A134643 -id:A134643 - cp's OEIS frontend

A062813 a(n) = Sum_{i=0..n-1} i*n^i.

0, 2, 21, 228, 2930, 44790, 800667, 16434824, 381367044, 9876543210, 282458553905, 8842413667692, 300771807240918, 11046255305880158, 435659737878916215, 18364758544493064720, 824008854613343261192, 39210261334551566857170, 1972313422155189164466189, 104567135734072022160664820

Offset: 1

Olivier Gérard, Jun 23 2001

Largest Katadrome (number with digits in strict descending order) in base n.
The largest permutational number (A134640) of order n. These numbers are isomorphic with antidiagonal permutation matrices of order n. Where diagonal matrices are a[i,1+n-i]=1 {i=1,n} a[i<>1+n-i]=0 for smallest permutational numbers of order n see A023811. - Artur Jasinski, Nov 07 2007
Permutational numbers A134640 isomorphic with permutation matrix generators of cyclic groups, n-th root of unity matrices. - Artur Jasinski, Nov 07 2007
Rephrasing: Largest pandigital number in base n (in the sense of A050278, which is base 10); e.g., a(10) = A050278(3265920), its final term. With a(1) = 1 instead of 0, also accommodates unary (A000042). - Rick L. Shepherd, Jul 10 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..400
Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 1.

Last elements of rows of A061845 (for n>1).
Cf. A134640, A134641, A134642, A134643, A134644, A023811, A062808.
Cf. A000142, A000042, A050278.

a062813 n = foldr (\dig val -> val * n + dig) 0 [0 .. n - 1]
-- Reinhard Zumkeller, Aug 29 2014

0,seq(n*((n-2)*n^n + 1)/(n-1)^2,n=2..100); # Robert Israel, Sep 03 2014

Table[Sum[i*n^i, {i, 0, -1 + n}], {n, 17}] (* Olivier Gérard, Jun 23 2001 *)
a[n_] := FromDigits[ Range[ n-1, 0, -1], n]; Array[a, 18] (* Robert G. Wilson v, Sep 03 2014 *)

PARI
```
a(n) = sum(i=0,n-1,i*n^i)
```

a(n) = if (n==1,0, my(t=n^n); t-(t-n)/(n-1)^2); \\ Joerg Arndt, Sep 03 2014

def A062813(n): return (m:=n**n)-(m-n)//(n-1)**2 if n>1 else 0 # Chai Wah Wu, Mar 18 2024

a(n) = n^n - (n^n-n)/(n-1)^2 for n>1. - Dean Hickerson, Jun 26 2001

a(n) = A134640(n, A000142(n)). - Reinhard Zumkeller, Aug 29 2014

A134644 Even permutational numbers A134640.

Original entry on oeis.org

0, 2, 30, 54, 78, 108, 114, 120, 156, 180, 198, 210, 216, 228, 194, 198, 214, 222, 238, 242, 294, 298, 334, 346, 358, 366, 414, 422, 434, 446, 482, 486, 538, 542, 558, 566, 582, 586, 694, 698, 714, 722, 738, 742, 894, 898, 954, 970, 978, 990, 1014, 1022, 1054

Offset: 1

Artur Jasinski, Nov 05 2007

nonn

Odd permutational numbers see A134643

Cf. A134640, A134641, A134642, A134643.

a = {}; b = {}; Do[AppendTo[b, n]; w =Permutations[b]; Do[j = FromDigits[w[[m]], n + 1]; If[OddQ[j], AppendTo[a, j]], {m, 1, Length[w]}], {n, 0, 10}]; a (*Artur Jasinski*)

A134741 Permutational numbers A134640 which are squares.

Original entry on oeis.org

0, 1, 225, 2500, 7225, 38025, 106929, 314721, 622521, 751689, 1750329, 3111696, 6002500, 7568001, 8168164, 8282884, 10323369, 11682724, 12517444, 23367556, 23483716, 25623844, 28536964, 33292900, 39513796, 61058596, 73513476, 74545956, 94517284, 105144516, 112572100, 112656996, 132756484

Offset: 1

Artur Jasinski, Nov 07 2007

nonn

Robert Israel, Table of n, a(n) for n = 1..261

Cf. A134640, A134641, A134642, A134643, A134644, A023811, A134742.

N:= 10^10: # for terms <= N
extend:= proc(x, N, S, b, k)
  local i, R;
  R:= NULL;
  for i in S while x + i*b^k <= N do
    if k = 0 then
       if issqr(x+i*b^k) then R:= R, x+i*b^k fi
    else
       R:= R, procname(x+i*b^k, N, subs(i=NULL, S), b, k-1)
    fi
  od;
  R
end proc:
f:= (b, N) -> extend(0, N, [$0..(b-1)], b, b-1):
R:= 0:
for b from 2 while b^(b-2) < N do
  R:= R, f(b, N);
od:
sort([R]); # Robert Israel, Sep 04 2020

a = {}; b = {}; Do[AppendTo[b, n]; w = Permutations[b]; Do[j = FromDigits[w[[m]], n + 1]; If[IntegerQ[j^(1/2)], AppendTo[a, j]], {m, 1, Length[w]}], {n, 0, 7}]; a

a(n) = A134742(n)^2.

Corrected and more terms from Robert Israel, Sep 04 2020

A062808 a(n) = Sum_{i=1..n} n^i * (n - i).

Original entry on oeis.org

0, 2, 15, 108, 970, 11190, 160125, 2739128, 54480996, 1234567890, 31384283755, 884241366756, 27342891567342, 920521275489998, 33512287529147385, 1311768467463790320, 54933923640889550728, 2450641333409472928554

Offset: 1

Olivier Gérard, Jun 23 2001

Permutational numbers A134640 isomorphic with permutation matrix generators of cyclic groups, n-th root of unity matrices. - Artur Jasinski, Nov 07 2007

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A134640, A134641, A134642, A134643, A134644, A023811.

Sum[n^i*(n - i), {i, 1, n}]
a = {}; b = {}; c = {}; Do[AppendTo[b, n]; c = b; AppendTo[c, 0]; AppendTo[a, FromDigits[c, n + 1]], {n, 1, 20}]; a (* Artur Jasinski, Nov 07 2007 *)

a(n) = sum(i=1, n, n^i*(n-i)); \\ Michel Marcus, Mar 26 2019

a(n) = (n^(n+1)-n^3+n^2-n)/(n-1)^2 for n>1. - Dean Hickerson, Jun 26 2001

A134742 Numbers whose square is a permutational number A134640.

Original entry on oeis.org

0, 1, 15, 50, 85, 195, 327, 561, 789, 867, 1323, 1764, 2450, 2751, 2858, 2878, 3213, 3418, 3538, 4834, 4846, 5062, 5342, 5770, 6286, 7814, 8574, 8634, 9722, 10254, 10610, 10614, 11522, 11702, 11826, 12363, 12543, 13490, 14246, 14502, 14538, 14676, 14818, 14902, 15186, 15434, 15681, 15874, 15963

Offset: 1

Artur Jasinski, Nov 07 2007

nonn

Robert Israel, Table of n, a(n) for n = 1..261

Cf. A134640, A134641, A134642, A134643, A134644, A023811, A134741.

N:= 10^5: # for terms <= N
extend:= proc(x,N,S,b,k)
  local i,R;
  R:= NULL;
  for i in S while x + i*b^k <= N^2 do
    if k = 0 then
       if issqr(x+i*b^k) then R:= R, sqrt(x+i*b^k) fi
    else
       R:= R, procname(x+i*b^k,N,subs(i=NULL,S),b,k-1)
    fi
  od;
  R
end proc:
f:= (b,N) -> extend(0,N,[$0..(b-1)],b,b-1):
R:= 0:
for b from 2 while b^(b-2) < N^2 do
  R:= R, f(b,N);
od:
sort([R]); # Robert Israel, Sep 04 2020

a = {}; b = {}; Do[AppendTo[b, n]; w =Permutations[b]; Do[j = FromDigits[w[[m]], n + 1]; If[IntegerQ[j^(1/2)], AppendTo[a, j]], {m, 1, Length[w]}], {n, 0, 7}]; Sqrt[a]

a(n) = sqrt(A134741(n)).

Corrected and more terms from Robert Israel, Sep 04 2020

A134745 Numbers which are not permutational numbers A134640.

Original entry on oeis.org

3, 4, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Artur Jasinski, Nov 07 2007

nonn

Cf. A134640, A134641, A134642, A134643, A134644, A134703, A134741, A134742, A134743, A134744.

a = {}; b = {}; Do[AppendTo[b, n]; w = Permutations[b]; Do[j = FromDigits[w[[m]], n + 1]; AppendTo[a, j], {m, 1, Length[w]}], {n, 0, 5}]; c = Table[n, {n, 0, 44790}]; k = Complement[c, a] (*Artur Jasinski*)

cp's OEIS Frontend

A062813 a(n) = Sum_{i=0..n-1} i*n^i.

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A134644 Even permutational numbers A134640.

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A134741 Permutational numbers A134640 which are squares.

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A062808 a(n) = Sum_{i=1..n} n^i * (n - i).

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A134742 Numbers whose square is a permutational number A134640.

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A134745 Numbers which are not permutational numbers A134640.

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