 97, 0, 101, 103, 159, 107, 109, 121, 113, 0, 177, 0, 143, 0, 0, 127, 255, 131, 161, 0, 137

Offset: 1

Jud McCranie

nonn

The zero values are easy to prove because of the bounds on the phi function.

			a(4) = 15 as phi(15) = 2*4 and no k < 15 has phi(k) = 2*4.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A002181, A072075, A079695. For records see A132012, A132115.

a(n)=n+=n;for(k=n+1, solve(x=n,if(n<20,99,5*n*log(log(n))), x/(exp(Euler)*log(log(x))+3/log(log(x)))-n), if(eulerphi(k)==n,return(k))); 0 \\ Charles R Greathouse IV, Dec 19 2011

a(10^n/2) = A072075(n). - R. J. Mathar, Dec 12 2024

a(A079695(n)) = 0. - David A. Corneth, Dec 12 2024

A306513 The number of unordered pairs of coprime integers q and r such that phi(q) + phi(r) = 2n.

Original entry on oeis.org

1, 1, 5, 7, 12, 10, 19, 18, 20, 21, 35, 32, 39, 42, 38, 37, 48, 46, 45, 58, 64, 63, 69, 73, 58, 93, 71, 70, 81, 92, 72, 113, 96, 94, 90, 100, 79, 158, 120, 95, 131, 153, 84, 147, 129, 132, 126, 150, 92, 179, 157, 150, 149, 187, 92, 224, 177, 166, 173, 207, 124

Offset: 1

Robert G. Wilson v, Feb 20 2019

nonn

Paul Erdős and Leo Moser conjectured that, for any even number 2n, there exist integers q and r such that phi(q) + phi(r) = 2n with gcd(q, r) = 1. Adding to this conjecture the requirement that q and r be prime yields the Goldbach Conjecture. The replacement of the requirement that q and r be prime with the relaxed requirement that they be coprime was done in an effort to solve the Goldbach Conjecture.

			a(1) = 1 with {q, r} = {1,2};
a(2) = 1 with {q, r} = {3,4};
a(3) = 5 because phi(q) + phi(r) = 6 for the pairs {q, r} = {3,5}, {3,8}, {3,10}, {4,5} & {5,6}; etc.

George E. Andrews, Number Theory, Chapter 6, Arithmetic Functions, Section 6-1, Combinatorial Study of Phi(n), pp. 75-82, Dover Publishing, NY, 1971.

Robert G. Wilson v, Table of n, a(n) for n = 1..1150
Eric W. Weisstein's World of Mathematics, Goldbach Conjecture.
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A000010, A005277, A079695, A002375.

f[n_] := Block[{c = 0, q = 1}, While[q < 12n, epq = EulerPhi[q]; r = 12n + 125; While[r > q, If[ GCD[q, r] == 1 && epq + EulerPhi[r] == 2 n, c++]; r--]; q++]; c]; Array[f, 61]

A207333 Allowed values of degrees of minimal polynomials of 2*cos(Pi/N).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 23, 24, 26, 28, 29, 30, 32, 33, 35, 36, 39, 27, 40, 41, 42, 44, 46, 48, 50, 51, 52, 53, 54, 56, 58, 55, 60, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 78, 81, 80, 82, 83, 84, 86, 88, 89, 90, 92, 95

Offset: 1

Wolfdieter Lang, Feb 19 2012

The coefficients of the minimal polynomials C(N,x) of the algebraic number 2*cos(Pi/N) are given in A187360, where N is n.
The degree of C(N,x) is delta(N) = 1 if N=1 and delta(N) = phi(2*N)/2 if N > 1, with Euler's totient function phi(n) = A000010(n).
The forbidden degree values are shown in the complement (relative to the positive integers) A079695.
The array of the values N (the indices) for which the degree delta(N) = a(n), n >= 1, is given in A207334.

			a(8) = 9 because there is at least one polynomial C(N,x) with degree delta(N)=9. In fact the only N values are 19 and 27.
7 is no member of this sequence (it belongs to the complement A079695).

Cf. A079695 (complement), A207334 (array of indices of C polynomials with degree a(n)).

a(n) gives the allowed degree values, called delta, of the minimal polynomials C ordered increasingly, For C and delta see the comment section.

A282160 Least k > 1 such that k*n is not a totient number.

Original entry on oeis.org

3, 7, 3, 17, 3, 15, 2, 19, 3, 5, 3, 43, 2, 7, 3, 19, 2, 5, 2, 17, 3, 7, 3, 167, 2, 7, 3, 11, 3, 3, 2, 19, 3, 2, 3, 67, 2, 2, 3, 17, 3, 17, 2, 7, 2, 5, 2, 211, 2, 7, 3, 7, 3, 11, 3, 13, 2, 3, 2, 139, 2, 2, 3, 31, 3, 9, 2, 5, 3, 5, 2, 109, 2, 5, 3, 2, 2, 3, 2, 85, 3, 3, 3, 61

Offset: 1

Altug Alkan, Feb 07 2017

nonn

First occurrence of odd k or zero if impossible: 0, 1, 10, 2, 66, 28, 56, 6, 4, 8, 5244, 460, 272, 0, 232, 64, 7788, 4180, 300, 348, 328, 12, etc. - Robert G. Wilson v, Feb 09 2017

			a(14) = 7 because 7 * 14 = 98 is not a totient number and 7 is the least number that is greater than 1 with this property.

Altug Alkan, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A007617, A067005, A071615, A079695.

TotientQ[m_] := Select[ Range[m +1, 2m*Product[(1 - 1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; (* after Jean-François Alcover, May 23 2011 in A002202 *) f[n_] := Block[{k = 2}, While[ TotientQ[k*n], k++]; k]; Array[f, 84] (* Robert G. Wilson v, Feb 09 2017 *)

a(n) = my(k = 2); while (istotient(k*n), k++); k;

a(A079695(n)) = 2. - Michel Marcus, Feb 08 2017

A328414 Numbers k such that (Z/mZ)* = C_2 X C_(2k) has no solutions m, where (Z/mZ)* is the multiplicative group of integers modulo m.

Original entry on oeis.org

7, 12, 13, 17, 19, 25, 28, 31, 34, 37, 38, 43, 47, 49, 52, 57, 59, 61, 62, 67, 71, 73, 76, 77, 79, 80, 84, 85, 91, 92, 93, 94, 97, 100, 101, 103, 104, 107, 108, 109, 112, 117, 118, 121, 122, 124, 127, 129, 133, 137, 139, 142, 143, 144, 148, 149, 151, 152, 157, 160, 161, 163, 164

Offset: 1

Jianing Song, Oct 14 2019

nonn

Indices of 0 in A328410, A328411 and A328412.

By definition, if there is no such m that psi(m) = 2k, psi = A002322, then m is a term of this sequence.

			12 is a term: if there exists m such that (Z/mZ)* = C_2 X C_24 = C_2 X C_8 X C_3, then m must have a factor q such that q is an odd prime power and phi(q) = 8 or phi(q) = 24, phi = A000010, which is impossible.
80 is a term: if there exists m such that (Z/mZ)* = C_2 X C_80 = C_2 X C_16 X C_5, then m must have a factor q such that q is an odd prime power and phi(q) = 80 or phi(q) = 16, which is impossible.

Wikipedia, Multiplicative group of integers modulo n

Cf. A328410, A328411, A328412. Complement of A328413.

Cf. also A000010, A002322, A005277, A079695.

isA328414(n) = my(r=4*n, N=floor(exp(Euler)*r*log(log(r^2))+2.5*r/log(log(r^2)))); for(k=r+1, N+1, if(eulerphi(k)==r && lcm(znstar(k)[2])==r/2, return(0)); if(k==N+1, return(1)))
for(n=1, 200, if(isA328414(n), print1(n, ", ")))

A333819 a(n) is the least integer q > 0 such that for some integer r, phi(q) + phi(r) = 2*n; where phi(n) is Euler's totient function (A000010).

Original entry on oeis.org

1, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 7, 3, 3, 3, 3, 5, 3, 5, 3, 5, 3, 5, 3, 3, 3, 3, 3, 3, 3, 5, 3, 5, 3, 5, 7, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 5, 3, 3, 5, 7, 3, 5, 3, 3, 3, 3, 3, 5, 3, 5, 3, 3, 3, 5, 3, 5, 7, 3, 3, 5, 3, 3, 3, 5, 3, 5, 3

Offset: 1

Robert G. Wilson v, Apr 06 2020

nonn

Paul Erdös and Leo Moser conjectured that, for any even numbers 2*n, there exist integers q and r such that phi(q) + phi(r) = 2*n.
The only time phi is odd, it equals 1. Therefore, the only time that phi(q) + phi(r) = 2*n-1 (for n>0) has no solution is when 2*n-2 is a member of A005277 = 2*A079695.
The first occurrence of 2*k-1, or 0 if not possible, is k=1,2,3,...: 1, 2, 8, 39, 0, 124, 204, 208, 2024, 3473, 0, 2983, 2023, ..., .

George E. Andrews, Number Theory, Chapter 6, Arithmetic Functions, 6-1 Combinatorial Study of Phi(n) page 75-82, Dover Publishing, NY, 1971.
Daniel Zwillinger, Editor-in-Chief, CRC Standard Mathematical Tables and Formulae, 31st Edition, 2.4.15 Euler Totient pages 128-130, Chapman & Hall/CRC, Boca Raton, 2003.

Eric W. Weisstein's World of Mathematics, Goldbach's Conjecture.
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A000010, A306513, A333820.

mbr = Union@ Array[ EulerPhi@# &, 500]; a[n_] := Block[{q = 1}, While[ !MemberQ[mbr, 2n - EulerPhi@ q], q++]; q]; Array[a, 105]

cp's OEIS Frontend

A005277 Nontotients: even numbers k such that phi(m) = k has no solution.

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A093819 Algebraic degree of sin(2*Pi/n).

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A032446 Number of solutions to phi(k) = 2n.

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A002180 Values taken by the half-totient function phi(m)/2.

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A051445 Smallest k such that phi(k) = 2n, or 0 if there is no such k.

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A306513 The number of unordered pairs of coprime integers q and r such that phi(q) + phi(r) = 2n.

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