A179243 -id:A179243 - cp's OEIS frontend

A052217 Numbers whose sum of digits is 3.

3, 12, 21, 30, 102, 111, 120, 201, 210, 300, 1002, 1011, 1020, 1101, 1110, 1200, 2001, 2010, 2100, 3000, 10002, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000, 20001, 20010, 20100, 21000, 30000, 100002, 100011, 100020, 100101

Offset: 1

Henry Bottomley, Feb 01 2000

From Joshua S.M. Weiner, Oct 19 2012: (Start)
Sequence is a representation of the "energy states" of "multiplex" notation of 3 quantum of objects in a juggling pattern.
0 = an empty site, or empty hand. 1 = one object resides in the site. 2 = two objects reside in the site. 3 = three objects reside in the site. (See A038447.) (End)
A007953(a(n)) = 3; number of repdigits = #{3,111} = A242627(3) = 2. - Reinhard Zumkeller, Jul 17 2014
Can be seen as a table whose n-th row holds the n-digit terms {10^(n-1) + 10^m + 10^k, 0 <= k <= m < n}, n >= 1. Row lengths are then (1, 3, 6, 10, ...) = n*(n+1)/2 = A000217(n). The first and the n last terms of row n are 10^(n-1) + 2 resp. 2*10^(n-1) + 10^k, 0 <= k < n. - M. F. Hasler, Feb 19 2020

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..84 from Vincenzo Librandi, terms 85..1140 from T. D. Noe)

Cf. A069521 to A069530, A069532 to A069537.
Cf. A007953, A218043 (subsequence).
Row n=3 of A245062.
Other digit sums: A011557 (1), A052216 (2), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Other bases: A014311 (binary), A226636 (ternary), A179243 (Zeckendorf).
Cf. A242614, A242627.
Cf. A003056, A002262 (triangular coordinates), A056556, A056557, A056558 (tetrahedral coordinates).

a052217 n = a052217_list !! (n-1)
a052217_list = filter ((== 3) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..100101] | &+Intseq(n) eq 3 ]; // Vincenzo Librandi, Mar 07 2013

Union[FromDigits/@Select[Flatten[Table[Tuples[Range[0,3],n],{n,6}],1],Total[#]==3&]] (* Harvey P. Dale, Oct 20 2012 *)
Select[Range[10^6], Total[IntegerDigits[#]] == 3 &] (* Vincenzo Librandi, Mar 07 2013 *)
Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 18]], {s, IntegerPartitions[3]}]]] (* T. D. Noe, Mar 08 2013 *)

isok(n) = sumdigits(n) == 3; \\ Michel Marcus, Dec 28 2015

apply( {A052217_row(n,s,t=-1)=vector(n*(n+1)\2,k,t++>s&&t=!s++;10^(n-1)+10^s+10^t)}, [1..5]) \\ M. F. Hasler, Feb 19 2020

from itertools import count, islice
def agen(): yield from (10**i + 10**j + 10**k for i in count(0) for j in range(i+1) for k in range(j+1))
print(list(islice(agen(), 40))) # Michael S. Branicky, May 14 2022

from math import comb, isqrt
from sympy import integer_nthroot
def A052217(n): return 10**((m:=integer_nthroot(6*n,3)[0])-(a:=n<=comb(m+2,3)))+10**((k:=isqrt(b:=(c:=n-comb(m-a+2,3))<<1))-((b<<2)<=(k<<2)*(k+1)+1))+10**(c-1-comb(k+(b>k*(k+1)),2)) # Chai Wah Wu, Dec 11 2024

T(n,k) = 10^(n-1) + 10^A003056(k) + 10^A002262(k) when read as a table with row lengths n*(n+1)/2, n >= 1, 0 <= k < n*(n+1)/2. - M. F. Hasler, Feb 19 2020

a(n) = 10^A056556(n-1) + 10^A056557(n-1) + 10^A056558(n-1). - Kevin Ryde, Apr 17 2021

Offset changed from 0 to 1 by Vincenzo Librandi, Mar 07 2013

A179242 Numbers that have two terms in their Zeckendorf representation.

Original entry on oeis.org

4, 6, 7, 9, 10, 11, 14, 15, 16, 18, 22, 23, 24, 26, 29, 35, 36, 37, 39, 42, 47, 56, 57, 58, 60, 63, 68, 76, 90, 91, 92, 94, 97, 102, 110, 123, 145, 146, 147, 149, 152, 157, 165, 178, 199, 234, 235, 236, 238, 241, 246, 254, 267, 288, 322, 378, 379, 380, 382, 385, 390

Offset: 1

Emeric Deutsch, Jul 05 2010

nonn

A007895(a(n)) = 2. - Reinhard Zumkeller, Mar 10 2013

			4 = 1+3;
6 = 1+5;
7 = 2+5;
9 = 1+8;
10 = 2+8;

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

Cf. A035517, A007895, A179243, A179244, A179245, A179246, A179247, A179248, A179249, A179250, A179251, A179252, A179253.

Cf. A000045.

import Data.List (inits)
a179242 n = a179242_list !! (n-1)
a179242_list = concatMap h $ drop 3 $ inits $ drop 2 a000045_list where
   h is = reverse $ map (+ f) fs where
          (f:_:fs) = reverse is
-- Reinhard Zumkeller, Mar 10 2013

with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i; for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: Q := {}: for i from fibonacci(5)-1 to 400 do if B(i) = 2 then Q := `union`(Q, {i}) else end if end do: Q;

f[n_] := (k = 1; ff = {}; While[(fi = Fibonacci[k]) <= n, AppendTo[ff, fi]; k++]; Drop[ff, 1]); z[n_] := If[n == 0, 0, r = n; s = {}; fr = f[n]; While[r > 0, lf = Last[fr]; If[lf <= r, r = r - lf; PrependTo[s, lf]]; fr = Drop[fr, -1]]; s]; Select[ Range[400], Length[z[#]] == 2 &] (* Jean-François Alcover, Sep 27 2011 *)
zeck = DigitCount[Select[Range[5000], BitAnd[#, 2*#] == 0&], 2, 1];
Position[zeck, 2] // Flatten (* Jean-François Alcover, Jan 25 2018 *)

from math import comb, isqrt
from sympy import fibonacci
def A179242(n): return fibonacci((m:=isqrt(n<<3)+1>>1)+3)+fibonacci(n+1-comb(m, 2)) # Chai Wah Wu, Apr 09 2025

A179253 Numbers k that have 13 terms in their Zeckendorf representation.

Original entry on oeis.org

196417, 271442, 300099, 311045, 315226, 316823, 317433, 317666, 317755, 317789, 317802, 317807, 317809, 317810, 392835, 421492, 432438, 436619, 438216, 438826, 439059, 439148, 439182, 439195, 439200, 439202, 439203, 467860, 478806, 482987

Offset: 1

Emeric Deutsch, Jul 05 2010

nonn

A007895(a(n)) = 13. - Reinhard Zumkeller, Mar 10 2013

			196417 = 1 + 3 + 8 + 21 + 55 + 144 + 377 + 987 + 2584 + 6765 + 17711 + 46368 + 121393;
271442 = 1 + 3 + 8 + 21 + 55 + 144 + 377 + 987 + 2584 + 6765 + 17711 + 46368 + 196418;

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A035517, A007895, A179242, A179243, A179244, A179245, A179246, A179247, A179248, A179249, A179250, A179251, A179252.

a179253 n = a179253_list !! (n-1)
a179253_list = filter ((== 13) . a007895) [1..]
-- Reinhard Zumkeller, Mar 10 2013

with(combinat): seq(add(fibonacci(2*k), k = 1 .. 13-m)+add(fibonacci(27-2*k+2), k = 1 .. m), m = 0 .. 13); # this program yields only the first 14 terms of the sequence
From R. J. Mathar, Jul 23 2010: (Start)
Lzto10 := proc(L) local i ; add( op(i,L)*combinat[fibonacci](i+1),i=1..nops(L) ) ; end proc:
zbits := proc(numbits,toset,upbits) local L,hibi ; if 2*toset-1 > numbits then return ; end if; if toset = 0 then L := [(seq(0,i=1..numbits)),op(upbits)] ; Lzto10(L); print(%) ; else for hibi from toset-1 to numbits -1 do if toset = 1 then procname(hibi,toset-1,[1,seq(0,i=1..numbits-hibi-1),op(upbits)]) ; else procname(hibi-1,toset-1,[0,1,seq(0,i=1..numbits-hibi-1),op(upbits)]) ; end if; end do; end if; return ; end proc:
ztot := 13 : for numbits from 2*ztot -1 to 50 do zbits(numbits-2,ztot-1,[0,1]) ; end do: (End)

Reap[For[m = 0; k = 1, k <= 10^8, k++, If[BitAnd[k, 2 k] == 0, m++; If[DigitCount[k, 2, 1] == 13, Print[m]; Sow[m]]]]][[2, 1]] (* Jean-François Alcover, Aug 20 2023 *)

More terms from R. J. Mathar, Jul 23 2010

A179245 Numbers that have 5 terms in their Zeckendorf representation.

Original entry on oeis.org

88, 122, 135, 140, 142, 143, 177, 190, 195, 197, 198, 211, 216, 218, 219, 224, 226, 227, 229, 230, 231, 266, 279, 284, 286, 287, 300, 305, 307, 308, 313, 315, 316, 318, 319, 320, 334, 339, 341, 342, 347, 349, 350, 352, 353, 354, 360, 362, 363, 365, 366, 367

Offset: 1

Emeric Deutsch, Jul 05 2010

A007895(a(n)) = 5. - Reinhard Zumkeller, Mar 10 2013

Numbers that are the sum of five non-consecutive Fibonacci numbers. Their Zeckendorf representation thus consists of five 1's with at least one 0 between each pair of 1's; for example, 122 is represented as 1001010101. - Alonso del Arte, Nov 17 2013

			88  = 55 + 21 + 8 + 3 + 1.
122 = 89 + 21 + 8 + 3 + 1.
135 = 89 + 34 + 8 + 3 + 1.
140 = 89 + 34 + 13 + 3 + 1.
142 = 89 + 34 + 13 + 5 + 1.
81 is not in the sequence because, although it is the sum of five Fibonacci numbers (81 = 5 + 8 + 13 + 21 + 34), its Zeckendorf representation only has three terms: 81 = 55 + 21 + 5.

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

Cf. A035517, A007895. Numbers that have m terms in their Zeckendorf representations: A179242 (m = 2), A179243 (m = 3), A179244 (m = 4), A179246 (m = 6), A179247 (m = 7), A179248 (m = 8), A179249 (m = 9), A179250 (m = 10), A179251 (m = 11), A179252 (m = 12), A179253 (m = 13).

a179245 n = a179245_list !! (n-1)
a179245_list = filter ((== 5) . a007895) [1..]
-- Reinhard Zumkeller, Mar 10 2013

with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i: for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: Q := {}: for i from fibonacci(11)-1 to 400 do if B(i) = 5 then Q := `union`(Q, {i}) else end if end do: Q;

zeck = DigitCount[Select[Range[3000], BitAnd[#, 2*#] == 0 &], 2, 1];
Position[zeck, 5] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

A048680(n)=my(k=1,s);while(n,if(n%2,s+=fibonacci(k++)); k++; n>>=1); s
[A048680(n)|n<-[1..100],hammingweight(n)==5] \\ Charles R Greathouse IV, Nov 17 2013

a(n) = A048680(A014313(n)). - Charles R Greathouse IV, Nov 17 2013

A179246 Numbers that have 6 terms in their Zeckendorf representation.

Original entry on oeis.org

232, 321, 355, 368, 373, 375, 376, 465, 499, 512, 517, 519, 520, 554, 567, 572, 574, 575, 588, 593, 595, 596, 601, 603, 604, 606, 607, 608, 698, 732, 745, 750, 752, 753, 787, 800, 805, 807, 808, 821, 826, 828, 829, 834, 836, 837, 839, 840, 841, 876, 889

Offset: 1

Emeric Deutsch, Jul 05 2010

nonn

A007895(a(n)) = 6. - Reinhard Zumkeller, Mar 10 2013

			232 = 144 + 55 + 21 +  8 + 3 + 1;
321 = 233 + 55 + 21 +  8 + 3 + 1;
355 = 233 + 89 + 21 +  8 + 3 + 1;
368 = 233 + 89 + 34 +  8 + 3 + 1;
373 = 233 + 89 + 34 + 13 + 3 + 1.

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

Cf. A035517, A007895, A179242, A179243, A179244, A179245, A179247, A179248, A179249, A179250, A179251, A179252, A179253.

a179246 n = a179246_list !! (n-1)
a179246_list = filter ((== 6) . a007895) [1..]
-- Reinhard Zumkeller, Mar 10 2013

with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i; for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: Q := {}: for i from fibonacci(13)-1 to 900 do if B(i) = 6 then Q := `union`(Q, {i}) else end if end do: Q;

zeck = DigitCount[Select[Range[12000], BitAnd[#, 2*#] == 0 &], 2, 1];
Position[zeck, 6] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

A179247 Numbers that have 7 terms in their Zeckendorf representation.

Original entry on oeis.org

609, 842, 931, 965, 978, 983, 985, 986, 1219, 1308, 1342, 1355, 1360, 1362, 1363, 1452, 1486, 1499, 1504, 1506, 1507, 1541, 1554, 1559, 1561, 1562, 1575, 1580, 1582, 1583, 1588, 1590, 1591, 1593, 1594, 1595, 1829, 1918, 1952, 1965, 1970, 1972, 1973

Offset: 1

Emeric Deutsch, Jul 05 2010

nonn

A007895(a(n)) = 7. - Reinhard Zumkeller, Mar 10 2013

			609 = 377+144+55+21+8+3+1;
842 = 610+144+55+21+8+3+1;
931 = 610+233+55+21+8+3+1;
965 = 610+233+89+21+8+3+1;
978 = 610+233+89+34+8+3+1.

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

Cf. A035517, A007895, A179242, A179243, A179244, A179245, A179246, A179248, A179249, A179250, A179251, A179252, A179253.

a179247 n = a179247_list !! (n-1)
a179247_list = filter ((== 7) . a007895) [1..]
-- Reinhard Zumkeller, Mar 10 2013

with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i: for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: Q := {}: for i from fibonacci(15)-1 to 2100 do if B(i) = 7 then Q := `union`(Q, {i}) else end if end do: Q;

zeck = DigitCount[Select[Range[4*10^4], BitAnd[#, 2*#] == 0 &], 2, 1];
Position[zeck, 7] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

A179248 Numbers that have 8 terms in their Zeckendorf representation.

Original entry on oeis.org

1596, 2206, 2439, 2528, 2562, 2575, 2580, 2582, 2583, 3193, 3426, 3515, 3549, 3562, 3567, 3569, 3570, 3803, 3892, 3926, 3939, 3944, 3946, 3947, 4036, 4070, 4083, 4088, 4090, 4091, 4125, 4138, 4143, 4145, 4146, 4159, 4164, 4166, 4167, 4172, 4174, 4175

Offset: 1

Emeric Deutsch, Jul 05 2010

nonn

A007895(a(n)) = 8. - Reinhard Zumkeller, Mar 10 2013

			1596 =  987+377+144+55+21+8+3+1;
2206 = 1597+377+144+55+21+8+3+1;
2439 = 1597+610+144+55+21+8+3+1;
2528 = 1597+610+233+55+21+8+3+1;
2562 = 1597+610+233+89+21+8+3+1.

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

Cf. A035517, A007895, A179242, A179243, A179244, A179245, A179246, A179247, A179249, A179250, A179251, A179252, A179253.

a179249 n = a179249_list !! (n-1)
a179249_list = filter ((== 9) . a007895) [1..]
-- Reinhard Zumkeller, Mar 10 2013

with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i: for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: Q := {}: for i from fibonacci(17)-1 to 5000 do if B(i) = 8 then Q := `union`(Q, {i}) else end if end do: Q;

zeck = DigitCount[Select[Range[10^5], BitAnd[#, 2*#] == 0 &], 2, 1];
Position[zeck, 8] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

A179249 Numbers that have 9 terms in their Zeckendorf representation.

Original entry on oeis.org

4180, 5777, 6387, 6620, 6709, 6743, 6756, 6761, 6763, 6764, 8361, 8971, 9204, 9293, 9327, 9340, 9345, 9347, 9348, 9958, 10191, 10280, 10314, 10327, 10332, 10334, 10335, 10568, 10657, 10691, 10704, 10709, 10711, 10712, 10801, 10835, 10848

Offset: 1

Emeric Deutsch, Jul 05 2010

nonn

A007895(a(n)) = 9. - Reinhard Zumkeller, Mar 10 2013

			4180 = 2584 +987+377+144+55+21+8+3+1;
5777 = 4181 +987+377+144+55+21+8+3+1;
6387 = 4181+1597+377+144+55+21+8+3+1;
6620 = 4181+1597+610+144+55+21+8+3+1;
6709 = 4181+1597+610+233+55+21+8+3+1.

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

Cf. A035517, A007895, A179242, A179243, A179244, A179245, A179246, A179247, A179248, A179250, A179251, A179252, A179253.

a179249 n = a179249_list !! (n-1)
a179249_list = filter ((== 9) . a007895) [1..]
-- Reinhard Zumkeller, Mar 10 2013

with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i: for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: Q := {}: for i from fibonacci(19)-1 to 10900 do if B(i) = 9 then Q := `union`(Q, {i}) else end if end do: Q;

zeck = DigitCount[Select[Range[4*10^5], BitAnd[#, 2*#] == 0 &], 2, 1];
Position[zeck, 9] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

A179250 Numbers that have 10 terms in their Zeckendorf representation.

Original entry on oeis.org

10945, 15126, 16723, 17333, 17566, 17655, 17689, 17702, 17707, 17709, 17710, 21891, 23488, 24098, 24331, 24420, 24454, 24467, 24472, 24474, 24475, 26072, 26682, 26915, 27004, 27038, 27051, 27056, 27058, 27059, 27669, 27902, 27991

Offset: 1

Emeric Deutsch, Jul 05 2010

A007895(a(n)) = 10. - Reinhard Zumkeller, Mar 10 2013

			10945=6765+2584+987+377+144+55+21+8+3+1;
15126=10946+2584+987+377+144+55+21+8+3+1;
16723=10946+4181+987+377+144+55+21+8+3+1;

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

Cf. A035517, A007895, A179242, A179243, A179244, A179245, A179246, A179247, A179248, A179249, A179251, A179252, A179253.

a179250 n = a179250_list !! (n-1)
a179250_list = filter ((== 10) . a007895) [1..]
-- Reinhard Zumkeller, Mar 10 2013

with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i: for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: Q := {}: for i from fibonacci(21)-1 to 28500 do if B(i) = 10 then Q := `union`(Q, {i}) else end if end do: Q;

zeck = DigitCount[Select[Range[2*10^6], BitAnd[#, 2*#] == 0 &], 2, 1];
Position[zeck, 10] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

A179251 Numbers that have 11 terms in their Zeckendorf representation.

Original entry on oeis.org

28656, 39602, 43783, 45380, 45990, 46223, 46312, 46346, 46359, 46364, 46366, 46367, 57313, 61494, 63091, 63701, 63934, 64023, 64057, 64070, 64075, 64077, 64078, 68259, 69856, 70466, 70699, 70788, 70822, 70835, 70840, 70842, 70843

Offset: 1

Emeric Deutsch, Jul 05 2010

nonn

A007895(a(n)) = 11. - Reinhard Zumkeller, Mar 10 2013

			28656=17711+6765+2584+987+377+144+55+21+8+3+1;
39602=28657+6765+2584+987+377+144+55+21+8+3+1;

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

Cf. A035517, A007895, A179242, A179243, A179244, A179245, A179246, A179247, A179248, A179249, A179250, A179252, A179253.

a179251 n = a179251_list !! (n-1)
a179251_list = filter ((== 11) . a007895) [1..]
-- Reinhard Zumkeller, Mar 10 2013

with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i: for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: Q := {}: for i from fibonacci(23)-1 to 73000 do if B(i) = 11 then Q := `union`(Q, {i}) else end if end do: Q;

Select[Range[6*10^6], BitAnd[#, 2*#] == 0&] // DigitCount[#, 2, 1]& // Position[#, 11]& // Flatten (* Jean-François Alcover, Feb 15 2018 *)

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A052217 Numbers whose sum of digits is 3.

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A179242 Numbers that have two terms in their Zeckendorf representation.

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A179253 Numbers k that have 13 terms in their Zeckendorf representation.

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A179245 Numbers that have 5 terms in their Zeckendorf representation.

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A179246 Numbers that have 6 terms in their Zeckendorf representation.

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A179247 Numbers that have 7 terms in their Zeckendorf representation.

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