A325780 -id:A325780 - cp's OEIS frontend

A325794 Number of divisors of n minus the sum of prime indices of n.

1, 1, 0, 1, -1, 1, -2, 1, -1, 0, -3, 2, -4, -1, -1, 1, -5, 1, -6, 1, -2, -2, -7, 3, -3, -3, -2, 0, -8, 2, -9, 1, -3, -4, -3, 3, -10, -5, -4, 2, -11, 1, -12, -1, -1, -6, -13, 4, -5, -1, -5, -2, -14, 1, -4, 1, -6, -7, -15, 5, -16, -8, -2, 1, -5, 0, -17, -3, -7

Offset: 1

Gus Wiseman, May 23 2019

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).

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

Positions of positive terms are A325795.
Positions of nonnegative terms are A325796.
Positions of negative terms are A325797.
Positions of nonpositive terms are A325798.
Positions of 1's are A325792.
Positions of 0's are A325793.
Positions of -1's are A325694.
Cf. A000005, A002033, A056239, A112798, A299702, A325780, A325781.

Table[DivisorSigma[0,n]-Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]],{n,100}]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A325794(n) = (numdiv(n)-A056239(n)); \\ Antti Karttunen, May 26 2019

a(n) = A000005(n) - A056239(n).

A325795 Numbers with more divisors than the sum of their prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 204, 210, 216, 220, 224, 234, 240, 252, 256, 260, 264, 270, 280, 288

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from A325781 in having 156.

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).

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

Positions of positive terms in A325794.
Heinz numbers of the partitions counted by A325831.
Cf. A000005, A002033, A056239, A112798, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325796, A325797, A325798.

Select[Range[100],DivisorSigma[0,#]>Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]

A325796 Numbers with at least as many divisors as the sum of their prime indices.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 80, 84, 88, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240

Offset: 1

Gus Wiseman, May 23 2019

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, with sum A056239(n).

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}

Positions of nonnegative terms in A325794.
Heinz numbers of the partitions counted by A325832.
Cf. A000005, A002033, A056239, A112798, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325795, A325797, A325798.

Select[Range[100],DivisorSigma[0,#]>=Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]

A325797 Numbers with fewer divisors than the sum of their prime indices.

Original entry on oeis.org

5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97

Offset: 1

Gus Wiseman, May 23 2019

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, with sum A056239(n).

			The sequence of terms together with their prime indices begins:
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  34: {1,7}
  35: {3,4}

Positions of negative terms in A325794.
Heinz numbers of the partitions counted by A325833.
Cf. A000005, A002033, A056239, A112798, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325795, A325796, A325798.

Select[Range[100],DivisorSigma[0,#]PrimePi[p]*k]]&]

A353743 Least number with run-sum trajectory of length k; a(0) = 1.

Original entry on oeis.org

1, 2, 4, 12, 84, 1596, 84588, 11081028, 3446199708, 2477817590052, 4011586678294188, 14726534696017964148, 120183249654202605411828, 2146833388573021140471483564, 83453854313999050793547980583372, 7011542477899258250521520684673165324

Offset: 0

Gus Wiseman, Jun 11 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832, A353847) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
     12: {1,1,2}
     84: {1,1,2,4}
   1596: {1,1,2,4,8}
  84588: {1,1,2,4,8,16}

The ordered version is A072639, for run-lengths A333629.
The version for run-lengths is A325278, firsts in A182850 or A323014.
The run-sum trajectory is the iteration of A353832.
The first length-k row of A353840 has index a(k).
Other sequences pertaining to this trajectory are A353841-A353846.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A300273 ranks collapsible partitions, counted by A275870.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A002033, A005117, A006939, A071625, A076954, A126796, A181819, A182857, A188431, A299702, A325780, A325781, A353834.

Join[{1,2},Table[2*Product[Prime[2^k],{k,0,n}],{n,0,6}]]

a(n > 1) = 2 * Product_{k=0..n-2} prime(2^k).

a(n > 0) = 2 * A325782(n).

A325789 Number of perfect necklace compositions of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 22 2019

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. A necklace composition of n is perfect if every positive integer from 1 to n is the sum of exactly one distinct circular subsequence.

			The a(1) = 1 , a(2) = 1, a(3) = 2, a(7) = 3, a(13) = 5, and a(31) = 11 perfect necklace compositions (A = 10, B = 11, C = 12, D = 13, E = 14):
  1  11  12   124      1264           12546D
         111  142      1327           1274C5
              1111111  1462           13278A
                       1723           13625E
                       1111111111111  15C472
                                      17324E
                                      1A8723
                                      1D6452
                                      1E4237
                                      1E5263
                                      1111111111111111111111111111111

Cf. A002033, A008965, A103300, A108917, A126796, A325676, A325679, A325685, A325780, A325782, A325786, A325787, A325789.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&Sort[Total/@subalt[#]]==Range[n]&]],{n,10}]

For n > 1, a(n) = A325787(n) + 1.

A325801 Number of divisors of n minus the number of distinct positive subset-sums of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, May 23 2019

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, with sum A056239(n). A positive subset-sum of an integer partition is any sum of a nonempty submultiset of it.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of 0's are A299702.
Positions of 1's are A325802.
Positions of positive integers are A299729.
Cf. A000005, A002033, A056239, A108917, A112798, A276024, A304793.
Cf. A325694, A325780, A325781, A325792, A325793, A325799.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p] k]];
Table[DivisorSigma[0,n]-Length[Union[hwt/@Divisors[n]]],{n,100}]

A325801(n) = (numdiv(n) - A299701(n));
A299701(n) = { my(f = factor(n), pids = List([])); for(i=1,#f~, while(f[i,2], f[i,2]--; listput(pids,primepi(f[i,1])))); pids = Vec(pids); my(sv=vector(vecsum(pids))); for(b=1,(2^length(pids))-1,sv[sumbybits(pids,b)] = 1); 1+vecsum(sv); }; \\ Not really an optimal way to count these.
sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); }; \\ Antti Karttunen, May 26 2019

a(n) = A000005(n) - A299701(n).

A325986 Heinz numbers of complete strict integer partitions.

Original entry on oeis.org

1, 2, 6, 30, 42, 210, 330, 390, 462, 510, 546, 714, 798, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7854, 8778, 8970, 9282, 9570, 9690, 10230, 10374, 10626, 11310, 11730, 12090, 12210, 12558, 13398, 13566, 14322, 14430

Offset: 1

Gus Wiseman, May 30 2019

nonn

Strict partitions are counted by A000009, while complete partitions are counted by A126796.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition of n is complete (A126796, A325781) if every number from 0 to n is the sum of some submultiset of the parts.
The enumeration of these partitions by sum is given by A188431.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      6: {1,2}
     30: {1,2,3}
     42: {1,2,4}
    210: {1,2,3,4}
    330: {1,2,3,5}
    390: {1,2,3,6}
    462: {1,2,4,5}
    510: {1,2,3,7}
    546: {1,2,4,6}
    714: {1,2,4,7}
    798: {1,2,4,8}
   2310: {1,2,3,4,5}
   2730: {1,2,3,4,6}
   3570: {1,2,3,4,7}
   3990: {1,2,3,4,8}
   4290: {1,2,3,5,6}
   4830: {1,2,3,4,9}
   5610: {1,2,3,5,7}

Cf. A002033, A056239, A103295, A112798, A126796, A188431, A299702, A304793.

Cf. A325780, A325781, A325782, A325788, A325790, A325988.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p] k]];
Select[Range[1000],SquareFreeQ[#]&&Union[hwt/@Divisors[#]]==Range[0,hwt[#]]&]

Intersection of A005117 (strict partitions) and A325781 (complete partitions).

A326018 Heinz numbers of knapsack partitions such that no addition of one part up to the maximum is knapsack.

Original entry on oeis.org

1925, 12155, 20995, 23375, 37145

Offset: 1

Gus Wiseman, Jun 03 2019

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition is knapsack if every submultiset has a different sum.
The enumeration of these partitions by sum is given by A326016.

			The sequence of terms together with their prime indices begins:
   1925: {3,3,4,5}
  12155: {3,5,6,7}
  20995: {3,6,7,8}
  23375: {3,3,3,5,7}
  37145: {3,7,8,9}

Cf. A002033, A108917, A275972, A299702, A299729, A304793.

Cf. A325780, A325782, A325857, A325862, A325878, A325880, A326015, A326016.

ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]];
Select[Range[2,200],With[{phm=If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]},ksQ[phm]&&Select[Table[Sort[Append[phm,i]],{i,Max@@phm}],ksQ]=={}]&]

A325787 Number of perfect strict necklace compositions of n.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, May 22 2019

nonn

A strict necklace composition of n is a finite sequence of distinct positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. In other words, it is a strict composition of n starting with its least part. A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. A necklace composition of n is perfect if every positive integer from 1 to n is the sum of exactly one distinct circular subsequence. For example, the composition (1,2,6,4) is perfect because it has the following circular subsequences and sums:
(1)
(2)
(1,2)
(4)
(4,1)
(6)
(4,1,2)
(2,6)
(1,2,6)
(6,4)
(6,4,1)
(2,6,4)
(1,2,6,4)
a(n) > 0 iff n = A002061(k) = A004136(k) for some k. - Bert Dobbelaere, Nov 11 2020

			The a(1) = 1 through a(31) = 10 perfect strict necklace compositions (empty columns not shown):
  (1)  (1,2)  (1,2,4)  (1,2,6,4)  (1,3,10,2,5)  (1,10,8,7,2,3)
              (1,4,2)  (1,3,2,7)  (1,5,2,10,3)  (1,13,6,4,5,2)
                       (1,4,6,2)                (1,14,4,2,3,7)
                       (1,7,2,3)                (1,14,5,2,6,3)
                                                (1,2,5,4,6,13)
                                                (1,2,7,4,12,5)
                                                (1,3,2,7,8,10)
                                                (1,3,6,2,5,14)
                                                (1,5,12,4,7,2)
                                                (1,7,3,2,4,14)
From _Bert Dobbelaere_, Nov 11 2020: (Start)
Compositions matching nonzero terms from a(57) to a(273), up to symmetry.
a(57) = 12:
  (1,2,10,19,4,7,9,5)
  (1,3,5,11,2,12,17,6)
  (1,3,8,2,16,7,15,5)
  (1,4,2,10,18,3,11,8)
  (1,4,22,7,3,6,2,12)
  (1,6,12,4,21,3,2,8)
a(73) = 8:
  (1,2,4,8,16,5,18,9,10)
  (1,4,7,6,3,28,2,8,14)
  (1,6,4,24,13,3,2,12,8)
  (1,11,8,6,4,3,2,22,16)
a(91) = 12:
  (1,2,6,18,22,7,5,16,4,10)
  (1,3,9,11,6,8,2,5,28,18)
  (1,4,2,20,8,9,23,10,3,11)
  (1,4,3,10,2,9,14,16,6,26)
  (1,5,4,13,3,8,7,12,2,36)
  (1,6,9,11,29,4,8,2,3,18)
a(133) = 36:
  (1,2,9,8,14,4,43,7,6,10,5,24)
  (1,2,12,31,25,4,9,10,7,11,16,5)
  (1,2,14,4,37,7,8,27,5,6,13,9)
  (1,2,14,12,32,19,6,5,4,18,13,7)
  (1,3,8,9,5,19,23,16,13,2,28,6)
  (1,3,12,34,21,2,8,9,5,6,7,25)
  (1,3,23,24,6,22,10,11,18,2,5,8)
  (1,4,7,3,16,2,6,17,20,9,13,35)
  (1,4,16,3,15,10,12,14,17,33,2,6)
  (1,4,19,20,27,3,6,25,7,8,2,11)
  (1,4,20,3,40,10,9,2,15,16,6,7)
  (1,5,12,21,29,11,3,16,4,22,2,7)
  (1,7,13,12,3,11,5,18,4,2,48,9)
  (1,8,10,5,7,21,4,2,11,3,26,35)
  (1,14,3,2,4,7,21,8,25,10,12,26)
  (1,14,10,20,7,6,3,2,17,4,8,41)
  (1,15,5,3,25,2,7,4,6,12,14,39)
  (1,22,14,20,5,13,8,3,4,2,10,31)
a(183) = 40:
  (1,2,13,7,5,14,34,6,4,33,18,17,21,8)
  (1,2,21,17,11,5,9,4,26,6,47,15,12,7)
  (1,2,28,14,5,6,9,12,48,18,4,13,16,7)
  (1,3,5,6,25,32,23,10,18,2,17,7,22,12)
  (1,3,12,7,20,14,44,6,5,24,2,28,8,9)
  (1,3,24,6,12,14,11,55,7,2,8,5,16,19)
  (1,4,6,31,3,13,2,7,14,12,17,46,8,19)
  (1,4,8,52,3,25,18,2,9,24,6,10,7,14)
  (1,4,20,2,12,3,6,7,33,11,8,10,35,31)
  (1,5,2,24,15,29,14,21,13,4,33,3,9,10)
  (1,5,23,27,42,3,4,11,2,19,12,10,16,8)
  (1,6,8,22,4,5,33,21,3,20,32,16,2,10)
  (1,8,3,10,23,5,56,4,2,14,15,17,7,18)
  (1,8,21,45,6,7,11,17,3,2,10,4,23,25)
  (1,9,5,40,3,4,21,35,16,18,2,6,11,12)
  (1,9,14,26,4,2,11,5,3,12,27,34,7,28)
  (1,9,21,25,3,4,8,5,6,16,2,36,14,33)
  (1,10,22,34,27,12,3,4,2,14,24,5,8,17)
  (1,10,48,9,19,4,8,6,7,17,3,2,34,15)
  (1,12,48,6,2,38,3,22,7,10,11,5,4,14)
a(273) = 12:
  (1,2,4,8,16,32,27,26,11,9,45,13,10,29,5,17,18)
  (1,3,12,10,31,7,27,2,6,5,19,20,62,14,9,28,17)
  (1,7,3,15,33,5,24,68,2,14,6,17,4,9,19,12,34)
  (1,7,12,44,25,41,9,17,4,6,22,33,13,2,3,11,23)
  (1,7,31,2,11,3,9,36,17,4,22,6,18,72,5,10,19)
  (1,21,11,50,39,13,6,4,14,16,25,26,3,2,7,8,27)
(End)

Bert Dobbelaere, Table of n, a(n) for n = 1..306

Cf. A002033, A002061, A004136, A008965, A032153, A103300, A126796, A325680, A325682, A325780, A325782, A325786, A325788, A325789.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],neckQ[#]&&Sort[Total/@subalt[#]]==Range[n]&]],{n,30}]

More terms from Bert Dobbelaere, Nov 11 2020

cp's OEIS Frontend

A325794 Number of divisors of n minus the sum of prime indices of n.

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A325795 Numbers with more divisors than the sum of their prime indices.

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A325796 Numbers with at least as many divisors as the sum of their prime indices.

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A325797 Numbers with fewer divisors than the sum of their prime indices.

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A353743 Least number with run-sum trajectory of length k; a(0) = 1.

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A325789 Number of perfect necklace compositions of n.

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A325801 Number of divisors of n minus the number of distinct positive subset-sums of the prime indices of n.

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A325986 Heinz numbers of complete strict integer partitions.

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