A005385 -id:A005385 - cp's OEIS frontend

A077065 Semiprimes of form prime - 1.

4, 6, 10, 22, 46, 58, 82, 106, 166, 178, 226, 262, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 862, 886, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2026, 2038, 2062, 2098, 2206, 2446, 2458, 2578, 2818, 2878, 2902

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

There are 670 semiprimes of form prime-1 below 10^5.

			A001358(16) = 46 = 2*23 is a term as 46 = A000040(15) - 1 = 47 - 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Intersection of A006093 and A001358.
Intersection of A006093 and A100484.
Cf. A077068, A232342, A064911.
Cf. A005384, A005385.
Cf. A000010, A023900, A194593.

a077065 n = a077065_list !! (n-1)
a077065_list = filter ((== 1) . a010051' . (`div` 2)) a006093_list
-- Reinhard Zumkeller, Nov 22 2013, Oct 27 2012

IsSemiprime:=func; [s: n in [2..500] | IsSemiprime(s) where s is NthPrime(n)-1]; // Vincenzo Librandi, Oct 17 2012

q:= n-> (n::even) and andmap(isprime, [n+1, n/2]):
select(q, [$1..5000])[];  # Alois P. Heinz, Jul 19 2023

Select[Range[6000],Plus@@Last/@FactorInteger[#]==2&&PrimeQ[#+1]&] (* Vladimir Joseph Stephan Orlovsky, May 08 2011 *)
Select[Range[3000],PrimeOmega[#]==2&&PrimeQ[#+1]&] (* Harvey P. Dale, Oct 16 2012 *)
Select[ Prime@ Range@ 430 - 1, PrimeOmega@# == 2 &] (* Robert G. Wilson v, Feb 18 2014 *)

[x-1|x<-primes(10^4),bigomega(x-1)==2] \\ Charles R Greathouse IV, Nov 22 2013

a(n) = A005385(n) - 1 = 2*A005384(n).
A010051(A006093(a(n))/2) = A064911(A006093(a(n))) = 1. - Reinhard Zumkeller, Nov 22 2013
a(n) = A077068(n) - A232342(n). - Reinhard Zumkeller, Dec 16 2013
a(n) = A000010(A194593(n+1)). - Torlach Rush, Aug 23 2018
A000010((a(n)*2)+2) = A023900((a(n)*2)+2). - Torlach Rush, Aug 23 2018

A336467 Fully multiplicative with a(2) = 1 and a(p) = A000265(p+1) for odd primes p, with A000265(k) giving the odd part of k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 7, 1, 3, 1, 9, 1, 5, 3, 1, 3, 3, 1, 9, 7, 1, 1, 15, 3, 1, 1, 3, 9, 3, 1, 19, 5, 7, 3, 21, 1, 11, 3, 3, 3, 3, 1, 1, 9, 9, 7, 27, 1, 9, 1, 5, 15, 15, 3, 31, 1, 1, 1, 21, 3, 17, 9, 3, 3, 9, 1, 37, 19, 9, 5, 3, 7, 5, 3, 1, 21, 21, 1, 27, 11, 15, 3, 45, 3, 7, 3, 1, 3, 15, 1, 49, 1, 3, 9, 51, 9, 13, 7, 3

Offset: 1

Antti Karttunen, Jul 22 2020

For the comment here, we extend the definition of the first kind of Cunningham chain (see Wikipedia-article) so that also isolated primes for which neither (p-1)/2 nor 2p+1 is a prime are considered to be in singular chains, that is, in chains of the length one. If we replace one or more instances of any particular odd prime factor p in n with any odd prime q of the same Cunningham chain, so that m = (q^k)*n / p^(e-k), where e is the exponent of p of n, and k <= e is the number of instances of p replaced with q, then it holds that a(m) = a(n), and by induction, the value stays invariant for any number of such replacements. Note also that A001222, but not necessarily A001221 will stay invariant in such changes.

For example, if some of the odd prime divisors p of n are Sophie Germain primes (in A005384), then replacing any of them with 2p+1 ("safe primes", i.e., the corresponding terms of A005385), gives a new number m, for which a(m) = a(n). And vice versa, the same is true for any safe prime factors > 5 of n (that are in A005385), then replacing any one of them with (p-1)/2 will not affect the result. For example, a(5*11*23*47) = a(11*11*23*23) = a(5^4) = a(11^4) = a(23^4) = 81, as 5, 11, 23 and 47 are in the same Cunningham chain of the first kind.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Wikipedia, Cunningham chain

Cf. A000265, A003959, A206787, A336390, A336392, A336651, A336921, A336922, A351461.

Cf. also A335915, A336466 (similar sequences).

A000265(n) = (n>>valuation(n,2));
A336467(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]+1))^f[k,2])); };

For all n >= 1, A331410(a(n)) = A336921(n).
From Antti Karttunen, Nov 21 2023: (Start)
a(n) = A335915(n) / A336466(n).
a(1) = 1, and for n > 1, a(n) = A000265(A206787(n)) * a(A336651(n)).
(End)

A063638 Primes p such that p-2 is a semiprime.

Original entry on oeis.org

11, 17, 23, 37, 41, 53, 59, 67, 71, 79, 89, 97, 113, 131, 157, 163, 179, 211, 223, 239, 251, 269, 293, 307, 311, 331, 337, 367, 373, 379, 383, 397, 409, 419, 439, 449, 487, 491, 499, 503, 521, 547, 593, 599, 613, 631, 673, 683, 691, 701, 709, 719, 733, 739

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

Primes of form p*q + 2, where p and q are primes.
11 is the only prime of this form where p=q. For prime p>3, 3 divides p^2+2. - T. D. Noe, Mar 01 2006
The asymptotic growth of this sequence is relevant for A204142. We have a(10^k) = (11, 79, 1571, 27961, 407741, 5647823, ...). - M. F. Hasler, Feb 13 2012

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A005385, A001358, A063637, A109611 (Chen primes), A204142, A064911, A040976, A241809.

a063638 n = a063638_list !! (n-1)
a063638_list = map (+ 2) $ filter ((== 1) . a064911) a040976_list
-- Reinhard Zumkeller, Feb 22 2012

Take[Select[ # + 2 & /@ Union[Flatten[Outer[Times, Prime[Range[100]], Prime[Range[100]]]]], PrimeQ], 60]
Select[Prime[Range[200]],PrimeOmega[#-2]==2&] (* Paolo Xausa, Oct 30 2023 *)

n=0; for (m=2, 10^9, p=prime(m); if (bigomega(p - 2) == 2, write("b063638.txt", n++, " ", p); if (n==1000, break))) \\ Harry J. Smith, Aug 26 2009

forprime(p=3,9999, bigomega(p-2)==2 & print1(p","))

p=2; for(n=1,1e4, until(bigomega(-2+p=nextprime(p+1))==2,); write("b063638.txt", n" "p)) \\ M. F. Hasler, Feb 13 2012

list(lim)=my(v=List(), t); forprime(p=3, (lim-2)\3, forprime(q=3, min((lim-2)\p, p), t=p*q+2; if(isprime(t), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Aug 05 2016

a(n) = A241809(n) + 2. - Hugo Pfoertner, Oct 30 2023

A059762 Initial primes of Cunningham chains of first type with length exactly 3. Primes in A059453 that survive as primes just two "2p+1 iterations", forming chains of exactly 3 terms.

Original entry on oeis.org

41, 1031, 1451, 1481, 1511, 1811, 1889, 1901, 1931, 3449, 3491, 3821, 3911, 5081, 5441, 5849, 6101, 6131, 7151, 7349, 7901, 8969, 9221, 10691, 10709, 11171, 11471, 11801, 12101, 12821, 12959, 13229, 14009, 14249, 14321, 14669, 14741, 15161

Offset: 1

Labos Elemer, Feb 20 2001

nonn

Primes p such that {(p-1)/2, p, 2p+1, 4p+3, 8p+7} = {composite, prime, prime, prime, composite}.

			41 is a term because 20 and 325 are composites, and 41, 83, and 167 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris Caldwell's Prime Glossary, Cunningham chains.
Warut Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains.
Eric Weisstein's World of Mathematics, Cunningham Chain.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059453, A059454, A059455, A007700.

ipccQ[n_]:=Module[{c=(n-1)/2},PrimeQ[NestList[2#+1&,c,4]]=={False, True, True, True, False}]; Select[Prime[Range[2000]],ipccQ] (* Harvey P. Dale, Nov 10 2014 *)

Definition corrected by Alexandre Wajnberg, Aug 31 2005

Offset corrected by Amiram Eldar, Jul 15 2024

A051254 Mills primes.

Original entry on oeis.org

2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499

Offset: 1

Simon Plouffe

Mills showed that there is a number A > 1 but not an integer, such that floor( A^(3^n) ) is a prime for all n = 1, 2, 3, ... A is approximately 1.306377883863... (see A051021).
a(1) = 2 and (for n > 1) a(n) is least prime > a(n-1)^3. - Jonathan Vos Post, May 05 2006, corrected by M. F. Hasler, Sep 11 2024
The name refers to the American mathematician William Harold Mills (1921-2007). - Amiram Eldar, Jun 23 2021

			a(3) = 1361 = 11^3 + 30 = a(2)^3 + 30 and there is no smaller k such that a(2)^3 + k is prime. - _Jonathan Vos Post_, May 05 2006

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.13, p. 130.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 137.

Robert G. Wilson v, Table of n, a(n) for n = 1..8
Chris K. Caldwell, Mills' Theorem - a generalization.
Chris K. Caldwell and Yuanyou Chen, Determining Mills' Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.
Steven R. Finch, Mills' Constant. [Broken link]
Steven R. Finch, Mills' Constant. [From the Wayback machine]
Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, Amer. Math. Monthly, Vol. 126, No. 1 (2019), pp. 72-73; ResearchGate link, arXiv preprint, arXiv:2010.15882 [math.NT], 2020.
James Grime and Brady Haran, Awesome Prime Number Constant, Numberphile video, 2013.
Brian Hayes, Pumping the Primes, bit-player, Aug 19 2015.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
William H. Mills, A prime-representing function, Bull. Amer. Math. Soc., Vol. 53, No. 6 (1947), p. 604; Errata, ibid., Vol. 53, No 12 (1947), p. 1196.
Simon Plouffe, The calculation of p(n) and pi(n), arXiv:2002.12137 [math.NT], 2020.
László Tóth, A Variation on Mills-Like Prime-Representing Functions, arXiv:1801.08014 [math.NT], 2018.
Juan L. Varona, A Couple of Transcendental Prime-Representing Constants, arXiv:2012.11750 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Mills' Prime.
Eric Weisstein's World of Mathematics, Prime Formulas.
Eric W. Weisstein, Table of n, a(n) for n = 1..13.

Cf. A001358, A055496, A076656, A006992, A005384, A005385, A118908, A118909, A118910, A118911, A118912, A118913.
Cf. A224845 (integer lengths of Mills primes).
Cf. A108739 (sequence of offsets b_n associated with Mills primes).
Cf. A051021 (decimal expansion of Mills constant).

floor(A^(3^n), n=1..10); # A is Mills's constant: 1.306377883863080690468614492602605712916784585156713644368053759966434.. (A051021).

p = 1; Table[p = NextPrime[p^3], {6}] (* T. D. Noe, Sep 24 2008 *)
NestList[NextPrime[#^3] &, 2, 5] (* Harvey P. Dale, Feb 28 2012 *)

a(n)=if(n==1, 2, nextprime(a(n-1)^3)) \\ Charles R Greathouse IV, Jun 23 2017

apply( {A051254(n, p=2)=while(n--, p=nextprime(p^3));p}, [1..6]) \\ M. F. Hasler, Sep 11 2024

a(1) = 2; a(n) is least prime > a(n-1)^3. - Jonathan Vos Post, May 05 2006

Edited by N. J. A. Sloane, May 05 2007

A058340 Primes p such that phi(x) = p-1 has only 2 solutions, namely x = p and x = 2p.

Original entry on oeis.org

11, 23, 29, 31, 47, 53, 59, 67, 71, 79, 83, 103, 107, 127, 131, 137, 139, 149, 151, 167, 173, 179, 191, 197, 199, 211, 223, 227, 229, 239, 251, 263, 269, 271, 283, 293, 307, 311, 317, 331, 347, 359, 367, 373, 379, 383, 389, 419, 431, 439, 443, 463, 467, 479

Offset: 1

Labos Elemer, Dec 14 2000

nonn

Two solutions, p and 2p, exist for all odd primes p; primes in sequence have no other solutions.
Conjecture: if q > 7 is in A005385, then q is in the sequence. - Thomas Ordowski, Jan 04 2017
Proof of conjecture: q'=(q-1)/2 is an odd prime > 3.  If phi(x)=2q', which has 2-adic order 1 but is not a power of 2, there must be exactly one odd prime r dividing x.  We could also have a factor of 2 (but no higher power, which would contribute more 2's to phi(x)).  If x = r^e or 2r^e, then phi(x) = (r-1) r^(e-1).  For this to be 2q' one possibility is r-1 = 2 and r^(e-1)=q', but then q'=r=3, ruled out by q > 7.  The only other possibility is r-1=2q' and e=1, which makes r=q and x=q or 2q. - Robert Israel, Jan 04 2017
Information from Carl Pomerance: It is known that almost all primes (in the sense of relative asymptotic density) are in the sequence. - Thomas Ordowski, Jan 08 2017

			For p=2, phi(x)=1 has only two solutions, but they are 1 and 2, not 2 and 4, so 2 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A005385, A006093, A058339, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080, A138537.

filter:= n -> isprime(n) and nops(numtheory:-invphi(n-1))=2:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Aug 12 2016

Take[Rest@ Keys@ Select[KeySelect[KeyMap[# + 1 &, PositionIndex@ Array[EulerPhi, 10^4]], PrimeQ], Length@ # == 2 &], 54] (* Michael De Vlieger, Dec 29 2017 *)

a(n) ~ n log . - Charles R Greathouse IV, Nov 18 2022

Edited by Ray Chandler, Jun 06 2008

A090866 Primes p == 1 (mod 4) such that (p-1)/4 is prime.

Original entry on oeis.org

13, 29, 53, 149, 173, 269, 293, 317, 389, 509, 557, 653, 773, 797, 1109, 1229, 1493, 1637, 1733, 1949, 1997, 2309, 2477, 2693, 2837, 2909, 2957, 3413, 3533, 3677, 3989, 4133, 4157, 4253, 4349, 4373, 4493, 4517, 5189, 5309, 5693, 5717, 5813, 6173, 6197

Offset: 1

Benoit Cloitre, Feb 12 2004

nonn

Same as Chebyshev's subsequence of the primes with primitive root 2, because Chebyshev showed that 2 is a primitive root of all primes p = 4*q+1 with q prime. If the sequence is infinite, then Artin's conjecture ("every nonsquare positive integer n is a primitive root of infinitely many primes q") is true for n = 2. - Jonathan Sondow, Feb 04 2013

Albert H. Beiler: Recreations in the theory of numbers. New York: Dover, (2nd ed.) 1966, p. 102, nr. 5.
P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Artin's conjecture

Cf. A001122, A005385, A005596, A023212, A221981, A222008.

f:=[n: n in [1..2000] | IsPrime(n) and IsPrime(4*n+1)]; [4*f[n] + 1: n in [1..50]]; // G. C. Greubel, Feb 08 2019

Select[Prime[Range[1000]], Mod[#, 4]==1 && PrimeQ[(#-1)/4] &] (* G. C. Greubel, Feb 08 2019 *)

isok(p) = isprime(p) && !frac(q=(p-1)/4) && isprime(q); \\ Michel Marcus, Feb 09 2019

a(n) = 4*A023212(n) + 1.

A152913 Primes of the form n^4 + (n+1)^4.

Original entry on oeis.org

17, 97, 337, 881, 3697, 10657, 16561, 49297, 66977, 89041, 149057, 847601, 988417, 1146097, 1972097, 2522257, 2836961, 3553777, 3959297, 4398577, 5385761, 7166897, 11073217, 17653681, 32530177, 41532497, 44048497, 55272097, 61627201

Offset: 1

Vincenzo Librandi, Dec 15 2008

Also primes in A008514.

Sequence is disjoint to A005385: If n^4 + (n+1)^4 is a prime p, then (p-1)/2 = n^4 + 2*n^3 + 3*n^2 + 2*n. (p-1)/2 = 8 for n = 1 and (p-1)/2 is divisible by n for n > 1. In each case, (p-1)/2 is not prime.

			For n=3, n^4 + (n+1)^4 = 337 is prime and (337-1)/2 = 168 = 3*56 is not prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A155211, A008514.

[ a: n in [1..80] | IsPrime(a) where a is n^4+(n+1)^4 ];

f[n_]:=n^4+(n+1)^4;lst={};Do[a=f[n];If[PrimeQ[a],AppendTo[lst,a]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, May 30 2009 *)
Select[Table[n^4+(n+1)^4,{n,0,700}],PrimeQ]
Select[Total/@Partition[Range[100]^4,2,1],PrimeQ] (* Harvey P. Dale, Sep 29 2023 *)

Edited and extended by Klaus Brockhaus, Dec 21 2008

A059761 Initial primes of Cunningham chains of first type with length exactly 2. Primes in A059453 that survive as primes only one "2p-1 iteration", forming chains of exactly 2 terms.

Original entry on oeis.org

3, 29, 53, 113, 131, 173, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 593, 641, 653, 659, 683, 743, 761, 809, 911, 953, 1013, 1049, 1103, 1223, 1289, 1499, 1559, 1583, 1601, 1733, 1973, 2003, 2069, 2129, 2141, 2273, 2339, 2351, 2393, 2399, 2543

Offset: 1

Labos Elemer, Feb 20 2001

nonn

Primes p such that {(p-1)/2, p, 2p+1, 4p+3} = {composite, prime, prime, composite}.

			53 is a term because 26 and 215 are composites, and 53 and 107 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Chris Caldwell's Prime Glossary, Cunningham chains.
Warut Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains. [Wayback Machine link]
Eric Weisstein's World of Mathematics, Cunningham Chain.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059453, A059454, A059455, A007700.

ccftQ[p_]:=Boole[PrimeQ[{(p-1)/2,p,2 p+1,4 p+3}]]=={0,1,1,0}; Select[ Prime[ Range[400]],ccftQ] (* Harvey P. Dale, Jun 19 2021 *)

A033631 Numbers k such that sigma(phi(k)) = sigma(k) where sigma is the sum of divisors function A000203 and phi is the Euler totient function A000010.

Original entry on oeis.org

1, 87, 362, 1257, 1798, 5002, 9374, 21982, 22436, 25978, 35306, 38372, 41559, 50398, 51706, 53098, 53314, 56679, 65307, 68037, 89067, 108946, 116619, 124677, 131882, 136551, 136762, 138975, 144014, 160629, 165554, 170037, 186231, 192394, 197806

Offset: 1

Jud McCranie

nonn

For corresponding values of phi(k) and sigma(k), see A115619 and A115620.
This sequence is infinite because for each positive integer k, 3^k*7*1979 and 3^k*7*2699 are in the sequence (the proof is easy). A108510 gives primes p like 1979 and 2699 such that for each positive integer k, 3^k*7*p is in this sequence. - Farideh Firoozbakht, Jun 07 2005
There is another class of [conjecturally] infinite subsets connected to A005385 (safe primes). Examples: Let s,t be safe primes, s<>t, then 3^2*5*251*s, 2^2*61*71*s, 2*61*s*t and 2*19*311*s are in this sequence. So is 3*s*A108510(m). (There are some obvious exceptions for small s, t.) - Vim Wenders, Dec 27 2006

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 87, p. 29, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B42.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books 1997.
David Wells, Curious and Interesting Numbers (Revised), Penguin Books, page 114.

T. D. Noe, Table of n, a(n) for n = 1..1000
J.-M. De Koninck, F. Luca, Positive integers n such that sigma(phi(n))=sigma(n), JIS 11 (2008) 08.1.5.
S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)

Cf. A000203, A000010, A006872, A115619, A115620.

[k:k in [1..200000]| DivisorSigma(1,EulerPhi(k)) eq DivisorSigma(1,k)]; // Marius A. Burtea, Feb 09 2020

Do[If[DivisorSigma[1, EulerPhi[n]]==DivisorSigma[1, n], Print[n]], {n, 1, 10^5}]

is(n)=sigma(eulerphi(n))==sigma(n) \\ Charles R Greathouse IV, Feb 13 2013

Entry revised by N. J. A. Sloane, Apr 10 2006

cp's OEIS Frontend

A077065 Semiprimes of form prime - 1.

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A336467 Fully multiplicative with a(2) = 1 and a(p) = A000265(p+1) for odd primes p, with A000265(k) giving the odd part of k.

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A063638 Primes p such that p-2 is a semiprime.

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A059762 Initial primes of Cunningham chains of first type with length exactly 3. Primes in A059453 that survive as primes just two "2p+1 iterations", forming chains of exactly 3 terms.

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A051254 Mills primes.

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A058340 Primes p such that phi(x) = p-1 has only 2 solutions, namely x = p and x = 2p.

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