A367908 -id:A367908 - cp's OEIS frontend

A367771 Number of ways to choose a different prime index of each prime index of 2n + 1.

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

Offset: 0

Gus Wiseman, Dec 12 2023

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 prime indices of 427 = 2*213 + 1 are {{1,1},{1,2,2}}, with four ways to choose (1,2), so a(213) = 4.
The prime indices of prime indices of 1469 = 2*734 + 1 are {{1,2},{1,2,3}}, with four choices (1,2), (1,3), (2,1), (2,3), so a(734) = 4.

The "extended" version below includes alternating zeros at even positions.
Extended positions of zeros are A355529, binary A367907.
The extended version for binary indices is A367905.
Extended positions of nonzeros are A368100, binary A367906.
Extended positions of ones are A368101, binary A367908.
The extended version without distinctness is A355741, for multisets A355744.
A058891 counts set-systems, covering A003465, connected A323818.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
Cf. A007716, A355739, A355740, A355745, A367901, A367902, A367903.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[prix/@prix[2n+1]], UnsameQ@@#&]],{n,0,100}]

A370584 Number of subsets of {1..n} such that only one set can be obtained by choosing a different prime factor of each element.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 18, 36, 48, 68, 104, 208, 284, 568, 888, 1296, 1548, 3096, 3968, 7936, 10736, 15440, 24008, 48016, 58848, 73680, 114368, 132608, 176240, 352480, 449824, 899648, 994976, 1399968, 2160720, 2859584, 3296048, 6592096, 10156672, 14214576, 16892352

Offset: 0

Gus Wiseman, Feb 26 2024

nonn

For example, the only choice of a different prime factor of each element of (4,5,6) is (2,5,3).

			The a(0) = 1 through a(6) = 18 subsets:
  {}  {}  {}   {}     {}     {}       {}
          {2}  {2}    {2}    {2}      {2}
               {3}    {3}    {3}      {3}
               {2,3}  {4}    {4}      {4}
                      {2,3}  {5}      {5}
                      {3,4}  {2,3}    {2,3}
                             {2,5}    {2,5}
                             {3,4}    {2,6}
                             {3,5}    {3,4}
                             {4,5}    {3,5}
                             {2,3,5}  {3,6}
                             {3,4,5}  {4,5}
                                      {4,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}

For divisors instead of factors we have A051026, cf. A368110, A355740.
The version for set-systems is A367904, ranks A367908.
Multisets of this type are ranked by A368101, cf. A368100, A355529.
For existence we have A370582, differences A370586.
For nonexistence we have A370583, differences A370587.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, cf. A370592, A370593.
For binary indices instead of factors we have A370638, cf. A370636, A370637.
The version for factorizations is A370645, cf. A368414, A368413.
For unlabeled multiset partitions we have A370646, cf. A368098, A368097.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts ways to choose a prime factor of each prime index.
Cf. A000040, A000720, A003963, A005117, A045778, A133686, A307984, A355739, A355744, A355745, A367905.

Table[Length[Select[Subsets[Range[n]], Length[Union[Sort/@Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]],{n,0,10}]

More terms from Jinyuan Wang, Mar 28 2025

A370637 Number of subsets of {1..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 25, 67, 134, 309, 709, 1579, 3420, 7240, 15077, 30997, 61994, 125364, 253712, 512411, 1032453, 2075737, 4166469, 8352851, 16731873, 33497422, 67038086, 134130344, 268328977, 536741608, 1073586022, 2147296425, 4294592850, 8589346462, 17179033384

Offset: 0

Gus Wiseman, Mar 08 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(0) = 0 through a(5) = 8 subsets:
  .  .  .  {1,2,3}  {1,2,3}    {1,2,3}
                    {1,2,3,4}  {1,4,5}
                               {1,2,3,4}
                               {1,2,3,5}
                               {1,2,4,5}
                               {1,3,4,5}
                               {2,3,4,5}
                               {1,2,3,4,5}

Wikipedia, Axiom of choice.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Simple graphs not of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A140637, complement A134964.
Simple graphs of this type are counted by A367867, covering A367868.
Set systems not of this type are counted by A367902, ranks A367906.
Set systems of this type are counted by A367903, ranks A367907.
Set systems uniquely not of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368097, complement A368098.
A version for MM-numbers of multisets is A355529, complement A368100.
Factorizations are counted by A368413/A370813, complement A368414/A370814.
The complement for prime indices is A370582, differences A370586.
For prime indices we have A370583, differences A370587.
First differences are A370589.
The complement is counted by A370636, differences A370639.
The case without ones is A370643.
The version for a unique choice is A370638, maxima A370640, diffs A370641.
The minimal case is A370642, without ones A370644.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A072639, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

a(2^n - 1) = A367903(n).

Partial sums of A370589.

a(21)-a(34) from Alois P. Heinz, Mar 09 2024

A370594 Number of integer partitions of n such that only one set can be obtained by choosing a different prime factor of each part.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 3, 4, 5, 5, 8, 10, 11, 7, 14, 13, 19, 23, 24, 20, 30, 33, 40, 47, 49, 55, 53, 72, 80, 90, 92, 110, 110, 132, 154, 169, 180, 201, 218, 246, 281, 302, 323, 348, 396, 433, 482, 530, 584, 618, 670, 754, 823, 903, 980, 1047, 1137

Offset: 0

Gus Wiseman, Feb 29 2024

nonn

			The partition (10,6,4) has unique choice (5,3,2) so is counted under a(20).
The a(0) = 1 through a(12) = 5 partitions:
()  .  (2)  (3)  (4)  (5)    .  (7)    (8)    (9)    (6,4)    (11)   (6,6)
                      (3,2)     (4,3)  (5,3)  (5,4)  (7,3)    (7,4)  (7,5)
                                (5,2)  (6,2)  (6,3)  (5,3,2)  (8,3)  (10,2)
                                              (7,2)           (9,2)  (5,4,3)
                                                                     (7,3,2)

The version for set-systems is A367904, ranks A367908.
Multisets of this type are ranked by A368101, cf. A368100, A355529.
The version for subsets is A370584, cf. A370582, A370583, A370586, A370587.
Maximal sets of this type are counted by A370585.
For existence we have A370592.
For nonexistence we have A370593.
For divisors instead of factors we have A370595.
For subsets and binary indices we have A370638, cf. A370636, A370637.
The version for factorizations is A370645, cf. A368414, A368413.
These partitions have ranks A370647.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts ways to choose a prime factor of each prime index.
Cf. A000040, A000041, A000720, A003963, A355739, A355740, A367905, A368110.

Table[Length[Select[IntegerPartitions[n], Length[Union[Sort/@Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]],{n,0,30}]

A368600 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is not possible to choose a different element from each.

Original entry on oeis.org

0, 0, 0, 3, 164, 18625, 5491851, 4649088885, 12219849683346

Offset: 0

Gus Wiseman, Jan 01 2024

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The a(3) = 3 set-systems:
  {{1},{2},{1,2}}
  {{1},{3},{1,3}}
  {{2},{3},{2,3}}

Wikipedia, Axiom of choice.

For a unique choice we have A003024, any length A367904 (ranks A367908).
Sets of n nonempty subsets of {1..n} are counted by A136556.
For any length we have A367903, ranks A367907, no singletons A367769.
The complement is A368601, any length A367902 (see also A367770, A367906).
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A003025, A088957, A133686, A334282, A355529, A355740, A367862, A367867, A367868, A367901, A368094, A368097.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]], {n}],Length[Select[Tuples[#], UnsameQ@@#&]]==0&]],{n,0,3}]

from itertools import combinations, product, chain
from scipy.special import comb
def v(c):
    for elements in product(*c):
        if len(set(elements)) == len(elements):
            return True
    return False
def a(n):
    if n == 0:
        return 1
    subsets = list(chain.from_iterable(combinations(range(1, n + 1), r) for r in range(1, n + 1)))
    cs = combinations(subsets, n)
    c = sum(1 for c in cs if v(c))
    return c
[print(int(comb(2**n-1,n) - a(n))) for n in range(7)] # Robert P. P. McKone, Jan 02 2024

a(n) = A136556(n) - A368601(n).

a(6) from Robert P. P. McKone, Jan 02 2024

a(7)-a(8) from Christian Sievers, Jul 25 2024

A368601 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is possible to choose a different element from each.

Original entry on oeis.org

1, 1, 3, 32, 1201, 151286, 62453670, 84707326890, 384641855115279

Offset: 0

Gus Wiseman, Jan 01 2024

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The a(2) = 3 set-systems:
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
Non-isomorphic representatives of the a(3) = 32 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Wikipedia, Axiom of choice.

For a unique choice we have A003024, any length A367904 (ranks A367908).
Sets of n nonempty subsets of {1..n} are counted by A136556.
For any length we have A367902, ranks A367906, no singletons A367770.
The complement is A368600, any length A367903 (see also A367907, A367769).
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A003025, A088957, A133686, A334282, A355529, A355740, A367862, A367867, A367901, A367905, A368094, A368097.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]], {n}],Length[Select[Tuples[#], UnsameQ@@#&]]>0&]],{n,0,3}]

from itertools import combinations, product, chain
def v(c):
    for elements in product(*c):
        if len(set(elements)) == len(elements):
            return True
    return False
def a(n):
    if n == 0:
        return 1
    subsets = list(chain.from_iterable(combinations(range(1, n + 1), r) for r in
range(1, n + 1)))
    cs = combinations(subsets, n)
    c = sum(1 for c in cs if v(c))
    return c
[print(a(n)) for n in range(7)] # Robert P. P. McKone, Jan 02 2024

a(n) + A368600(n) = A136556(n).

a(6) from Robert P. P. McKone, Jan 02 2024

a(7)-a(8) from Christian Sievers, Jul 25 2024

A370638 Number of subsets of {1..n} such that a unique set can be obtained by choosing a different binary index of each element.

Original entry on oeis.org

1, 2, 4, 6, 12, 19, 30, 45, 90, 147, 230, 343, 504, 716, 994, 1352, 2704, 4349, 6469, 9162, 12585, 16862, 22122, 28617, 36653, 46431, 58075, 72097, 88456, 107966, 130742, 157647, 315294, 494967, 704753, 950080, 1234301, 1565165, 1945681, 2387060, 2890368, 3470798

Offset: 0

Gus Wiseman, Mar 09 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The set {3,4} has binary indices {{1,2},{3}}, with two choices {1,3}, {2,3}, so is not counted under a(4).
The a(0) = 1 through a(5) = 19 subsets:
  {}  {}   {}     {}     {}       {}
      {1}  {1}    {1}    {1}      {1}
           {2}    {2}    {2}      {2}
           {1,2}  {1,2}  {4}      {4}
                  {1,3}  {1,2}    {1,2}
                  {2,3}  {1,3}    {1,3}
                         {1,4}    {1,4}
                         {2,3}    {1,5}
                         {2,4}    {2,3}
                         {1,2,4}  {2,4}
                         {1,3,4}  {4,5}
                         {2,3,4}  {1,2,4}
                                  {1,2,5}
                                  {1,3,4}
                                  {1,3,5}
                                  {2,3,4}
                                  {2,3,5}
                                  {2,4,5}
                                  {3,4,5}

Wikipedia, Axiom of choice.

Set systems of this type are counted by A367904, ranks A367908.
A version for MM-numbers of multisets is A368101.
For prime indices we have A370584.
This is the unique version of A370636, complement A370637.
The maximal case is A370640, differences A370641.
Factorizations of this type are counted by A370645.
The case A370818 is the restriction to A000225.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A133686, A134964, A326031, A326702, A367772, A367867, A367905, A367909, A367912, A368109.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]],Length[Union[Sort /@ Select[Tuples[bpe/@#],UnsameQ@@#&]]]==1&]],{n,0,10}]

a(2^n - 1) = A370818(n).

More terms from Jinyuan Wang, Mar 28 2025

A367909 Numbers n such that there is more than one way to choose a different binary index of each binary index of n.

Original entry on oeis.org

4, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 65, 66, 68, 72, 76, 80, 82, 84, 96, 97, 100, 112, 132, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 193, 194, 196, 200, 204, 208, 210, 212, 224, 225, 228, 240, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288

Offset: 1

Gus Wiseman, Dec 11 2023

nonn

Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in more than one way.
A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary digits (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.
The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The set-system {{1},{1,2},{1,3}} with BII-number 21 satisfies the axiom in only one way (1,2,3), so 21 is not in the sequence.
The terms together with the corresponding set-systems begin:
   4: {{1,2}}
  12: {{1,2},{3}}
  16: {{1,3}}
  18: {{2},{1,3}}
  20: {{1,2},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}
  36: {{1,2},{2,3}}
  48: {{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  72: {{3},{1,2,3}}

Wikipedia, Axiom of choice.

These set-systems are counted by A367772.
Positions of terms > 1 in A367905, firsts A367910, sorted firsts A367911.
If there is at least one choice we get A367906, counted by A367902.
If there are no choices we get A367907, counted by A367903.
If there is one unique choice we get A367908, counted by A367904.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
A368098 counts unlabeled multiset partitions per axiom, complement A368097.
Cf. A000612, A055621, A072639, A309326, A326702, A326753, A355529, A368100.
BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers), A326783 (uniform), A326784 (regular), A326788 (simple), A330217 (achiral).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100], Length[Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]]>1&]

A367907 U A367908 U A367909 = A000027.

A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.

Original entry on oeis.org

0, 0, 1, 23, 1105, 154941, 66072394, 88945612865, 396990456067403

Offset: 0

Gus Wiseman, Dec 12 2023

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Non-isomorphic representatives of the a(3) = 23 set-systems:
  {{1,2}}
  {{1,2,3}}
  {{1},{2,3}}
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Wikipedia, Axiom of choice.

For at least one choice we have A367902.
For no choices we have A367903, no singletons A367769, ranks A367907.
For a unique choice we have A367904, ranks A367908.
These set-systems have ranks A367909.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
Cf. A059201, A102896, A133686, A283877, A306445, A323818, A355741, A367770, A367862, A367869, A367901, A367905.

Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Select[Tuples[#], UnsameQ@@#&]]>1&]], {n,0,3}]

A367903(n) + A367904(n) + a(n) = A058891(n).

a(5)-a(8) from Christian Sievers, Jul 26 2024

A367910 Least number k such that there are exactly n ways to choose a different binary index of each binary index of k.

Original entry on oeis.org

7, 1, 4, 20, 68, 320, 352, 1088, 3136, 13376, 16704, 5184, 82240, 70720, 17472

Offset: 0

Gus Wiseman, Dec 16 2023

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
      7: {{1},{2},{1,2}}
      1: {{1}}
      4: {{1,2}}
     20: {{1,2},{1,3}}
     68: {{1,2},{1,2,3}}
    320: {{1,2,3},{1,4}}
    352: {{2,3},{1,2,3},{1,4}}
   1088: {{1,2,3},{1,2,4}}
   3136: {{1,2,3},{1,2,4},{3,4}}
  13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  16704: {{1,2,3},{1,4},{1,2,3,4}}
   5184: {{1,2,3},{1,2,4},{1,3,4}}
  82240: {{1,2,3},{1,4},{1,2,3,4},{1,5}}
  70720: {{1,2,3},{1,2,4},{1,3,4},{1,5}}

Wikipedia, Axiom of choice.

Positions of first appearances in A367905.
The sorted version is A367911.
For multisets w/o distinctness: A367913, firsts of A367912, sorted A367915.
Not requiring distinctness gives A368111, firsts of A368109, sorted A368112.
For multisets of indices we have A368184, firsts of A368183, sorted A368185.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A367771, A367906, A367907, A367908, A367909.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Select[Tuples[bpe/@bpe[n]],UnsameQ@@#&]],{n,1000}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
Table[Position[c,n][[1,1]],{n,0,spnm[c]}]

cp's OEIS Frontend

A367771 Number of ways to choose a different prime index of each prime index of 2n + 1.

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A370584 Number of subsets of {1..n} such that only one set can be obtained by choosing a different prime factor of each element.

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A370637 Number of subsets of {1..n} such that it is not possible to choose a different binary index of each element.

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A370594 Number of integer partitions of n such that only one set can be obtained by choosing a different prime factor of each part.

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A368600 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is not possible to choose a different element from each.

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A368601 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is possible to choose a different element from each.

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A370638 Number of subsets of {1..n} such that a unique set can be obtained by choosing a different binary index of each element.

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A367909 Numbers n such that there is more than one way to choose a different binary index of each binary index of n.

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