A297113 -id:A297113 - cp's OEIS frontend

A329644 Möbius transform of A323244, the deficiency of A156552(n).

0, 1, 1, 1, 1, 2, 1, 4, -1, 3, 1, 5, 1, 14, 0, 0, 1, 9, 1, 12, -5, 16, 1, 8, -5, 44, 4, 5, 1, 2, 1, 24, 12, 80, -4, -4, 1, 254, -14, 0, 1, 22, 1, 47, 7, 224, 1, 24, -13, 19, 6, 83, 1, 12, -21, 44, -14, 746, 1, 14, 1, 1360, 20, -8, 8, 9, 1, 131, 252, 24, 1, 12, 1, 3836, 13, 149, -12, 71, 1, 56, -16, 5456, 1, -21, -74, 12248, -350, -40, 1

Offset: 1

Antti Karttunen, Nov 21 2019

sign

The first eleven zeros occur at n = 1, 15, 16, 40, 96, 119, 120, 160, 893, 2464, 6731. There are 3091 negative terms among the first 10000 terms.
Applying this function to the divisors of the first four terms of A324201 reveals the following pattern:
------------------------------------------------------------------------------------
A324201(n) divisors                             a(n) applied       Sum of positive
                                                 to each:          terms, A329610
        9: [1, 3, 9]                         -> [0, 1, -1]                     1
      125: [1, 5, 25, 125]                   -> [0, 1, -5, 4]                  5
   161051: [1, 11, 121, 1331, 14641, 161051] -> [0, 1, -29, 4, -240, 264]    269
410338673: [1, 17, 289, 4913, 83521, 1419857, 24137569, 410338673]
                             -> [0, 1, -125, 4, -1008, 1032, -5048, 5144]   6181
The positive and negative terms seem to alternate, and the fourth term (from case n=125 onward) is always 4. See also array A329637.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A008683, A036563, A156552, A297112, A297113, A323243, A323244, A324543, A329610, A329637, A329638, A329639, A329645, A329646.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));
A329644(n) = sumdiv(n,d,moebius(n/d)*A323244(d));

a(n) = Sum_{d|n} A008683(n/d) * A323244(d).
a(n) = Sum_{d|n} A008683(n/d) * (2*A156552(d) - A323243(d)).
a(1) = 0; for n > 1, a(n) = 2*A297112(n) - A324543(n) = 2^A297113(n) - A324543(n).
a(n) = A329642(n) - A329643(n).
For all n >= 1, a(A000040(n)^2) = A323244(A000040(n)^2)-1 = -A036563(n).
For all primes p, a(p^3) = A323244(p^3) - A323244(p^2) = 4.

A297168 Difference between A156552 and its Moebius transform: a(n) = A156552(n) - A297112(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 3, 2, 5, 0, 7, 0, 9, 6, 7, 0, 9, 0, 11, 10, 17, 0, 15, 4, 33, 6, 19, 0, 17, 0, 15, 18, 65, 12, 19, 0, 129, 34, 23, 0, 29, 0, 35, 14, 257, 0, 31, 8, 17, 66, 67, 0, 21, 20, 39, 130, 513, 0, 35, 0, 1025, 22, 31, 36, 53, 0, 131, 258, 33, 0, 39, 0, 2049, 18, 259, 24, 101, 0, 47, 14, 4097, 0, 59, 68, 8193, 514, 71, 0, 37, 40

Offset: 1

Antti Karttunen, Feb 27 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A008683, A033265, A156552, A297112, A297112, A297113, A297167, A297169, A300827.

With[{s = Array[Total@ MapIndexed[#1 2^(First@ #2 - 1) &, Flatten@ Map[ConstantArray[2^(PrimePi@ #1 - 1), #2] & @@ # &, FactorInteger@ #]] - Boole[# == 1]/2 &, 91]}, Table[-DivisorSum[n, MoebiusMu[n/#] s[[#]] &, # < n &], {n, Length@ s}]] (* Michael De Vlieger, Mar 13 2018 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A297112(n) = sumdiv(n,d,moebius(n/d)*A156552(d));
A297168(n) = (A156552(n)-A297112(n));
\\ Or alternatively as:
A297168(n) = -sumdiv(n,d,(dA156552(d));

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n,0,2^A297167(n));
A297168(n) = sumdiv(n,d,(dA297112(d)); \\ Antti Karttunen, Mar 13 2018

(define (A297168 n) (- (A156552 n) (A297112 n)))
(define (A297168 n) (if (= 1 n) 0 (- (A156552 n) (A000079 (A297167 n)))))

a(n) = -Sum_{d|n, dA008683(n/d)*A156552(d).
a(n) = Sum_{d|n, dA297112(d).
For n > 1, a(n) = Sum_{d|n, 1A033265(A156552(d)).
a(n) = A156552(n) - A297112(n).
a(1) = 0, for n > 1, a(n) = A156552(n) - 2^A297167(n).

A325170 Heinz numbers of integer partitions with origin-to-boundary graph-distance equal to 2.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 33, 34, 35, 36, 38, 39, 40, 44, 46, 48, 49, 51, 52, 54, 55, 56, 57, 58, 62, 65, 68, 69, 72, 74, 76, 77, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 104, 106, 108, 111, 112, 115, 116, 118, 119

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps East or South from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the minimum triangular partition contained inside the diagram.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   6: {1,2}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  25: {3,3}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}

Gus Wiseman, Young diagrams corresponding to the first 50 terms.

Cf. A001221, A001222, A006918, A056239, A065770, A112798, A174090, A257990, A297113, A325167, A325169.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Select[Range[200],otb[Reverse[primeMS[#]]]==2&]

A033265 Number of i such that d(i) >= d(i-1), where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4, 3, 3, 3, 3, 2, 3, 3, 4, 3, 3, 3, 4, 3, 4, 4, 5, 4, 4, 4, 4, 3, 4, 4, 4, 3, 3, 3, 4, 3, 4, 4, 5, 4, 4, 4, 4, 3, 4, 4, 5, 4, 4, 4, 5, 4, 5, 5, 6, 5, 5, 5, 5, 4, 5, 5, 5, 4, 4, 4, 5, 4, 5, 5, 5, 4, 4, 4, 4, 3, 4, 4, 5, 4, 4

Offset: 1

Clark Kimberling

			The base-2 representation of n=4 is 100 with d(0)=0, d(1)=0, d(2)=1. There are two rise-or-equal, one from d(0) to d(1) and one from d(1) to d(2), so a(4)=2. - _R. J. Mathar_, Oct 16 2015

Antti Karttunen, Table of n, a(n) for n = 1..65537
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences related to binary expansion of n

Cf. A014081, A023416, A037800, A037809, A005940, A156552, A297113, A364567 [= 2^a(n)].

A033265 := proc(n)
    a := 0 ;
    dgs := convert(n,base,2);
    for i from 2 to nops(dgs) do
        if op(i,dgs)>=op(i-1,dgs) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 16 2015

A033265(n) = { my(i=0); while(n>1, if((n%4)!=1, i++); n >>= 1); (i); }; \\ Antti Karttunen, Aug 06 2023

From Ralf Stephan, Oct 05 2003: (Start)
a(0) = 0, a(2n) = a(n) + 1, a(2n+1) = a(n) + [n odd].
a(n) = A014081(n) + A023416(n).
G.f.: 1/(1-x) * Sum_{k>=0} (t^2 + t^3 + t^4)/((1+t)*(1+t^2)), t=x^2^k. (End)
a(n) = -1 + A297113(A005940(1+n)). - Antti Karttunen, Dec 30 2017

Sign in Name corrected by R. J. Mathar, Oct 16 2015

A325168 Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 121

Offset: 0

Gus Wiseman, Apr 05 2019

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps left or down from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the maximum triangular partition contained inside it.

			The a(3) = 1 through a(10) = 16 partitions:
  (21)  (22)   (32)    (33)     (43)      (44)       (54)        (55)
        (31)   (41)    (42)     (52)      (53)       (63)        (64)
        (211)  (221)   (51)     (61)      (62)       (72)        (73)
               (311)   (222)    (511)     (71)       (81)        (82)
               (2111)  (411)    (2221)    (611)      (711)       (91)
                       (2211)   (4111)    (2222)     (6111)      (811)
                       (3111)   (22111)   (5111)     (22221)     (7111)
                       (21111)  (31111)   (22211)    (51111)     (22222)
                                (211111)  (41111)    (222111)    (61111)
                                          (221111)   (411111)    (222211)
                                          (311111)   (2211111)   (511111)
                                          (2111111)  (3111111)   (2221111)
                                                     (21111111)  (4111111)
                                                                 (22111111)
                                                                 (31111111)
                                                                 (211111111)

Colin Barker, Table of n, a(n) for n = 0..1000
N. Guru Sharan, Rook decomposition of the Partition function, arXiv:2507.20260 [math.CO], 2025. See p. 5.
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 10.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A006918, A065770, A115994, A117485, A257990, A297113, A325135, A325166, A325169, A325170.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otb[#]==2&]],{n,0,30}]

concat([0,0,0], Vec(x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)) + O(x^80))) \\ Colin Barker, Apr 08 2019

From Colin Barker, Apr 08 2019: (Start)
G.f.: x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>7.
a(n) = 2*n - 4 for n>4 and even.
a(n) = 2*n - 5 for n>4 and odd.
(End)

A325183 Heinz number of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 10, 10, 11, 10, 13, 14, 15, 7, 17, 15, 19, 14, 21, 22, 23, 14, 21, 26, 21, 22, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 22, 41, 42, 43, 26, 42, 46, 47, 22, 55, 42, 51, 34, 53, 35, 55, 26, 57, 58, 59, 42, 61, 62, 66, 13, 65, 66, 67

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

The k-th part of the origin-to-boundary partition of a Young diagram is the number of squares graph-distance k from the lower-right boundary.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with origin-to-boundary graph-distances
  4 4 4 3 2 1
  3 3 3 2 1
  2 2 2 1 1
  1 1 1
giving the origin-to-boundary partition (7,5,4,3) with Heinz number 6545, so a(7865) = 6545.

Eric Weisstein's World of Mathematics, Graph Distance.

The only terms appearing only once are the primorials A002110.
The union consists of all squarefree numbers A005117.
Cf. A000245, A056239, A065770, A112798, A174090, A297113.
Cf. A325166, A325167, A325169, A325184, A325188, A325189, A325195.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
ptnmat[ptn_]:=PadRight[(ConstantArray[1,#]&)/@Sort[ptn,Greater],{Length[ptn],Max@@ptn}+1];
corpos[mat_]:=ReplacePart[mat,Select[Position[mat,1],Times@@Extract[mat,{#+{1,0},#+{0,1}}]==0&]->0];
Table[Times@@Prime/@If[n==1,{},-Differences[Map[Total,Drop[FixedPointList[corpos,ptnmat[primeptn[n]]],-1],2]]],{n,30}]

A297155 a(1) = a(2) = 0, after which, a(n) = 1+a(n/2) if n is of the form 4k+2, otherwise a(n) = a(A252463(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 27 2017

nonn

Consider the binary tree illustrated in A005940: If we start from any vertex containing n, computing successive iterations of A252463 until 1 is reached, a(n) gives the number of the numbers of the form 4k+2 (with k >= 1) encountered on the path (i.e., excluding 2 from the count but including the starting n if it is of the form 4k+2).

Antti Karttunen, Table of n, a(n) for n = 1..12000
Index entries for sequences computed from indices in prime factorization

Cf. A001221, A005940, A037800, A156552, A252463, A252464, A297113.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A297155(n) = if(n<=2,0,if(n%2,A297155(A064989(n)),(2==(n%4))+A297155(n/2)));

;; With memoization-macro definec.
(definec (A297155 n) (cond ((<= n 2) 0) ((= 2 (modulo n 4)) (+ 1 (A297155 (/ n 2)))) (else (A297155 (A252463 n)))))

a(n) = A252464(n) - A297113(n).
a(n) = A037800(A156552(n)).
a(n) = A001221(n) - 1 for all n > 1. - Velin Yanev, Mar 26 2019

A364557 Möbius transform of A005941.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 8, 4, 4, 4, 16, 4, 32, 8, 4, 8, 64, 4, 128, 8, 8, 16, 256, 8, 8, 32, 8, 16, 512, 4, 1024, 16, 16, 64, 8, 8, 2048, 128, 32, 16, 4096, 8, 8192, 32, 8, 256, 16384, 16, 16, 8, 64, 64, 32768, 8, 16, 32, 128, 512, 65536, 8, 131072, 1024, 16, 32, 32, 16, 262144, 128, 256, 8, 524288, 16, 1048576, 2048

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A005941, A008683, A297112, A297113, A297167, A364558.

A364557(n) = if(1==n, 1, 2^(primepi(vecmax(factor(n)[, 1]))+(bigomega(n)-omega(n))-1));

A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
A364557(n) = sumdiv(n,d,moebius(n/d)*A005941(d));

from sympy import factorint, primepi
def A364557(n): return 1<1 else 1 # Chai Wah Wu, Jul 29 2023

a(n) = Sum_{d|n} A008683(n/d) * A005941(d).

a(1) = 1; for n > 1, a(n) = A297112(n) = 2^(A297113(n)-1) = 2^A297167(n).

A325167 Heinz number of the internal portion of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 3, 1, 1, 4, 1, 2, 3, 2, 1, 2, 3, 2, 4, 2, 1, 6, 1, 1, 3, 2, 5, 4, 1, 2, 3, 2, 1, 6, 1, 2, 6, 2, 1, 2, 5, 6, 3, 2, 1, 8, 5, 2, 3, 2, 1, 6, 1, 2, 6, 1, 5, 6, 1, 2, 3, 10, 1, 4, 1, 2, 9, 2, 7, 6, 1, 2, 8, 2, 1, 6, 5, 2, 3

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The internal portion of an integer partition consists of all squares in the Young diagram that have a square both directly below and directly to the right.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with internal portion
  o o o o o
  o o o o
  o o o
which is the partition (5,4,3), with Heinz number 385, so a(7865) = 385.

Cf. A052126, A056239, A064989, A065770, A093641, A112798, A257990, A297113, A325133, A325166, A325169.

A329638 Sum of A329644(d) for all such divisors d of n for which that value is positive. Here A329644 is the Möbius transform of A323244, the deficiency of A156552(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 6, 1, 5, 1, 10, 1, 16, 2, 6, 1, 13, 1, 18, 2, 18, 1, 22, 1, 46, 5, 22, 1, 10, 1, 30, 14, 82, 2, 19, 1, 256, 2, 22, 1, 41, 1, 66, 9, 226, 1, 46, 1, 24, 8, 130, 1, 29, 2, 70, 2, 748, 1, 42, 1, 1362, 22, 30, 10, 42, 1, 214, 254, 44, 1, 43, 1, 3838, 15, 406, 2, 120, 1, 78, 5, 5458, 1, 52, 2, 12250, 2, 70, 1, 26, 2, 934

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A156552, A323243, A323244, A329639, A329641, A329644.

Cf. also A318878.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323243(n) = if(1==n,0,sigma(A156552(n)));
A324543(n) = sumdiv(n,d,moebius(n/d)*A323243(d));
A297113(n) = if(1==n, 0, (primepi(vecmax(factor(n)[, 1])) + (bigomega(n)-omega(n))));
A329644(n) = if(1==n,0, 2^A297113(n) - A324543(n));
A329638(n) = sumdiv(n,d,if((d=A329644(d))>0,d,0));

a(n) = Sum_{d|n} [A329644(d) > 0] * A329644(d), where [ ] is Iverson bracket.

a(n) = A323244(n) + A329639(n).

cp's OEIS Frontend

A329644 Möbius transform of A323244, the deficiency of A156552(n).

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A297168 Difference between A156552 and its Moebius transform: a(n) = A156552(n) - A297112(n).

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A325170 Heinz numbers of integer partitions with origin-to-boundary graph-distance equal to 2.

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A033265 Number of i such that d(i) >= d(i-1), where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

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A325168 Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.

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A325183 Heinz number of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.

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A297155 a(1) = a(2) = 0, after which, a(n) = 1+a(n/2) if n is of the form 4k+2, otherwise a(n) = a(A252463(n)).

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Formula

A364557 Möbius transform of A005941.

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