A054800 -id:A054800 - cp's OEIS frontend

A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression.

9843019, 37772429, 53868649, 71427757, 78364549, 79080577, 98150021, 99591433, 104436889, 106457509, 111267419, 121174811, 121174841, 168236119, 199450099, 203908891, 207068803, 216618187, 230952859, 234058871, 235524781, 253412317, 263651161, 268843033, 294485363, 296239787

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000

nonn

Each set has a constant difference of 30, for all of the terms listed so far.
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000, the record is 10 primes.
The first CPAP-5 with common difference 60 starts at 6182296037 ~ 6e9, cf. A210727. This sequence consists of first members of pairs of consecutive primes in A054800 (see also formula): a(1..6) = A054800({1555, 4555, 6123, 7695, 8306, 8371}). Conversely, pairs of consecutive primes in this sequence yield a term of A058362, i.e., they start a sequence of 6 consecutive primes in arithmetic progression (CPAP-6): e.g., the nearby values a(12) = 121174811, a(13) = 121174841 = a(12) + 30 indicate such a term, whence A006560(6) = A058362(1) = a(12). The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3e11, cf. A210727. Longer CPAP's must have common difference >= 210. - M. F. Hasler, Oct 26 2018
About 500 initial terms of this sequence are the same as for the sequence "First of 5 consecutive primes separated by gaps of 30". The first 10^4 terms of A052243 give 281 terms of this sequence (up to ~ 3.34e9) with the same formula as the one using A054800, but as the above comment says, this will miss terms beyond twice that range. - M. F. Hasler, Jan 02 2020

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 181.

Zak Seidov, Table of n, a(n) for n = 1..241 (all terms up to 3*10^9)
Jens K. Andersen, The Largest Known CPAP's, updated Sept. 2018
OEIS wiki, Consecutive primes in arithmetic progression, updated Jan. 2020
Index entries for sequences related to primes in arithmetic progressions

Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.
Cf. A052239: start of first CPAP-4 with common difference 6n.
Cf. A058362: start of 6 consecutive primes in arithmetic progression.
Cf. A006560: first prime to start a CPAP-n.

Select[Partition[Prime[Range[14000000]],5,1],Length[Union[ Differences[ #]]]==1&] (* Harvey P. Dale, Jun 22 2013 *)

A059044(n,p=2,c,g,P)={forprime(q=p+1,, if(p+g!=p+=g=q-p, next, q!=P+2*g, c=3, c++>4, print1(P-2*g,",");n--||break);P=q-g);P-2*g} \\ This does not impose the gap to be 30, but it happens to be the case for the first values. - M. F. Hasler, Oct 26 2018

Found by exhaustive search for 5 primes in arithmetic progression with all other intermediate numbers being composite.

A059044 = { A054800(i) | A054800(i+1) - A151800(A054800(i)) } with the nextprime function A151800(prime(k)) = prime(k+1) = prime(k) + A001223(k). - M. F. Hasler, Oct 27 2018, edited Jan 02 2020.

a(16)-a(22) from Donovan Johnson, Sep 05 2008
Reference added by Harvey P. Dale, Jun 22 2013
Edited (definition clarified, cross-references corrected and extended) by M. F. Hasler, Oct 26 2018

A333216 Lengths of maximal subsequences without adjacent equal terms in the sequence of prime gaps.

Original entry on oeis.org

2, 13, 21, 3, 7, 8, 1, 18, 29, 5, 3, 8, 11, 31, 4, 20, 3, 7, 5, 19, 21, 32, 1, 19, 48, 19, 29, 32, 7, 38, 1, 43, 12, 33, 46, 6, 16, 8, 4, 34, 15, 1, 19, 7, 1, 23, 28, 30, 22, 8, 1, 7, 1, 52, 14, 56, 10, 26, 2, 30, 65, 5, 71, 12, 44, 39, 37, 6, 19, 47, 11, 10

Offset: 1

Gus Wiseman, Mar 15 2020

Prime gaps are differences between adjacent prime numbers.

Essentially the same as A145024. - R. J. Mathar, Mar 16 2020

			The prime gaps split into the following subsequences without adjacent equal terms: (1,2), (2,4,2,4,2,4,6,2,6,4,2,4,6), (6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6), (6,4,6), (6,2,10,2,4,2,12), (12,4,2,4,6,2,10,6), ...

First differences of A064113.
The version for the Kolakoski sequence is A306323.
The weakly decreasing version is A333212.
The weakly increasing version is A333215.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
The equal version is A333254.
Cf. A000040, A001223, A084758, A106356, A124762, A124767, A333214.

Length/@Split[Differences[Array[Prime,100]],UnsameQ]//Most

Ones correspond to balanced prime quartets (A054800), so the sum of terms up to but not including the n-th one is A000720(A054800(n - 1)) = A090832(n).

A052239 Smallest prime p in set of 4 consecutive primes in arithmetic progression with common difference 6n.

Original entry on oeis.org

251, 111497, 74453, 1397609, 642427, 5321191, 23921257, 55410683, 400948369, 253444777, 1140813701, 491525857, 998051413, 2060959049, 4480114337, 55140921491, 38415872947, 315392068463, 15162919459, 60600021611, 278300877401, 477836574947, 1486135570643

Offset: 1

Labos Elemer, Jan 31 2000

See also the less restrictive A054701 where the gaps are multiples 6n. - M. F. Hasler, Nov 06 2018

			a(5) = 642427, 642457, 642487, 642517 are the smallest consecutive primes with 3 consecutive gaps of 30, cf. A052243.
From _M. F. Hasler_, Nov 06 2018: (Start)
Other terms are also initial terms of corresponding sequences:
a(1) = 251 = A033451(1) = A054800(1), start of first CPAP-4 with common gap of 6,
a(2) = 111497 = A033447(1), start of first CPAP-4 with common gap of 12,
a(3) = 74453 = A033448(1) = A054800(25), first CPAP-4 with common gap of 18,
a(4) = 1397609 = A052242(1), start of first CPAP-4 with common gap of 24,
a(5) = 642427 = A052243(1) = A052195(16), first CPAP-4 with common gap of 30,
a(6) = 5321191 = A058252(1) = A161534(26), first CPAP-4 with common gap 36 = 6^2,
a(7) = 23921257 = A058323(1), start of first CPAP-4 with common gap of 42,
a(8) = 55410683 = A067388(1), start of first CPAP-4 with common gap of 48,
a(9) = 400948369 = A259224(1), start of first CPAP-4 with common gap of 54,
a(10) = 253444777 = A210683(1) = A089234(417), CPAP-4 with common gap of 60,
a(11) = 1140813701 = A287547(1), start of first CPAP-4 with common gap of 66,
a(12) = 491525857 = A287550(1), start of first CPAP-4 with common gap of 72,
a(13) = 998051413 = A287171(1), start of first CPAP-4 with common gap of 78,
a(14) = 2060959049 = A287593(1), start of first CPAP-4 with common gap of 84,
a(15) = 4480114337 = A286817(1) = A204852(444), common distance 90. (End)

Jerry M Lagrou, Table of n, a(n) for n = 1..42
Index entries for sequences related to primes in arithmetic progressions

Initial terms of sequences A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388, A259224, A210683.
Range is a subset of A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
Cf. A054701: gaps are possibly distinct multiples of 6n (not CPAP's).

Transpose[Flatten[Table[Select[Partition[Prime[Range[2000000]],4,1], Union[ Differences[ #]] =={6n}&,1],{n,7}],1]][[1]] (* Harvey P. Dale, Aug 12 2012 *)

a(n, p=[2, 0, 0], d=6*[n, n, n])={while(p+d!=p=[nextprime(p[1]+1), p[1], p[2]], ); p[3]-d[3]} \\ after M. F. Hasler in A052243; Graziano Aglietti (mg5055(AT)mclink.it), Aug 22 2010, Corrected by M. F. Hasler, Nov 06 2018

A052239(n, p=2, c, o)={n*=6; forprime(q=p+1, , if(p+n!=p=q, next, q!=o+2*n, c=2, c++>3, break); o=q-n); o-n} \\ M. F. Hasler, Nov 06 2018

More terms from Labos Elemer, Jan 04 2002
a(7) corrected and more terms added by Graziano Aglietti (mg5055(AT)mclink.it), Aug 22 2010
a(15)-a(20) from Donovan Johnson, Oct 05 2010
a(21)-a(23) from Donovan Johnson, May 23 2011

A210683 Primes p such that p, p+60, p+120, p+180 are consecutive primes.

Original entry on oeis.org

253444777, 271386581, 286000489, 415893013, 475992773, 523294549, 620164949, 794689481, 838188877, 840725323, 846389227, 884106599, 884951807, 908725507, 941796223, 952288331, 971614151, 1002290693, 1003166771, 1006976797, 1053792359, 1097338313, 1163141201

Offset: 1

Zak Seidov, May 09 2012

nonn

Subsequence of A089234 which itself is a subsequence of A126771:
a(1) = 253444777 = A089234(417) = A126771(81526),
a(36) = 1998782563 = A089234(5579) = A126771(788920).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A089234, A126771.
Analogous sequences (start of CPAP-4, with common difference in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54].
Cf. A054800: union of all sequences of this type (start of CPAP-4).

A210683(n, p=2, v=1, g=60, c, o)={forprime(q=p+1, , if(p+g!=p=q, next, q!=o+2*g, c=2, c++>3, v&& print1(o-g", "); n--||break); o=q-g); o-g} \\ Can be used as next(p)=A210683(1, p) to get the next term, e.g.:
p=0; A210683_vec=vector(10,i,p=A210683(1,p)) \\ Will take a long time! - M. F. Hasler, Oct 26 2018

A082080 Smallest balanced prime of order n.

Original entry on oeis.org

2, 5, 79, 17, 491, 53, 71, 29, 37, 983, 5503, 173, 157, 353, 5297, 263, 179, 383, 137, 2939, 2083, 751, 353, 5501, 1523, 149, 4561, 1259, 397, 787, 8803, 8803, 607, 227, 3671, 17443, 57097, 3607, 23671, 12539, 1217, 11087, 1087, 21407, 19759, 953

Offset: 0

Labos Elemer, Apr 08 2003

nonn

Or, smallest (2n+1)-balanced prime number.

Prime(k) is a balanced prime of order n if it is the average of the 2n+1 primes from prime(k-n) to prime(k+n).

			a(1) = 5 = (3 + 5 + 7)/3 = 15/3.
a(5) = 53 = (31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73)/11 = 583/11.
a(6) = 71 = (43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101)/13 = 923/13.

Giovanni Resta, Table of n, a(n) for n = 0..10000 (first 1375 terms from Robert G. Wilson v)

Cf. A096693, A006562, A082077, A082078, A082079, A096697, A096698, A096699, A096700, A096701, A096702, A096703, A096704.

Cf. A006562, A051795, A081415, A096710, A055380, A082312, A075540, A054800.

f[n_] := Block[{p = Prime@ Range[2n +1]}, While[ Total[p] != (2n +1) p[[n +1]], p = Join[Rest@ p, {NextPrime[ p[[-1]]] }]]; p[[n +1]]]; Array[f, 46, 0] (* Robert G. Wilson v, Jun 21 2004 and modified Apr 11 2017 *)

for(n=0, 50, i=2*n+1;f=0;forprime(p=2, 10^7, s=0;c=i;pr=p-1;t=0;while(c>0, c=c-1;pr=nextprime(pr+1);s=s+pr; if(c==(i-1)/2, t=pr)); if(s/i==t, print1(t", ");f=1;break)); if(!f, print1("0, ")))

Corrected and extended by Ralf Stephan, Apr 09 2003

A259224 Initial primes in sets of 4 consecutive primes with common gap 54.

Original entry on oeis.org

400948369, 473838319, 583946599, 678953059, 816604199, 972598819, 1136526949, 1466715139, 1475790529, 1499794999, 1502149559, 1610895679, 1643313869, 1673057219, 1686181579, 1845792019, 1867046639, 1907478889, 1992202439, 2011077869, 2030490479, 2207714969

Offset: 1

Zak Seidov, Jun 21 2015

nonn

All terms are == {19,29} mod 30.

Lars Blomberg, Table of n, a(n) for n = 1..10000 (The first 102 terms from Zak Seidov)

Start of CPAP-4 with given common difference (in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [this: 54], A210683 [60].

Subsequence of A054800: start of a CPAP-4 with arbitrary common difference.

A259224(n, p=2, v=1, g=54, c, o)={forprime(q=p+1, , if(p+g!=p=q, next, q!=o+2*g, c=2, c++>3, v&& print1(o-g", "); n--||break); o=q-g); o-g} \\ Can be used as next(p)=A259224(1,p+1) to get the next term, e.g.:
p=0; A259224_vec=vector(10,i,p=A259224(1,p+1)) \\ Will be slow! - M. F. Hasler, Oct 26 2018

A333383 First index of weakly increasing prime quartets.

Original entry on oeis.org

1, 2, 7, 13, 14, 22, 28, 35, 38, 45, 49, 54, 60, 64, 69, 70, 75, 78, 85, 89, 95, 104, 109, 116, 117, 122, 123, 144, 148, 152, 155, 159, 160, 163, 164, 173, 178, 182, 183, 184, 187, 194, 195, 198, 201, 206, 212, 215, 218, 219, 225, 226, 230, 236, 237, 238, 244

Offset: 1

Gus Wiseman, May 14 2020

nonn

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) <= g(k + 1) <= g(k + 2).

			The first 10 weakly increasing prime quartets:
    2   3   5   7
    3   5   7  11
   17  19  23  29
   41  43  47  53
   43  47  53  59
   79  83  89  97
  107 109 113 127
  149 151 157 163
  163 167 173 179
  197 199 211 223
For example, 43 is the 14th prime, and the primes (43,47,53,59) have differences (4,6,6), which are weakly increasing, so 14 is in the sequence.

Prime gaps are A001223.
Second prime gaps are A036263.
Strictly decreasing prime quartets are A054804.
Strictly increasing prime quartets are A054819.
Equal prime quartets are A090832.
Weakly increasing prime quartets are A333383 (this sequence).
Weakly decreasing prime quartets are A333488.
Unequal prime quartets are A333490.
Partially unequal prime quartets are A333491.
Positions of adjacent equal prime gaps are A064113.
Positions of strict ascents in prime gaps are A258025.
Positions of strict descents in prime gaps are A258026.
Positions of adjacent unequal prime gaps are A333214.
Positions of weak ascents in prime gaps are A333230.
Positions of weak descents in prime gaps are A333231.
Indices of weakly increasing rows of A066099 are A225620.
Lengths of maximal weakly increasing subsequences of prime gaps: A333215.
Lengths of maximal strictly decreasing subsequences of prime gaps: A333252.
Cf. A000040, A006560, A031217, A054800, A059044, A084758, A089180, A333253.

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x<=z-y<=t-z:>PrimePi[x]]

A333490 First index of unequal prime quartets.

Original entry on oeis.org

7, 8, 10, 11, 13, 17, 18, 19, 20, 22, 23, 24, 28, 30, 31, 32, 34, 40, 42, 44, 47, 49, 50, 51, 52, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 75, 76, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 94, 95, 96, 97, 98, 99, 104, 111, 112, 113, 114, 115, 116, 119

Offset: 1

Gus Wiseman, May 15 2020

nonn

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k), g(k + 1), and g(k + 2) are all different.

			The first 10 unequal prime quartets:
  17  19  23  29
  19  23  29  31
  29  31  37  41
  31  37  41  43
  41  43  47  53
  59  61  67  71
  61  67  71  73
  67  71  73  79
  71  73  79  83
  79  83  89  97
For example, 83 is the 23rd prime, and the primes (83,89,97,101) have differences (6,8,4), which are all distinct, so 23 is in the sequence.

Primes are A000040.
Prime gaps are A001223.
Second prime gaps are A036263.
Indices of unequal rows of A066099 are A233564.
Lengths of maximal anti-run subsequences of prime gaps are A333216.
Lengths of maximal runs of prime gaps are A333254.
Maximal anti-runs in standard compositions are counted by A333381.
Indices of anti-run rows of A066099 are A333489.
Strictly decreasing prime quartets are A054804.
Strictly increasing prime quartets are A054819.
Equal prime quartets are A090832.
Weakly increasing prime quartets are A333383.
Weakly decreasing prime quartets are A333488.
Unequal prime quartets are A333490 (this sequence).
Partially unequal prime quartets are A333491.
Positions of adjacent equal prime gaps are A064113.
Positions of strict ascents in prime gaps are A258025.
Positions of strict descents in prime gaps are A258026.
Positions of adjacent unequal prime gaps are A333214.
Positions of weak ascents in prime gaps are A333230.
Positions of weak descents in prime gaps are A333231.
Cf. A006560, A031217, A054800, A059044, A084758, A089180, A124767, A333215.

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x!=z-y!=t-z:>PrimePi[x]]

A333491 First index of partially unequal prime quartets.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 37, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Gus Wiseman, May 15 2020

nonn

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) != g(k + 1) != g(k + 2), but we may have g(k) = g(k + 2).

			The first 10 partially unequal prime quartets:
   5  7 11 13
   7 11 13 17
  11 13 17 19
  13 17 19 23
  17 19 23 29
  19 23 29 31
  23 29 31 37
  29 31 37 41
  31 37 41 43
  37 41 43 47

Primes are A000040.
Prime gaps are A001223.
Second prime gaps are A036263.
Indices of unequal rows of A066099 are A233564.
Lengths of maximal anti-runs of prime gaps are A333216.
Lengths of maximal runs of prime gaps are A333254.
Maximal anti-runs in standard compositions are counted by A333381.
Indices of anti-run rows of A066099 are A333489.
Strictly decreasing prime quartets are A054804.
Strictly increasing prime quartets are A054819.
Equal prime quartets are A090832.
Weakly increasing prime quartets are A333383.
Weakly decreasing prime quartets are A333488.
Unequal prime quartets are A333490.
Partially unequal prime quartets are A333491 (this sequence).
Positions of adjacent equal prime gaps are A064113.
Positions of strict ascents in prime gaps are A258025.
Positions of strict descents in prime gaps are A258026.
Positions of adjacent unequal prime gaps are A333214.
Positions of weak ascents in prime gaps are A333230.
Positions of weak descents in prime gaps are A333231.
Cf. A006560, A031217, A054800, A059044, A084758, A089180, A124767, A333215.

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x!=z-y&&z-y!=t-z:>PrimePi[x]]
PrimePi[#]&/@(Select[Partition[Prime[Range[90]],4,1],#[[2]]-#[[1]]!=#[[3]]-#[[2]]&&#[[3]]-#[[2]]!=#[[4]]-#[[3]]&][[;;,1]]) (* Harvey P. Dale, Aug 05 2025 *)

A333488 First index of weakly decreasing prime quartets.

Original entry on oeis.org

11, 15, 18, 24, 36, 39, 46, 47, 53, 54, 55, 58, 62, 72, 73, 87, 91, 101, 102, 106, 107, 110, 111, 114, 118, 127, 128, 129, 132, 146, 150, 157, 180, 186, 193, 199, 210, 217, 223, 228, 232, 239, 242, 259, 260, 263, 269, 270, 271, 274, 275, 282, 283, 284, 290

Offset: 1

Gus Wiseman, May 15 2020

nonn

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) >= g(k + 1) >= g(k + 2).

			The first 10 weakly decreasing prime quartets:
   31  37  41  43
   47  53  59  61
   61  67  71  73
   89  97 101 103
  151 157 163 167
  167 173 179 181
  199 211 223 227
  211 223 227 229
  241 251 257 263
  251 257 263 269
For example, 241 is the 53rd prime, and the primes (241,251,257,263) have differences (10,6,6), which are weakly decreasing, so 53 is in the sequence.

Prime gaps are A001223.
Second prime gaps are A036263.
Strictly decreasing prime quartets are A054804.
Strictly increasing prime quartets are A054819.
Equal prime quartets are A090832.
Weakly increasing prime quartets are A333383.
Weakly decreasing prime quartets are A333488 (this sequence).
Unequal prime quartets are A333490.
Partially unequal prime quartets are A333491.
Positions of adjacent equal prime gaps are A064113.
Positions of strict ascents in prime gaps are A258025.
Positions of strict descents in prime gaps are A258026.
Positions of adjacent unequal prime gaps are A333214.
Positions of weak ascents in prime gaps are A333230.
Positions of weak descents in prime gaps are A333231.
Indices of weakly decreasing rows of A066099 are A114994.
Lengths of maximal weakly decreasing subsequences of prime gaps: A333212.
Lengths of maximal strictly increasing subsequences of prime gaps: A333253.
Cf. A000040, A006560, A031217, A054800, A059044, A084758, A089180.

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x>=z-y>=t-z:>PrimePi[x]]

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A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression.

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A333216 Lengths of maximal subsequences without adjacent equal terms in the sequence of prime gaps.

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A052239 Smallest prime p in set of 4 consecutive primes in arithmetic progression with common difference 6n.

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A210683 Primes p such that p, p+60, p+120, p+180 are consecutive primes.

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A082080 Smallest balanced prime of order n.

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A259224 Initial primes in sets of 4 consecutive primes with common gap 54.

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A333383 First index of weakly increasing prime quartets.

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A333490 First index of unequal prime quartets.

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