A134640 -id:A134640 - 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

A061845 Numbers that have one of every digit in some base.

Original entry on oeis.org

2, 11, 15, 19, 21, 75, 78, 99, 108, 114, 120, 135, 141, 147, 156, 177, 180, 198, 201, 210, 216, 225, 228, 694, 698, 714, 722, 738, 742, 894, 898, 954, 970, 978, 990, 1014, 1022, 1054, 1070, 1102, 1110, 1138, 1142, 1178, 1190, 1202, 1210, 1294, 1298, 1334

Offset: 2

Erich Friedman, Jun 23 2001

Also known as pandigital numbers, especially in base 10.

			Base 3 values are 102_3 = 11, 120_3 = 15, 201_3 = 19, 210_3 = 21.
Triangle begins:
    2;
   11,  15,  19,  21;
   75,  78,  99, 108, 114, 120, 135, 141, 147, 156, 177, 180,  198,  201, ...
  694, 698, 714, 722, 738, 742, 894, 898, 954, 970, 978, 990, 1014, 1022, ...
  ...

Alois P. Heinz, Rows n = 2..8, flattened

Column k=1 gives A049363 (for n>1).
Last elements of rows give A062813.
Cf. A050278, A134640, A001563 (row lengths).

dtn[ L_, base_ ] := Fold[ base*#1+#2&, 0, L ] f[ n_ ] := Map[ dtn[ #, n ]&, Select[ Permutations[ Range[ 0, n-1 ] ], First[ # ]>0& ] ] Flatten[ Join[ Table[ f[ i ], {i, 2, 5} ] ] ]

A051845 Triangle T(n,k) read by rows, in which row n gives all permutations of digits 1..n interpreted in base n+1.

Original entry on oeis.org

1, 5, 7, 27, 30, 39, 45, 54, 57, 194, 198, 214, 222, 238, 242, 294, 298, 334, 346, 358, 366, 414, 422, 434, 446, 482, 486, 538, 542, 558, 566, 582, 586, 1865, 1870, 1895, 1905, 1930, 1935, 2045, 2050, 2105, 2120, 2140, 2150, 2255, 2265, 2285, 2300, 2355, 2360

Offset: 1

Antti Karttunen, Dec 13 1999

All terms in any odd row 2m+1 are divisible by 2m+1
The n-th row has n! elements.
Variant of permutational numbers with shifted digits 0->1->2->...->p+1 in p+1 positional system -- see A134750. - Artur Jasinski, Nov 08 2007
From Alexander R. Povolotsky, Oct 22 2022: (Start)
All terms in any even-indexed row n=2m are divisible by m, where m>0.
Row n starts with T(n,1) = ((n+1)^(n+1)-n^2-n-1)/n^2 = A023811(n+1).
Row n ends with T(n,n!) = ((n+1)^(n+1)*(n-1)+1)/n^2 = A051846(n).
(End)

			Triangle begins:
         k=1   k=2   k=3  ...
  n=1:     1;
  n=2      5,    7;
  n=3:    27,   30,  ...,   57;
  n=4:   194,  198,  214,  ..., 586;
  n=5:  1865, 1870, 1905, 1930, ..., 7465;
E.g., the permutations of digits 1, 2 and 3 in lexicographic order are 123, 132, 213, 231, 312, 321, which interpreted in base 4 give the third row of the table: 27, 30, 39, 45, 54, 57.

Samuel Harkness, Table of n, a(n) for n = 1..10000

Left edge = A023811, right edge = A051846.

Cf. A134640, A134750.

with(combinat,permute); compute_u_rows := proc(u) local a,n; a := []; for n from 1 to u do a := [op(a),op(map(list_in_base_b,permute(n),(n+1)))]; od; RETURN(a); end; list_in_base_b := proc(l,b) local k; add(l[nops(l)-k]*(b^k), k=0..(nops(l)-1)); end;

a = {}; b = {}; Do[AppendTo[b, n]; w = Permutations[b]; Do[j = FromDigits[1 + w[[m]], n + 2]; AppendTo[a, j], {m, 1, Length[w]}], {n, 0, 5}]; a (* Artur Jasinski, Nov 08 2007 *)

from itertools import permutations
def fd(d, b): return sum(di*b**i for i, di in enumerate(d[::-1]))
def row(n): return [fd(d, n+1) for d in permutations(range(1, n+1))]
print([an for r in range(1, 6) for an in row(r)]) # Michael S. Branicky, Oct 21 2022

A134750 Variant of permutational numbers with shifted digits 0->1->2->...->k+1 in k+1 positional system.

Original entry on oeis.org

1, 4, 5, 18, 20, 24, 28, 32, 34, 112, 115, 124, 130, 139, 142, 160, 163, 184, 193, 199, 205, 220, 226, 232, 241, 262, 265, 283, 286, 295, 301, 310, 313, 975, 979, 995, 1003, 1019, 1023, 1075, 1079, 1115, 1127, 1139, 1147, 1195, 1203, 1215, 1227, 1263, 1267

Offset: 1

Artur Jasinski, Nov 08 2007

Variant of permutational numbers with shifted digits 0->1->2->...->k+1 in k+2 positional system see A051845

Cf. A134640, A051845.

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

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

A134766 Primes among variant of permutational numbers A134750.

Original entry on oeis.org

5, 139, 163, 193, 199, 241, 283, 313, 1019, 1319, 1367, 1523, 1759, 2179, 2251, 2539, 2767, 2803, 2851, 2927, 3011, 3163, 3167, 3191, 3319, 3323, 3347, 3491, 3539, 3547, 3559, 3571, 3607, 3691, 11261, 11471, 11801, 11831, 12281, 12641, 13121, 13721

Offset: 1

Artur Jasinski, Nov 10 2007

nonn

For indices where primes occurs in A134750, see A134767.

Cf. A134640, A134641, A134750, A134767.

k = {}; a = {}; b = {}; Do[AppendTo[b, n]; w = Permutations[b]; Do[j = FromDigits[1 + w[[m]], n + 1]; AppendTo[a, j], {m, 1, Length[w]}], {n, 0, 6}]; Do[If[PrimeQ[a[[n]]], AppendTo[k, a[[n]]]], {n, 1, Length[a]}]; k

A134767 Indices where primes occur in variant of permutational numbers A134750.

Original entry on oeis.org

3, 14, 17, 19, 20, 25, 28, 33, 38, 52, 57, 63, 68, 89, 91, 100, 111, 113, 115, 119, 123, 124, 125, 127, 130, 131, 133, 141, 142, 143, 144, 145, 146, 151, 158, 164, 172, 174, 179, 185, 198, 209, 212, 256, 258, 275, 294, 299, 307, 309, 323, 326, 331, 332, 337

Offset: 1

Artur Jasinski, Nov 10 2007

nonn

For primes in A134750 see A134766

Cf. A134640, A134641, A134750, A134766.

k = {}; a = {}; b = {}; Do[AppendTo[b, n]; w = Permutations[b]; Do[j = FromDigits[1 + w[[m]], n + 1]; AppendTo[a, j], {m, 1, Length[w]}], {n, 0, 6}]; Do[If[PrimeQ[a[[n]]], AppendTo[k, n]], {n, 1, Length[a]}]; k

A134768 First differences of variant of permutational numbers A134750 .

Original entry on oeis.org

3, 1, 13, 2, 4, 4, 4, 2, 78, 3, 9, 6, 9, 3, 18, 3, 21, 9, 6, 6, 15, 6, 6, 9, 21, 3, 18, 3, 9, 6, 9, 3, 662, 4, 16, 8, 16, 4, 52, 4, 36, 12, 12, 8, 48, 8, 12, 12, 36, 4, 52, 4, 16, 8, 16, 4, 108, 4, 16, 8, 16, 4, 152, 4, 56, 16, 8, 12, 24, 8, 32, 16, 32, 8, 28, 4, 36, 12, 12, 8, 84, 4, 36, 12

Offset: 1

Artur Jasinski, Nov 10 2007

nonn

Cf. A134640, A134641, A134750, A134766, A134767.

k = {}; a = {}; b = {}; Do[AppendTo[b, n]; w = Permutations[b]; Do[j = FromDigits[1 + w[[m]], n + 1]; AppendTo[a, j], {m, 1, Length[w]}], {n, 0, 6}]; Do[AppendTo[k, a[[n + 1]] - a[[n]]], {n, 1, Length[a] - 1}]; k

cp's OEIS Frontend

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

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A061845 Numbers that have one of every digit in some base.

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A051845 Triangle T(n,k) read by rows, in which row n gives all permutations of digits 1..n interpreted in base n+1.

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A134750 Variant of permutational numbers with shifted digits 0->1->2->...->k+1 in k+1 positional system.

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

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A134766 Primes among variant of permutational numbers A134750.

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A134767 Indices where primes occur in variant of permutational numbers A134750.

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A134768 First differences of variant of permutational numbers A134750 .

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