A299701 -id:A299701 - cp's OEIS frontend

A301934 Number of positive subset-sum trees of weight n.

1, 3, 14, 85, 586, 4331, 33545, 268521, 2204249

Offset: 1

Gus Wiseman, Mar 28 2018

A positive subset-sum tree with root x is either the symbol x itself, or is obtained by first choosing a positive subset-sum x <= (y_1,...,y_k) with k > 1 and then choosing a positive subset-sum tree with root y_i for each i = 1...k. The weight is the sum of the leaves. We write positive subset-sum trees in the form rootsum(branch,...,branch). For example, 4(1(1,3),2,2(1,1)) is a positive subset-sum tree with composite 4(1,1,1,2,3) and weight 8.

			The a(3) = 14 positive subset-sum trees:
3           3(1,2)       3(1,1,1)     3(1,2(1,1))
2(1,2)      2(1,1,1)     2(1,1(1,1))  2(1(1,1),1)  2(1,2(1,1))
1(1,2)      1(1,1,1)     1(1,1(1,1))  1(1(1,1),1)  1(1,2(1,1))

Cf. A000108, A000712, A108917, A122768, A262671, A262673, A275972, A276024, A284640, A299701, A301854, A301855, A301856, A301935.

A325800 Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.

Original entry on oeis.org

3, 10, 28, 66, 88, 156, 208, 306, 340, 408, 544, 570, 684, 760, 912, 966, 1216, 1242, 1288, 1380, 1656, 1840, 2208, 2436, 2610, 2900, 2944, 3132, 3248, 3480, 3906, 4092, 4176, 4340, 4640, 4650, 5022, 5208, 5456, 5568, 5580, 6200, 6696, 6944, 7326, 7424, 7440

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from A325793 in lacking 70.
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, with sum A056239(n). A subset-sum of an integer partition is any sum of a submultiset of it.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose sum is equal to their number of distinct subset-sums. The enumeration of these partitions by sum is given by A126796 interlaced with zeros.

			340 has prime indices {1,1,3,7} which sum to 12 and have 12 distinct subset-sums: {0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12}, so 340 is in the sequence.
The sequence of terms together with their prime indices begins:
     3: {2}
    10: {1,3}
    28: {1,1,4}
    66: {1,2,5}
    88: {1,1,1,5}
   156: {1,1,2,6}
   208: {1,1,1,1,6}
   306: {1,2,2,7}
   340: {1,1,3,7}
   408: {1,1,1,2,7}
   544: {1,1,1,1,1,7}
   570: {1,2,3,8}
   684: {1,1,2,2,8}
   760: {1,1,1,3,8}
   912: {1,1,1,1,2,8}
   966: {1,2,4,9}
  1216: {1,1,1,1,1,1,8}
  1242: {1,2,2,2,9}
  1288: {1,1,1,4,9}
  1380: {1,1,2,3,9}

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 1's in A325799.
Includes A239885 except for 1.
Cf. A002033, A056239, A108917, A112798, A126796, A299701, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325801, A325802.

filter:= proc(n) local F,t,S,i,r;
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], ifactors(n)[2]);
  S:= {0}:
  for t in F do
   S:= map(s -> seq(s + i*t[1],i=0..t[2]),S);
  od;
  nops(S) = add(t[1]*t[2],t=F)
end proc:
select(filter, [$1..10000]); # Robert Israel, Oct 30 2024

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],hwt[#]==Length[Union[hwt/@Divisors[#]]]&]

A056239(a(n)) = A299701(a(n)) = A304793(a(n)) + 1.

A371955 Numbers with triquanimous prime indices.

Original entry on oeis.org

8, 27, 36, 48, 64, 125, 150, 180, 200, 216, 240, 288, 320, 343, 384, 441, 490, 512, 567, 588, 630, 700, 729, 756, 784, 810, 840, 900, 972, 1000, 1008, 1080, 1120, 1200, 1296, 1331, 1344, 1440, 1600, 1694, 1728, 1792, 1815, 1920, 2156, 2178, 2197, 2304, 2310

Offset: 1

Gus Wiseman, Apr 19 2024

nonn

A finite multiset of numbers is defined to be triquanimous iff it can be partitioned into three multisets with equal sums.

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 terms together with their prime indices begin:
     8: {1,1,1}
    27: {2,2,2}
    36: {1,1,2,2}
    48: {1,1,1,1,2}
    64: {1,1,1,1,1,1}
   125: {3,3,3}
   150: {1,2,3,3}
   180: {1,1,2,2,3}
   200: {1,1,1,3,3}
   216: {1,1,1,2,2,2}
   240: {1,1,1,1,2,3}
   288: {1,1,1,1,1,2,2}
   320: {1,1,1,1,1,1,3}
   343: {4,4,4}
   384: {1,1,1,1,1,1,1,2}
   441: {2,2,4,4}
   490: {1,3,4,4}
   512: {1,1,1,1,1,1,1,1,1}
   567: {2,2,2,2,4}
   588: {1,1,2,4,4}

Robert Israel, Table of n, a(n) for n = 1..500

These are the Heinz numbers of the partitions counted by A002220.
For biquanimous we have A357976, counted by A002219.
For non-biquanimous we have A371731, counted by A371795, even case A006827.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A371783 counts k-quanimous partitions.
Cf. A000005, A002221, A035470, A064914, A232466, A299701, A321142, A321454, A321455, A326534, A357879, A371781.

tripart:= proc(L) local t,X,Y,n,cons,i,R;
  t:= convert(L,`+`)/3;
  n:= nops(L);
  if not t::integer then return false fi;
  cons:= [add(L[i]*X[i],i=1..n)=t,
          add(L[i]*Y[i],i=1..n)=t,
          seq(X[i] + Y[i] <= 1, i=1..n)];
  R:= traperror(Optimization:-Maximize(0, cons, assume=binary));
  R::list
end proc:
primeindices:= proc(n) local F,t;
  F:= ifactors(n)[2];
  map(t -> numtheory:-pi(t[1])$t[2], F)
end proc:
select(tripart @ primindices, [$2..3000]); # Robert Israel, May 19 2025

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#], Length[#]==3&&SameQ@@hwt/@#&]!={}&]

A304796 Number of special sums of integer partitions of n.

Original entry on oeis.org

1, 2, 5, 10, 18, 32, 51, 82, 122, 188, 262, 392, 529, 750, 997, 1404, 1784, 2452, 3123, 4164, 5239, 6916, 8499, 11112, 13693, 17482, 21257, 27162, 32581, 41114, 49606, 61418, 73474, 91086, 107780, 132874, 157359, 191026, 225159, 274110, 320691, 386722, 453875

Offset: 0

Gus Wiseman, May 18 2018

nonn

A special sum of an integer partition y is a number n >= 0 such that exactly one submultiset of y sums to n.

			The a(4) = 18 special positive subset-sums:
0<=(4), 4<=(4),
0<=(22), 2<=(22), 4<=(22),
0<=(31), 1<=(31), 3<=(31), 4<=(31),
0<=(211), 1<=(211), 3<=(211), 4<=(211),
0<=(1111), 1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111).

Cf. A000712, A108917, A122768, A275972, A276024, A284640, A299701, A299702, A299729, A301829, A301830, A301854.

uqsubs[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Total],Length[#]===1&];
Table[Total[Length/@uqsubs/@IntegerPartitions[n]],{n,25}]

a(n) = A301854(n) + A000041(n).

More terms from Alois P. Heinz, May 18 2018

a(36)-a(42) from Chai Wah Wu, Sep 26 2023

A321144 Irregular triangle where T(n,k) is the number of divisors of n whose prime indices sum to k.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 28 2018

The rows are all palindromes.

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.

			Triangle begins:
  1
  1  1
  1  0  1
  1  1  1
  1  0  0  1
  1  1  1  1
  1  0  0  0  1
  1  1  1  1
  1  0  1  0  1
  1  1  0  1  1
  1  0  0  0  0  1
  1  1  2  1  1
  1  0  0  0  0  0  1
  1  1  0  0  1  1
  1  0  1  1  0  1
  1  1  1  1  1
  1  0  0  0  0  0  0  1
  1  1  1  1  1  1
  1  0  0  0  0  0  0  0  1
  1  1  1  1  1  1
  1  0  1  0  1  0  1
  1  1  0  0  0  1  1
  1  0  0  0  0  0  0  0  0  1
  1  1  2  2  1  1
  1  0  0  1  0  0  1
  1  1  0  0  0  0  1  1
  1  0  1  0  1  0  1
  1  1  1  0  1  1  1
  1  0  0  0  0  0  0  0  0  0  1
  1  1  1  2  1  1  1

Row lengths are A056239. Number of nonzero entries in row n is A299701(n).

Cf. A000005, A112798, A276024, A301854, A301899.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
Table[Count[Total/@primeMS/@Divisors[n],k],{n,20},{k,0,Total[primeMS[n]]}]

A343943 Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 19 2021

nonn

First differs from A096825 at a(525) = 3, A096825(525) = 4.
First differs from A345926 at a(90) = 4, A345926(90) = 3.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime factors is also the reverse-alternating sum of reversed prime factors.
Also the number of distinct "sums of prime factors" of divisors d|n such that bigomega(d) = bigomega(n)/2 rounded up.

			The divisors of 525 with 2 prime factors are: 15, 21, 25, 35, with prime factors {3,5}, {3,7}, {5,5}, {5,7}, with distinct sums {8,10,12}, so a(525) = 3.

The half-length submultisets are counted by A114921.
Including all multisets of prime factors gives A305611(n) + 1.
The strict rounded version appears to be counted by A342343.
The version for prime indices instead of prime factors is A345926.
A000005 counts divisors, which add up to A000203.
A001414 adds up prime factors, row sums of A027746.
A056239 adds up prime indices, row sums of A112798.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A276024 and A299701 count positive subset-sums of partitions.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A334968 counts subsequence-sums of standard compositions.
Cf. A008549, A032443, A083399, A096825, A344609, A345957.

prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Length[Union[Total/@Subsets[prifac[n],{Ceiling[PrimeOmega[n]/2]}]]],{n,100}]

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A343943(n):
    fs = factorint(n)
    return len(set(sum(d) for d in multiset_combinations(fs,(sum(fs.values())+1)//2))) # Chai Wah Wu, Aug 23 2021

A365660 Number of integer partitions of 2n with exactly n distinct sums of nonempty submultisets.

Original entry on oeis.org

1, 1, 1, 3, 2, 6, 6, 16, 12, 20, 26, 59, 45, 79, 94, 186, 142, 231, 244, 442, 470, 616, 746, 1340, 1053, 1548, 1852, 2780, 2826, 3874, 4320, 6617, 6286, 7924, 9178, 13180, 13634, 17494, 20356, 28220, 29176, 37188, 41932, 56037

Offset: 0

Gus Wiseman, Sep 16 2023

Are n = 1, 2, 4 the only n such that none of these partitions has 1?

Are n = 2, 4, 5, 8, 9 the only n such that none of these partitions is strict?

			The partition (433) has sums 3, 4, 6, 7, 10 so is counted under a(5).
The a(1) = 1 through a(7) = 16 partitions:
(2)  (2,2)  (4,2)    (4,2,2)    (4,3,3)      (6,4,2)        (6,5,3)
            (5,1)    (2,2,2,2)  (4,4,2)      (6,5,1)        (8,4,2)
            (2,2,2)             (6,2,2)      (4,4,2,2)      (8,5,1)
                                (8,1,1)      (6,2,2,2)      (9,3,2)
                                (4,2,2,2)    (4,2,2,2,2)    (9,4,1)
                                (2,2,2,2,2)  (2,2,2,2,2,2)  (10,3,1)
                                                            (11,2,1)
                                                            (4,4,4,2)
                                                            (5,3,3,3)
                                                            (6,4,2,2)
                                                            (8,2,2,2)
                                                            (11,1,1,1)
                                                            (4,4,2,2,2)
                                                            (6,2,2,2,2)
                                                            (4,2,2,2,2,2)
                                                            (2,2,2,2,2,2,2)

For n instead of 2n we have A126796.
Central column n = 2k of A365658.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A002219 counts partitions of 2n with a submultiset summing to n.
A046663 counts partitions of n w/o a submultiset of sum k, strict A365663.
A122768 counts distinct nonempty submultisets of partitions.
A299701 counts sums of submultisets of prime indices, of partitions A304792.
A364272 counts sum-full strict partitions, sum-free A364349.
A365543 counts partitions of n w/ a submultiset of sum k, strict A365661.
Cf. A000041, A008967, A095944, A364911, A365376, A365377.

msubs[y_]:=primeMS/@Divisors[Times@@Prime/@y];
Table[Length[Select[IntegerPartitions[2n], Length[Union[Total/@Rest[msubs[#]]]]==n&]],{n,0,10}]

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_combinations
def A365660(n):
    c = 0
    for p in partitions(n<<1):
        q, s = list(Counter(p).elements()), set()
        for l in range(1,len(q)+1):
            for k in multiset_combinations(q,l):
                s.add(sum(k))
                if len(s) > n:
                    break
            else:
                continue
            break
        if len(s)==n:
            c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(21)-a(38) from Chai Wah Wu, Sep 20 2023

a(39)-a(43) from Chai Wah Wu, Sep 21 2023

A301970 Heinz numbers of integer partitions with more subset-products than subset-sums.

Original entry on oeis.org

165, 273, 325, 351, 495, 525, 561, 595, 675, 741, 765, 819, 825, 931, 1045, 1053, 1155, 1173, 1425, 1485, 1495, 1575, 1625, 1653, 1683, 1771, 1785, 1815, 1911, 2025, 2139, 2145, 2223, 2275, 2277, 2295, 2310, 2415, 2457, 2465, 2475, 2625, 2639, 2695, 2805, 2945

Offset: 1

Gus Wiseman, Mar 29 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A subset-sum (or subset-product) of a multiset y is any number equal to the sum (or product) of some submultiset of y.

Numbers n such that A301957(n) > A299701(n).

			Sequence of partitions begins: (532), (642), (633), (6222), (5322), (4332), (752), (743), (33222), (862), (7322), (6422), (5332), (844), (853), (62222), (5432), (972), (8332), (53222), (963), (43322), (6333).

Cf. A000712, A003963, A056239, A108917, A276024, A284640, A292886, A296150, A299701, A299702, A299729, A301854, A301855, A301856, A301957, A301979.

Select[Range[1000],With[{ptn=If[#===1,{},Join@@Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Length[Union[Times@@@Subsets[ptn]]]>Length[Union[Plus@@@Subsets[ptn]]]]&]

A304795 Number of positive special sums of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 18 2018

nonn

A positive special sum of y is a number n > 0 such that exactly one submultiset of y sums to n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(36) = 4 special sums are 1, 3, 5, 6, corresponding to the submultisets (1), (21), (221), (2211), with Heinz numbers 2, 6, 18, 36.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000712, A056239, A108917, A122768, A276024, A284640, A296150, A299701, A299702, A301854, A301855, A301957, A304793, A304796.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uqsubs[y_]:=Join@@Select[GatherBy[Union[Rest[Subsets[y]]],Total],Length[#]===1&];
Table[Length[uqsubs[primeMS[n]]],{n,100}]

up_to = 65537;
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
v056239 = vector(up_to,n,A056239(n));
A304795(n) = { my(m=Map(),s,k=0,c); fordiv(n,d,if(!mapisdefined(m,s = v056239[d],&c), mapput(m,s,1), mapput(m,s,c+1))); sumdiv(n,d,(1==mapget(m,v056239[d])))-1; }; \\ Antti Karttunen, Jul 02 2018

More terms from Antti Karttunen, Jul 02 2018

A316398 Number of distinct subset-averages of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 01 2018

nonn

Although the average of an empty set is technically indeterminate, we consider it to be distinct from the other subset-averages.

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

			The a(60) = 9 distinct subset-averages of (3,2,1,1) are 0/0, 1, 4/3, 3/2, 5/3, 7/4, 2, 5/2, 3.

Cf. A032302, A056239, A108917, A275972, A276024, A296150, A299701, A301899, A301957, A316313.

One more than A316314.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Mean/@Subsets[primeMS[n]]]],{n,100}]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A316398(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s = A056239(d)/bigomega(d)), mapput(m,s,s); k++)); (1+k); }; \\ Antti Karttunen, Sep 23 2018

a(n) = A316314(n) + 1.

More terms from Antti Karttunen, Sep 23 2018

cp's OEIS Frontend

A301934 Number of positive subset-sum trees of weight n.

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A325800 Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.

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Maple

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A371955 Numbers with triquanimous prime indices.

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Maple

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A304796 Number of special sums of integer partitions of n.

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A321144 Irregular triangle where T(n,k) is the number of divisors of n whose prime indices sum to k.

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A343943 Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.

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A365660 Number of integer partitions of 2n with exactly n distinct sums of nonempty submultisets.

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A301970 Heinz numbers of integer partitions with more subset-products than subset-sums.

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