A073445 -id:A073445 - cp's OEIS frontend

A377035 Antidiagonal-sums of the absolute value of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

4, 8, 10, 12, 14, 18, 21, 28, 34, 40, 47, 74, 96, 110, 138, 286, 715, 2393, 8200, 25731, 72468, 184716, 431575, 934511, 1892267, 3605315, 6494464, 11116110, 18134549, 28348908, 42701927, 62290660, 88313069, 120999433, 159769475, 221775851, 483797879

Offset: 1

Gus Wiseman, Oct 18 2024

nonn

			The fourth antidiagonal of A377033 is (9, 1, -1, -1), so a(4) = 12.

The version for prime instead of composite is A376681, absolute version of A140119.
The version for noncomposite is A376684, absolute version of A376683.
This is the antidiagonal-sums of absolute value of the array A377033.
For squarefree instead of composite we have A377040, absolute version of A377039.
For nonsquarefree instead of composite we have A377048, absolute version of A377047.
For prime-power instead of composite we have A377053, absolute version of A377052.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, seconds A036263.
A002808 lists the composite numbers, differences A073783, seconds A073445.
A008578 lists the noncomposites, differences A075526.
Cf. A018252, A065310, A065890, A333254, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377036.

q=Select[Range[120],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}];
Total/@Table[Abs[t[[j,i-j+1]]],{i,Length[q]/2},{j,i}]

A376653 Sorted positions of first appearances in the second differences of consecutive prime-powers inclusive (A000961).

Original entry on oeis.org

1, 4, 5, 10, 12, 18, 25, 45, 47, 48, 60, 68, 69, 71, 80, 118, 121, 178, 179, 199, 206, 207, 216, 244, 245, 304, 325, 327, 402, 466, 484, 605, 801, 880, 939, 1033, 1055, 1077, 1234, 1281, 1721, 1890, 1891, 1906, 1940, 1960, 1962, 2257, 2290, 2410, 2880, 3150

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with first appearances (A376653):
  1, 4, 5, 10, 12, 18, 25, 45, 47, 48, 60, 68, 69, 71, 80, 118, 121, 178, 179, 199, ...

For first differences we had A057820, sorted firsts A376340(n)+1 (except first term).
These are the sorted positions of first appearances in A376596.
The exclusive version is a(n) - 1 = A376654(n), except first term.
For squarefree instead of prime-power we have A376655.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A053707, A069623, A093555, A174965, A251092, A333254, A361102.

q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A378972 Second differences of the strict partition numbers A000009.

Original entry on oeis.org

0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, 3, 2, 3, 4, 3, 4, 6, 4, 6, 8, 6, 9, 10, 9, 12, 14, 13, 16, 19, 18, 22, 26, 24, 30, 34, 34, 40, 45, 46, 53, 60, 62, 70, 79, 82, 93, 104, 108, 122, 136, 142, 160, 176, 186, 208, 228, 243, 268

Offset: 0

Gus Wiseman, Dec 14 2024

sign

			The strict partition numbers begin (A000009):
  1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, ...
with differences (A087897 without first term):
  0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, ...
with differences (a(n)):
  0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, ...

For primes we have A036263.
The version for partitions is A053445.
For composites we have A073445.
For squarefree numbers we have A376590.
For nonsquarefree numbers we have A376593.
For powers of primes (inclusive) we have A376596.
For non powers of primes (inclusive) we have A376599.
Second row of A378622. See also:
- A293467 gives first column (up to sign).
- A377285 gives position of first zero in each row.
- A378970 gives row-sums.
- A378971 gives absolute value row-sums.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A225485, A325242, A325268.

Differences[Table[PartitionsQ[n],{n,0,100}],2]

A379300 Number of prime indices of n that are composite.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1

Offset: 1

Gus Wiseman, Dec 25 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 39 are {2,6}, so a(39) = 1.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 3.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.

Positions of first appearances are A000420.
Positions of zero are A302540, counted by A034891 (strict A036497).
Positions of one are A379301, counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A023895, A113646, A175804, A204389, A320629, A376680, A378373, A378456.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],CompositeQ]],{n,100}]

Totally additive with a(prime(k)) = A066247(k).

A376560 Points of upward concavity in the sequence of perfect-powers (A001597). Positives of A376559.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 10, 11, 12, 15, 16, 17, 20, 22, 23, 26, 27, 28, 31, 32, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 46, 47, 48, 49, 50, 53, 54, 55, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91

Offset: 1

Gus Wiseman, Sep 30 2024

nonn

These are points at which the second differences are positive.
Perfect-powers (A001597) are numbers with a proper integer root.
Note that, for some sources, upward concavity is negative curvature.

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
with positive positions (A376560):
  1, 3, 4, 6, 8, 9, 10, 11, 12, 15, 16, 17, 20, 22, 23, 26, 27, 28, 31, 32, 33, 34, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Points of upward concavity in the perfect-powers.

The version for A000002 is A022297, complement A025505. See also A054354, A376604.
For first differences we have A053289, union A023055, firsts A376268, A376519.
For primes instead of perfect-powers we have A258025.
These are positions of positive terms in A376559.
For downward concavity we have A376561 (probably the complement).
A001597 lists the perfect-powers.
A064113 lists positions of adjacent equal prime gaps.
A333254 gives run-lengths of differences between consecutive primes.
Second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A006549, A025475, A030173, A045542, A052410, A053707, A054819, A069623, A174965, A216765, A251092, A376308.

N:= 10^6: # to use perfect powers <= N
S:= {1,seq(seq(i^j,j=2..floor(log[i](N))),i=2..isqrt(N))}:
L:= sort(convert(S,list)):
DL:= L[2..-1]-L[1..-2]:
D2L:= DL[2..-1]-DL[1..-2]:
select(i -> D2L[i]>0, [$1..nops(D2L)]); # Robert Israel, Dec 01 2024

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],1]

A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

Original entry on oeis.org

3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, 205, 206, 215, 243, 244, 303, 324, 326, 401, 465, 483, 604, 800, 879, 938, 1032, 1054, 1076, 1233, 1280, 1720, 1889, 1890, 1905, 1939, 1959, 1961, 2256, 2289, 2409, 2879, 3149

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

			The prime-powers exclusive (A246655) are:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, ...
with first differences (A057820 except first term) :
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 3, ...
with first differences (A376596 except first term):
  0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, ...
with first appearances (A376654):
  1, 3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, ...

For first differences we have A376340.
These are the sorted positions of first appearances in A376596 except first term.
The inclusive version is a(n) + 1 = A376653(n), except first term.
For squarefree instead of prime-power we have A376655.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A053707, A057820, A064113, A069623, A093555, A174965, A333254, A336416, A361102.

q=Differences[Select[Range[1000],PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A379303 Number of strict integer partitions of n with a unique composite part.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 3, 6, 6, 8, 10, 10, 13, 15, 17, 20, 22, 24, 28, 31, 36, 40, 44, 50, 55, 62, 70, 75, 83, 89, 97, 108, 115, 128, 136, 146, 161, 172, 188, 203, 215, 233, 249, 269, 291, 309, 331, 353, 376, 405, 433, 459, 490, 518, 554, 592, 629, 670, 705

Offset: 0

Gus Wiseman, Dec 25 2024

nonn

			The a(4) = 1 through a(11) = 8 partitions:
  (4)  (4,1)  (6)    (4,3)    (8)      (9)      (10)       (6,5)
              (4,2)  (6,1)    (6,2)    (5,4)    (8,2)      (7,4)
                     (4,2,1)  (4,3,1)  (6,3)    (9,1)      (8,3)
                                       (8,1)    (5,4,1)    (9,2)
                                       (4,3,2)  (6,3,1)    (10,1)
                                       (6,2,1)  (4,3,2,1)  (5,4,2)
                                                           (6,3,2)
                                                           (8,2,1)

If no parts are composite we have A036497,  non-strict A034891 (ranks A302540).
If all parts are composite we have A204389, non-strict A023895 (ranks A320629).
The non-strict version is A379302, ranks A379301 (positions of 1 in A379300).
For a unique prime we have A379305, non-strict A379304 (ranks A331915).
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Cf. A000070, A000586, A000607, A002095, A038348, A096258, A113646, A376680, A379308, A379309, A379314, A379315.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?CompositeQ]==1&]],{n,0,30}]

A376600 Inflection or undulation points in the sequence of non-prime-powers inclusive (A024619).

Original entry on oeis.org

2, 7, 9, 10, 11, 14, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 39, 41, 43, 44, 45, 47, 48, 50, 51, 52, 55, 56, 57, 58, 59, 62, 64, 66, 68, 70, 73, 74, 75, 76, 77, 80, 86, 87, 88, 90, 92, 93, 94, 95, 96, 97, 98, 100, 102, 103, 104, 107, 108, 109, 112, 114, 116

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376599) are zero.

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, add 1 to all terms.

			The non-prime-powers inclusive are (A024619):
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...
with zeros at (A376600):
  2, 7, 9, 10, 11, 14, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 39, 41, 43, 44, ...

Gus Wiseman, Inflection and undulation points in the non-prime-powers inclusive.

For first differences we had A375735, ones A375713(n)-1.
These are the zeros of A376599.
The complement is A376601.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A024619/A361102 list non-prime-powers inclusive.
A321346/A321378 count integer partitions into non-prime-powers, factorizations A322452.
For non-prime-powers: A375735/A375708 (first differences), A376599 (second differences), A376601 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power).
Cf. A025475, A053707, A057820, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Join@@Position[Differences[Select[Range[100], !(#==1||PrimePowerQ[#])&],2],0]

A376655 Sorted positions of first appearances in the second differences of consecutive squarefree numbers (A005117).

Original entry on oeis.org

1, 2, 3, 5, 6, 30, 61, 150, 514, 1025, 5153, 13390, 13391, 131964, 502651, 664312, 4387185, 5392318, 20613826

Offset: 1

Gus Wiseman, Oct 07 2024

Warning: Do not confuse with A246655 (prime-powers exclusive).

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, ...
with sorted first appearances at (A376655):
  1, 2, 3, 5, 6, 30, 61, 150, 514, 1025, 5153, 13390, 13391, ...

For first differences we had A376311 (first appearances in A076259).
These are the sorted positions of first appearances in A376590.
For prime-powers instead of squarefree numbers we have A376653/A376654.
For primes instead of squarefree numbers we have A376656.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
For squarefree: A376591 (inflections and undulations), A376592 (nonzero curvature).
Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A251092, A373198, A376342.

q=Differences[Select[Range[1000],SquareFreeQ],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

a(14)-a(19) from Chai Wah Wu, Oct 07 2024

A377036 First term of the n-th differences of the composite numbers. Inverse zero-based binomial transform of A002808.

Original entry on oeis.org

4, 2, 0, -1, 2, -2, 0, 4, -8, 8, 0, -16, 32, -32, -1, 78, -233, 687, -2363, 8160, -25670, 72352, -184451, 430937, -933087, 1888690, -3597221, 6479696, -11086920, 18096128, -28307626, 42644791, -62031001, 86466285, -110902034, 110907489, -52325, -483682930

Offset: 0

Gus Wiseman, Oct 18 2024

sign

The version for prime instead of composite is A007442.
For noncomposite numbers we have A030016.
This is the first column (n=1) of A377033.
For row-sums we have A377034, absolute version A377035.
First zero positions are A377037, cf. A376678, A376855, A377042, A377050, A377055.
For squarefree instead of composite we have A377041, nonsquarefree A377049.
For prime-power instead of composite we have A377054.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, seconds A036263.
A002808 lists the composite numbers, differences A073783, seconds A073445.
A008578 lists the noncomposites, differences A075526.
Cf: A018252, A065310, A065890, A140119, A173390, A333214, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

q=Select[Range[100],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[1+k]],{k,0,j}],{j,0,Length[q]-1}]

The inverse zero-based binomial transform of a sequence (q(0), q(1), ..., q(m)) is the sequence p given by:

p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)

cp's OEIS Frontend

A377035 Antidiagonal-sums of the absolute value of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

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A376653 Sorted positions of first appearances in the second differences of consecutive prime-powers inclusive (A000961).

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A378972 Second differences of the strict partition numbers A000009.

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A379300 Number of prime indices of n that are composite.

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Formula

A376560 Points of upward concavity in the sequence of perfect-powers (A001597). Positives of A376559.

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Maple

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A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

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A379303 Number of strict integer partitions of n with a unique composite part.

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A376600 Inflection or undulation points in the sequence of non-prime-powers inclusive (A024619).

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A376655 Sorted positions of first appearances in the second differences of consecutive squarefree numbers (A005117).

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