cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A035310 Let f(n) = number of ways to factor n = A001055(n); a(n) = sum of f(k) over all terms k in A025487 that have n factors.

Original entry on oeis.org

1, 4, 12, 47, 170, 750, 3255, 16010, 81199, 448156, 2579626, 15913058, 102488024, 698976419, 4976098729, 37195337408, 289517846210, 2352125666883, 19841666995265, 173888579505200, 1577888354510786, 14820132616197925, 143746389756336173, 1438846957477988926
Offset: 1

Views

Author

Alford Arnold

Keywords

Comments

Ways of partitioning an n-multiset with multiplicities some partition of n.
Number of multiset partitions of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities. The (weakly) normal version is A255906. - Gus Wiseman, Dec 31 2019

Examples

			a(3) = 12 because there are 3 terms in A025487 with 3 factors, namely 8, 12, 30; and f(8)=3, f(12)=4, f(30)=5 and 3+4+5 = 12.
From _Gus Wiseman_, Dec 31 2019: (Start)
The a(1) = 1 through a(3) = 12 multiset partitions of strongly normal multisets:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1,2}}    {{1,1,2}}
         {{1},{1}}  {{1,2,3}}
         {{1},{2}}  {{1},{1,1}}
                    {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,1}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}
(End)

Links

Crossrefs

Cf. A025487, A000041, A000110, A035098, A080688.
Sequence A035341 counts the ordered cases. Tables A093936 and A095705 distribute the values; e.g. 81199 = 30 + 536 + 3036 + 6181 + 10726 + 11913 + 14548 + 13082 + 21147.
Cf. A035341, A093936, A095705.
Row sums of A317449.
The uniform case is A317584.
The case with empty intersection is A317755.
The strict case is A317775.
The constant case is A047968.
The set-system case is A318402.
The case of strict parts is A330783.
Multiset partitions of integer partitions are A001970.
Unlabeled multiset partitions are A007716.
Cf. A001055, A255906, A269134, A316652, A317654, A318284, A330467, A330471, A330475, A330675.

Programs

  • Maple
    with(numtheory):
g:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, g(n/d, d)), d=divisors(n) minus {1, n}))
    end:
b:= proc(n, i, l)
      `if`(n=0, g(mul(ithprime(t)^l[t], t=1..nops(l))$2),
      `if`(i<1, 0, add(b(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> b(n$2, []):
seq(a(n), n=1..10);  # Alois P. Heinz, May 26 2013
  • Mathematica
    g[n_, k_] := g[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, g[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]]; b[n_, i_, l_] := If[n == 0, g[p = Product[Prime[t]^l[[t]], {t, 1, Length[l]}], p], If[i < 1, 0, Sum[b[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_] := b[n, n, {}]; Table[Print[an = a[n]]; an, {n, 1, 13}] (* Jean-François Alcover, Dec 12 2013, after Alois P. Heinz *)
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Dec 30 2020
  • Python
    from sympy.core.cache import cacheit
from sympy import divisors, isprime, prime
from operator import mul
@cacheit
def g(n, k):
    return (0 if n > k else 1) + (0 if isprime(n) else sum(g(n//d, d) for d in divisors(n)[1:-1] if d <= k))
@cacheit
def b(n, i, l):
    if n==0:
        p = reduce(mul, (prime(t + 1)**l[t] for t in range(len(l))))
        return g(p, p)
    else:
        return 0 if i<1 else sum([b(n - i*j, i - 1, l + [i]*j) for j in range(n//i + 1)])
def a(n):
    return b(n, n, [])
for n in range(1, 11): print(a(n)) # Indranil Ghosh, Aug 19 2017, after Maple code

Extensions

More terms from Erich Friedman.
81199 from Alford Arnold, Mar 04 2008
a(10) from Alford Arnold, Mar 31 2008
a(10) corrected by Alford Arnold, Aug 07 2008
a(11)-a(13) from Alois P. Heinz, May 26 2013
a(14) from Alois P. Heinz, Sep 27 2014
a(15) from Alois P. Heinz, Jan 10 2015
Terms a(16) and beyond from Andrew Howroyd, Dec 30 2020

A317755 Number of multiset partitions of strongly normal multisets of size n such that the blocks have empty intersection.

Original entry on oeis.org

0, 1, 6, 30, 130, 629, 2930, 15019, 78224, 438626, 2548481
Offset: 1

Views

Author

Gus Wiseman, Aug 06 2018

Keywords

Comments

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

Examples

			The a(3) = 6 strongly normal multiset partitions with empty intersection:
  {{2},{1,1}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1},{1},{2}}
  {{1},{2},{3}}

Crossrefs

Cf. A007716, A035310, A255906, A317073, A281116, A317077, A317078, A317532, A317533.
Cf. A317748, A317751, A317752, A317757, A317775.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,strnorm[n]}]],{n,6}]

Extensions

a(10)-a(11) from Robert Price, May 08 2021

A381452 Number of multisets that can be obtained by partitioning the prime indices of n into a set of multisets and taking their sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 4, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 7, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 6, 1, 2, 3, 3, 2, 5, 1, 6, 2, 2, 1, 8, 2, 2, 2
Offset: 1

Views

Author

Gus Wiseman, Mar 06 2025

Keywords

Comments

First differs from A045778 at a(24) = 4, A045778(24) = 5.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into distinct factors > 1.
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.
A multiset partition can be regarded as an arrow in the poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Sets of multisets are generally not transitive. For example, we have arrows: {{1},{2},{1,2}}: {1,1,2,2} -> {1,2,3} and {{1,2},{3}}: {1,2,3} -> {3,3}, but there is no set of multisets {1,1,2,2} -> {3,3}.

Examples

			The prime indices of 24 are {1,1,1,2}, with 5 partitions into a set of multisets:
  {{1,1,1,2}}
  {{1},{1,1,2}}
  {{2},{1,1,1}}
  {{1,1},{1,2}}
  {{1},{2},{1,1}}
with block-sums: {5}, {1,4}, {2,3}, {2,3}, {1,2,2}, of which 4 are distinct, so a(24) = 4.

Crossrefs

Before taking sums we had A045778.
If each block is a set we have A381441, before sums A050326.
For distinct block-sums instead of blocks we have A381637, before sums A321469.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For sets of constant multisets (A050361) see A381715.
- For set systems with distinct sums (A381633) see A381634, zeros A293243.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
More on sets of multisets: A261049, A317776, A317775, A296118, A318286.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A001970, A002846, A066328, A213385, A213427, A299200, A299202, A300385, A317142.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@#&]]],{n,100}]

Formula

a(A002110(n)) = A066723(n).

A317776 Number of strict multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers.

Original entry on oeis.org

1, 1, 3, 13, 59, 313, 1847, 11977, 84483, 642405, 5228987, 45297249, 415582335, 4021374193, 40895428051, 435721370413, 4850551866619, 56282199807401, 679220819360775, 8508809310177481, 110454586096508563, 1483423600240661781, 20581786429087269819
Offset: 0

Views

Author

Gus Wiseman, Aug 06 2018

Keywords

Examples

			The a(3) = 13 strict multiset partitions:
  {{1,1,1}}, {{1},{1,1}},
  {{1,2,2}}, {{1},{2,2}}, {{2},{1,2}},
  {{1,1,2}}, {{1},{1,2}}, {{2},{1,1}},
  {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}.

Links

Crossrefs

Cf. A001055, A007716, A045778, A255906, A281116, A317449, A317532, A317583, A317653, A317752, A317757, A317775.
Row sums of A327116.

Programs

  • Maple
    C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 16 2019
  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@#&]],{n,9}]
(* Second program: *)
c := Binomial;
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k] c[c[k+i-1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[b[n, n, i] (-1)^(k-i) c[k, i], {k, 0, n}, {i, 0, k}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Extensions

a(0), a(8)-a(22) from Alois P. Heinz, Sep 16 2019

A330783 Number of set multipartitions (multisets of sets) of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

Original entry on oeis.org

1, 1, 3, 8, 27, 94, 385, 1673, 8079, 41614, 231447, 1364697, 8559575, 56544465, 393485452, 2867908008, 21869757215, 173848026202, 1438593095272, 12360614782433, 110119783919367, 1015289796603359, 9674959683612989, 95147388659652754, 964559157655032720, 10067421615492769230
Offset: 0

Views

Author

Gus Wiseman, Jan 02 2020

Keywords

Comments

The (weakly) normal version is A116540.

Examples

			The a(1) = 1 through a(3) = 8 set multipartitions:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{1}}  {{1},{1,2}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}
The a(4) = 27 set multipartitions:
  {{1},{1},{1},{1}}  {{1},{1},{1,2}}  {{1},{1,2,3}}  {{1,2,3,4}}
  {{1},{1},{1},{2}}  {{1},{1},{2,3}}  {{1,2},{1,2}}
  {{1},{1},{2},{2}}  {{1},{2},{1,2}}  {{1,2},{1,3}}
  {{1},{1},{2},{3}}  {{1},{2},{1,3}}  {{1},{2,3,4}}
  {{1},{2},{3},{4}}  {{1},{2},{3,4}}  {{1,2},{3,4}}
                     {{1},{3},{1,2}}  {{1,3},{2,4}}
                     {{1},{3},{2,4}}  {{1,4},{2,3}}
                     {{1},{4},{2,3}}  {{2},{1,3,4}}
                     {{2},{3},{1,4}}  {{3},{1,2,4}}
                     {{2},{4},{1,3}}  {{4},{1,2,3}}
                     {{3},{4},{1,2}}

Links

Crossrefs

Allowing edges to be multisets gives is A035310.
The strict case is A318402.
The constant case is A000005.
The (weakly) normal version is A116540.
Unlabeled set multipartitions are A049311.
Set multipartitions of prime indices are A050320.
Set multipartitions of integer partitions are A089259.
Cf. A001055, A047968, A255906, A269134, A283877, A296119, A317775, A318360, A318362, A330625, A330628.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],And@@UnsameQ@@@#&]],{n,0,5}]
  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=WeighT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n)))/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Dec 30 2020

Extensions

Terms a(10) and beyond from Andrew Howroyd, Dec 30 2020

A317449 Regular triangle where T(n,k) is the number of multiset partitions of strongly normal multisets of size n into k blocks, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 5, 21, 16, 5, 7, 52, 72, 32, 7, 11, 141, 306, 216, 65, 11, 15, 327, 1113, 1160, 512, 113, 15, 22, 791, 4033, 6052, 3737, 1154, 199, 22, 30, 1780, 13586, 28749, 24325, 10059, 2317, 323, 30, 42, 4058, 45514, 133642, 151994, 82994, 24854, 4493, 523, 42
Offset: 1

Views

Author

Gus Wiseman, Aug 06 2018

Keywords

Examples

			The T(3,2) = 6 multiset partitions are {{1},{1,1}}, {{1},{1,2}}, {{2},{1,1}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}.
Triangle begins:
    1
    2    2
    3    6    3
    5   21   16    5
    7   52   72   32    7
   11  141  306  216   65   11
   15  327 1113 1160  512  113   15
   ...

Links

Crossrefs

Row sums are A035310. First and last columns are both A000041.
Cf. A001055, A007716, A045778, A255906, A281116, A317584, A317654, A317755, A317775, A317776.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],Length[#]==k&]],{n,6},{k,n}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}
U(m, n)={my(s=0); forpart(p=m, s+=D(p,n)); s}
M(n)={Mat(vector(n,k,(U(k,n)-U(k-1,n))~))}
{ my(A=M(8)); for(n=1, #A~, print(A[n,1..n])) } \\ Andrew Howroyd, Dec 30 2020

Extensions

Terms a(46) and beyond from Andrew Howroyd, Dec 30 2020

A368099 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element sets of finite nonempty multisets with cardinalities summing to n, or strict multiset partitions of weight n and length k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 5, 1, 0, 7, 28, 22, 5, 1, 0, 11, 66, 83, 31, 5, 1, 0, 15, 134, 252, 147, 34, 5, 1, 0, 22, 280, 726, 620, 203, 35, 5, 1, 0, 30, 536, 1946, 2283, 1069, 235, 35, 5, 1, 0, 42, 1043, 4982, 7890, 5019, 1469, 248, 35, 5, 1
Offset: 0

Views

Author

Gus Wiseman, Dec 31 2023

Keywords

Examples

			Triangle begins:
    1
    0    1
    0    2    1
    0    3    4    1
    0    5   12    5    1
    0    7   28   22    5    1
    0   11   66   83   31    5    1
    0   15  134  252  147   34    5    1
    0   22  280  726  620  203   35    5    1
    0   30  536 1946 2283 1069  235   35    5    1
    0   42 1043 4982 7890 5019 1469  248   35    5    1
    ...
Row n = 4 counts the following representatives:
  .  {{1,1,1,1}}  {{1},{1,1,1}}  {{1},{2},{1,1}}  {{1},{2},{3},{4}}
     {{1,1,1,2}}  {{1},{1,1,2}}  {{1},{2},{1,2}}
     {{1,1,2,2}}  {{1},{1,2,2}}  {{1},{2},{1,3}}
     {{1,1,2,3}}  {{1},{1,2,3}}  {{1},{2},{3,3}}
     {{1,2,3,4}}  {{1},{2,2,2}}  {{1},{2},{3,4}}
                  {{1},{2,2,3}}
                  {{1},{2,3,4}}
                  {{1,1},{1,2}}
                  {{1,1},{2,2}}
                  {{1,1},{2,3}}
                  {{1,2},{1,3}}
                  {{1,2},{3,4}}

Links

Crossrefs

Row sums are A316980, connected case A319557.
For multiset partitions we have A317533, connected A322133.
Counting connected components instead of edges gives A321194.
For normal multiset partitions we have A330787, row sums A317776.
For set multipartitions we have A334550.
For set-systems we have A368096, row-sums A283877 (connected A300913).
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A049311 counts non-isomorphic set multipartitions, connected A056156.
A058891 counts set-systems, unlabeled A000612, connected A323818.
Cf. A255903, A296122, A302545, A306005, A317532, A317775, A317794, A317795, A319560, A368094, A368095.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024
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