A154945 -id:A154945 - cp's OEIS frontend

A084920 a(n) = (prime(n)-1)*(prime(n)+1).

3, 8, 24, 48, 120, 168, 288, 360, 528, 840, 960, 1368, 1680, 1848, 2208, 2808, 3480, 3720, 4488, 5040, 5328, 6240, 6888, 7920, 9408, 10200, 10608, 11448, 11880, 12768, 16128, 17160, 18768, 19320, 22200, 22800, 24648, 26568, 27888, 29928

Offset: 1

Reinhard Zumkeller, Jun 11 2003

Squares of primes minus 1. - Wesley Ivan Hurt, Oct 11 2013

Integers k for which there exist exactly two positive integers b such that (k+1)/(b+1) is an integer. - Benedict W. J. Irwin, Jul 26 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.
Nik Lygeros and Olivier Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010), Article 10.7.4.
Lực Ta, Enumeration of virtual quandles up to isomorphism, arXiv:2506.16536 [math.GT], 2025. See p. 6.

Cf. A000040, A005563, A049001, A154945, A166010, A182200, A182174.
Cf. A006093, A008864, A084921, A084922.
Cf. A066872, A001248, A024702, A013661, A065469.

a084920 n = (p - 1) * (p + 1) where p = a000040 n
-- Reinhard Zumkeller, Aug 27 2013

[p^2-1: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 30 2015

A084920:=n->ithprime(n)^2-1; seq(A084920(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Table[Prime[n]^2 - 1, {n, 50}] (* Wesley Ivan Hurt, Oct 11 2013 *)
Prime[Range[50]]^2-1 (* Harvey P. Dale, Oct 02 2021 *)

a(n) = (prime(n)-1)*(prime(n)+1); \\ Michel Marcus, Jul 28 2016

[(p-1)*(p+1) for p in primes(200)] # Bruno Berselli, Mar 30 2015

a(n) = A006093(n) * A008864(n);
a(n) = A084921(n)*2, for n > 1; a(n) = A084922(n)*6, for n > 2.
Product_{n > 0} a(n)/A066872(n) = 2/5. a(n) = A001248(n) - 1. - R. J. Mathar, Feb 01 2009
a(n) = prime(n)^2 - 1 = A001248(n) - 1. - Vladimir Joseph Stephan Orlovsky, Oct 17 2009
a(n) ~ n^2*log(n)^2. - Ilya Gutkovskiy, Jul 28 2016
a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^2*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime. - Seiichi Manyama, Dec 31 2017
a(n) = 24 * A024702(n) for n > 2. - Jianing Song, Apr 28 2019
Sum_{n>=1} 1/a(n) = A154945. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Nov 07 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = Pi^2/6 (A013661).
Product_{n>=1} (1 - 1/a(n)) = A065469. (End)

A190641 Numbers having exactly one non-unitary prime factor.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164

Offset: 1

Reinhard Zumkeller, Dec 29 2012

nonn

Numbers k such that the powerful part of k, A057521(k), is a composite prime power (A246547). - Amiram Eldar, Aug 01 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio.
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, Iss. 1 (2011), pp. 52-66. See p. 61.

Subsequence of A013929 and of A327877.

Cf. A056170, A057521, A154945, A246547, A359466 (characteristic function).

a190641 n = a190641_list !! (n-1)
a190641_list = map (+ 1) $ elemIndices 1 a056170_list

Select[Range[164],Count[FactorInteger[#][[All, 2]], 1] == Length[FactorInteger[#]] - 1 &] (* Geoffrey Critzer, Feb 05 2015 *)

list(lim)=my(s=lim\4, v=List(), u=vectorsmall(s, i, 1), t, x); forprime(k=2, sqrtint(s), t=k^2; forstep(i=t, s, t, u[i]=0)); forprime(k=2, sqrtint(lim\1), for(e=2,logint(lim\1,k), t=k^e; for(i=1, #u, if(u[i] && gcd(k, i)==1, x=t*i; if(x>lim, break); listput(v, x))))); Set(v) \\ Charles R Greathouse IV, Aug 02 2016

isok(n) = my(f=factor(n)); #select(x->(x>1), f[,2]) == 1; \\ Michel Marcus, Jul 30 2017

A056170(a(n)) = 1.

a(n) ~ k*n, where k = Pi^2/(6*A154945) = 2.9816096.... - Charles R Greathouse IV, Aug 02 2016

A056798 Prime powers with even nonnegative exponents.

Original entry on oeis.org

1, 4, 9, 16, 25, 49, 64, 81, 121, 169, 256, 289, 361, 529, 625, 729, 841, 961, 1024, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4096, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 16384

Offset: 1

Labos Elemer, Aug 28 2000

nonn

Also numbers whose geometric mean of divisors is an integer. - Ctibor O. Zizka, Sep 29 2008

This is just a special case. In fact, the numbers whose geometric mean of divisors is an integer are all the squares of integers (A000290). - Daniel Lignon, Nov 29 2014

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

Cf. A000290, A000961, A025475, A154945.

Take[Union[Flatten[Table[Prime[n]^k, {n, 31}, {k, 0, 14, 2}]]], 45] (* Alonso del Arte, Jul 05 2011 *)

is(n)=my(e=isprimepower(n)); if(e, e%2==0, n==1) \\ Charles R Greathouse IV, Sep 18 2015

from sympy import primepi, integer_nthroot
def A056798(n):
    if n==1: return 1
    def f(x): return int(n-2+x-sum(primepi(integer_nthroot(x,k)[0])for k in range(2,x.bit_length(),2)))
    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 13 2024

a(n) = A025473(n)^(2*A025474(n)) = A000961(n)^2;
A001222(a(n)) mod 2 = 0;
A003415(a(n)) = A192083(n); A068346(a(n)) = A192084(n). - Reinhard Zumkeller, Jun 26 2011
Sum_{n>=2} 1/a(n) = A154945. - Amiram Eldar, Sep 21 2020

A036785 Numbers divisible by the squares of two distinct primes.

Original entry on oeis.org

36, 72, 100, 108, 144, 180, 196, 200, 216, 225, 252, 288, 300, 324, 360, 392, 396, 400, 432, 441, 450, 468, 484, 500, 504, 540, 576, 588, 600, 612, 648, 675, 676, 684, 700, 720, 756, 784, 792, 800, 828, 864, 882, 900, 936, 968, 972, 980, 1000, 1008, 1044

Offset: 1

Alford Arnold

Not squarefree, not a nontrivial prime power and not in {squarefree} times {nontrivial prime powers}.

Numbers k such that A056170(k) > 1. The asymptotic density of this sequence is 1 - (6/Pi^2) * (1 + A154945) = 0.05668359058... - Amiram Eldar, Nov 01 2020

CRC Standard Mathematical Tables and Formulae, 30th ed., (1996) page 102-105.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005117, A025475, A056170, A154945.
Equivalent sequence for 3 distinct primes: A318720.
Cf. A085986, A338539, A339245 (subsequences).
Subsequence of A038838.

Select[Range@ 1050, And[Length@ # > 1, Total@ Boole@ Map[# > 1 &, #[[All, -1]]] > 1] &@ FactorInteger@ # &] (* Michael De Vlieger, Apr 25 2017 *)
dstdpQ[n_]:=Length[Select[Sqrt[#]&/@Divisors[n],PrimeQ]]>1; Select[ Range[ 1100],dstdpQ] (* Harvey P. Dale, Jan 15 2020 *)

is(n)=my(f=vecsort(factor(n)[,2],,4));#f>1&&f[2]>1 \\ Charles R Greathouse IV, Nov 15 2012

More terms from Larry Reeves (larryr(AT)acm.org), Apr 03 2000

New name from Charles R Greathouse IV, Nov 15 2012

A380840 Decimal expansion of Sum_{p prime} 1/(p-1)^3.

Original entry on oeis.org

1, 1, 4, 7, 5, 2, 9, 0, 9, 7, 7, 5, 8, 5, 8, 0, 0, 4, 6, 9, 3, 3, 2, 8, 3, 8, 0, 6, 2, 8, 2, 1, 3, 0, 4, 0, 1, 6, 4, 4, 7, 6, 4, 7, 3, 5, 5, 2, 5, 1, 1, 2, 2, 5, 5, 2, 7, 5, 8, 2, 4, 1, 2, 3, 9, 5, 0, 5, 3, 3, 5, 9, 0, 4, 5, 5, 0, 4, 5, 4, 3, 1, 4, 7, 2, 6, 5, 2, 2, 8, 7, 3, 7, 2, 6, 9, 0, 9, 4, 6, 7, 5, 1, 6, 8, 0

Offset: 1

Artur Jasinski, Mar 19 2025

nonn

			1.1475290977585800469332838..,

Cf. A085541, A086242, A085548, A136141, A152441, A154945, A179119, A369632, A324833.

PARI
```
sumeulerrat(1/(p-1)^3)
```

A116512 a(n) is the number of positive integers each of which is <= n and is divisible by exactly one prime dividing n (but is coprime to every other prime dividing n). a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 6, 8, 1, 9, 1, 10, 8, 11, 1, 12, 5, 13, 9, 14, 1, 14, 1, 16, 12, 17, 10, 18, 1, 19, 14, 20, 1, 20, 1, 22, 18, 23, 1, 24, 7, 25, 18, 26, 1, 27, 14, 28, 20, 29, 1, 28, 1, 31, 24, 32, 16, 32, 1, 34, 24, 34, 1, 36, 1, 37, 30, 38, 16, 38, 1, 40, 27, 41, 1

Offset: 1

Leroy Quet, Mar 23 2006

nonn

a(n) = number of m's, 1 <= m <= n, where gcd(m,n) is a power of a prime (> 1).

We could also have taken a(1) = 1, but a(1) = 0 is better since there are no numbers <= 1 with the desired property. - N. J. A. Sloane, Sep 16 2006

			12 is divisible by 2 and 3. The positive integers which are <= 12 and which are divisible by 2 or 3, but not by both 2 and 3, are: 2,3,4,8,9,10. Since there are six such integers, a(12) = 6.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000010, A013661, A069513, A119790, A119794, A120499, A131233, A154945.

Cf. A095112 (Inverse Möbius transform), A354109 (positions of even terms).

with(numtheory): a:=proc(n) local c,j: c:=0: for j from 1 to n do if nops(factorset(gcd(j,n)))=1 then c:=c+1 else c:=c: fi od: c; end: seq(a(n),n=1..90); # Emeric Deutsch, Apr 01 2006

Table[Length@Select[GCD[n, Range@n], MatchQ[FactorInteger@#, {{, }}] && # != 1 &], {n, 93}] (* Giovanni Resta, Apr 04 2006; corrected by Ilya Gutkovskiy, Sep 26 2021 *)
a[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, n * Times @@ (1-1/p) * Total[1/(p-1)]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2025 *)

{ for(n=1,60, hav=0; for(i=1,n, g = gcd(i,n); d = factor(g); dec=matsize(d); if( dec[1] == 1, hav++; ); ); print1(hav,","); ); } \\ R. J. Mathar, Mar 29 2006

a(n) = sumdiv(n, d, eulerphi(n/d) * (isprimepower(d) >= 1)); \\ Daniel Suteu, Jun 27 2018

a(n) = {my(p = factor(n)[,1]); n * vecprod(apply(x -> 1-1/x, p)) * vecsum(apply(x -> 1/(x-1), p));} \\ Amiram Eldar, Jun 21 2025

Dirichlet g.f.: A(s)*zeta(s-1)/zeta(s) where A(s) is the Dirichlet g.f. for A069513. - Geoffrey Critzer, Feb 22 2015
a(n) = Sum_{d|n, d is a prime power} phi(n/d), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 27 2018
a(n) = phi(n)*Sum_{p|n} 1/(p-1), where p is a prime and phi(k) is the Euler totient function. - Ridouane Oudra, Apr 29 2019
a(n) = Sum_{k=1..n, gcd(n,k) = 1} omega(gcd(n,k-1)). - Ilya Gutkovskiy, Sep 26 2021
a(n) = Sum_{p|n, p prime} p^(v(n,p)-1)*phi(n/p^v(n,p)), where p^v(n,p) is the highest power of p dividing n. - Ridouane Oudra, Oct 06 2023
From Amiram Eldar, Jun 21 2025: (Start)
a(n) = A131233(n) - A000010(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c =  Sum_{p prime} (1/(p^2-1)) / zeta(2) = A154945 / A013661 = 0.3353893075569103736099... . (End)

More terms from R. J. Mathar, Emeric Deutsch and Giovanni Resta, Apr 01 2006

Edited by N. J. A. Sloane, Sep 16 2006

A324833 Decimal expansion of eta_2, a constant related to the asymptotic density of certain sets of residues.

Original entry on oeis.org

1, 2, 9, 0, 3, 8, 9, 2, 5, 8, 9, 7, 8, 0, 7, 5, 5, 6, 4, 9, 7, 4, 3, 4, 8, 6, 3, 4, 8, 1, 7, 7, 5, 8, 7, 7, 6, 3, 8, 4, 9, 3, 2, 1, 4, 1, 9, 9, 2, 0, 5, 6, 8, 8, 3, 0, 0, 4, 1, 2, 7, 0, 4, 5, 6, 3, 9, 8, 0, 6, 6, 5, 7, 3, 0, 9, 1, 7, 0, 3, 9, 8, 9, 9, 9, 7, 1, 6, 7, 7, 8, 3, 5, 9, 8, 1, 9, 3, 4, 3, 8

Offset: 0

Jean-François Alcover, Mar 17 2019

			0.12903892589780755649743486348177587763849321419920568830041270456398...

Carl Pomerance, Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory. 2011. Vol. 1. Iss. 1. pp. 52-66. See p. 62.
Index to constants which are prime zeta sums {0,2,2}

Cf. A154945 (eta_1), A324834 (eta_3), A324835 (eta_4), A324836 (eta_5).

digits = 101; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[eta2 = Sum[n PrimeZetaP[2n + 2], {n, 1, m}], 10, digits][[1]]; rd[m0]; rd[m = 2m0]; While[rd[m] != rd[m-m0], Print[m]; m = m+m0]; Print[N[eta2, digits] ]; rd[m]

Sum_{p prime} 1/(p^2-1)^2.
Sum_{n>0} n P(2n+2) where P is the prime zeta P function.
Equals - A136141/4 + A086242/4 - A179119/4 + A382554/4. - Artur Jasinski, Mar 31 2025

A095112 a(n) is the sum of n/k over all prime powers k > 1 which divide n.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 4, 7, 1, 13, 1, 9, 8, 15, 1, 17, 1, 19, 10, 13, 1, 29, 6, 15, 13, 25, 1, 31, 1, 31, 14, 19, 12, 43, 1, 21, 16, 43, 1, 41, 1, 37, 29, 25, 1, 61, 8, 37, 20, 43, 1, 53, 16, 57, 22, 31, 1, 77, 1, 33, 37, 63, 18, 61, 1, 55, 26, 59, 1, 95, 1, 39, 43, 61, 18, 71, 1, 91, 40

Offset: 1

Dean Hickerson, following a suggestion of Leroy Quet, May 28 2004

nonn

A073093(n)-1 terms are added to produce a(n). - Michel Marcus, Aug 29 2013

			The prime power divisors of 24 are 2, 4, 8 and 3, so a(24) = 24/2 + 24/4 + 24/8 + 24/3 = 29.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Jon Maiga, Computer-generated formulas for A095112, Sequence Machine.

Cf. A000010, A001221, A001222, A046337 (positions of even terms), A073093, A154945, A366265.

Inverse Möbius transform of A116512.

with(numtheory): seq(add(bigomega(d)*phi(n/d),d in divisors(n)), n=1..60); # Ridouane Oudra, Oct 30 2023

a[n_]:=Plus@@(n/Flatten[ #[[1]]^Range[ #[[2]]]&/@FactorInteger[n]])

A095112(n) = sumdiv(n,d,(1==omega(d))*(n/d)); \\ Antti Karttunen, Feb 25 2018

a(n) = Sum_{k=1..n} bigomega(gcd(n,k)). - Lechoslaw Ratajczak, Jun 18 2017
Sum_{k=1..n} a(k) ~ A154945 * n*(n+1)/2. - Daniel Suteu, Apr 01 2019
a(n) = Sum_{d|n} bigomega(d)*phi(n/d). - Ridouane Oudra, Oct 30 2023
a(n) = Sum_{d|n} A116512(d). [From Sequence Machine] - Antti Karttunen, Nov 22 2023

A318720 Numbers k such that there exists a strict relatively prime factorization of k in which no pair of factors is relatively prime.

Original entry on oeis.org

900, 1764, 1800, 2700, 3528, 3600, 4356, 4500, 4900, 5292, 5400, 6084, 6300, 7056, 7200, 8100, 8712, 8820, 9000, 9800, 9900, 10404, 10584, 10800, 11025, 11700, 12100, 12168, 12348, 12600, 12996, 13068, 13500, 14112, 14400, 14700, 15300, 15876, 16200, 16900

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

From Amiram Eldar, Nov 01 2020: (Start)
Also, numbers with more than two non-unitary prime divisors, i.e., numbers k such that A056170(k) > 2, or equivalently, numbers divisible by the squares of three distinct primes.
The complement of the union of A005117, A190641 and A338539.
The asymptotic density of this sequence is 1 - 6/Pi^2 - (6/Pi^2)*A154945 - (3/Pi^2)*(A154945^2 - A324833) = 0.0033907041... (End)

			900 is in the sequence because the factorization 900 = (6*10*15) is relatively prime (since the GCD of (6,10,15) is 1) but each of the pairs (6,10), (6,15), (10,15) has a common divisor > 1. Larger examples are:
1800 = (6*15*20) = (10*12*15).
9900 = (6*10*165) = (6*15*110) = (10*15*66).
5400 = (6*20*45) = (10*12*45) = (10*15*36) = (15*18*20).
60 is not in the sequence because all its possible factorizations (4 * 15, 3 * 4 * 5, etc.) contain at least one pair that is coprime, if not more than one prime.

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

Cf. A001055, A001221, A001222, A007716, A045778, A051185, A078374, A281116, A303140, A303283, A305843, A305854, A317748, A318715, A318717, A318721.

Cf. A005117, A036785, A056170, A154945, A190641, A324833, A338539.

strfacs[n_] := If[n <= 1, {{}}, Join@@Table[(Prepend[#1, d] &)/@Select[strfacs[n/d], Min@@#1 > d &], {d, Rest[Divisors[n]]}]]; Select[Range[10000], Function[n, Select[strfacs[n], And[GCD@@# == 1, And@@(GCD[##] > 1 &)@@@Select[Tuples[#, 2], Less@@# &]] &] != {}]]
Select[Range[20000], Count[FactorInteger[#][[;;,2]], ?(#1 > 1 &)] > 2 &] (* _Amiram Eldar, Nov 01 2020 *)

A338539 Numbers having exactly two non-unitary prime factors.

Original entry on oeis.org

36, 72, 100, 108, 144, 180, 196, 200, 216, 225, 252, 288, 300, 324, 360, 392, 396, 400, 432, 441, 450, 468, 484, 500, 504, 540, 576, 588, 600, 612, 648, 675, 676, 684, 700, 720, 756, 784, 792, 800, 828, 864, 882, 936, 968, 972, 980, 1000, 1008, 1044, 1080, 1089

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 2.
Numbers divisible by the squares of exactly two distinct primes.
Subsequence of A036785 and first differs from it at n = 44.
The asymptotic density of this sequence is (3/Pi^2)*(eta_1^2 - eta_2) = 0.0532928864..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			36 = 2^2 * 3^2 is a term since it has exactly 2 prime factors, 2 and 3, that are non-unitary.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, No. 1 (2011), pp. 52-66. See pp. 61-62.

Subsequence of A013929 and A036785.
Cf. A001221, A056170, A057521, A190641, A338540, A338541, A338542, A347892 (subsequence).
Cf. A154945 (eta_1), A324833 (eta_2).

Select[Range[1000], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 2 &]

cp's OEIS Frontend

A084920 a(n) = (prime(n)-1)*(prime(n)+1).

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A190641 Numbers having exactly one non-unitary prime factor.

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A056798 Prime powers with even nonnegative exponents.

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A036785 Numbers divisible by the squares of two distinct primes.

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A380840 Decimal expansion of Sum_{p prime} 1/(p-1)^3.

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A116512 a(n) is the number of positive integers each of which is <= n and is divisible by exactly one prime dividing n (but is coprime to every other prime dividing n). a(1) = 0.

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A324833 Decimal expansion of eta_2, a constant related to the asymptotic density of certain sets of residues.

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