A134750 -id:A134750 - cp's OEIS frontend

A134766 Primes among variant of permutational numbers A134750.

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

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

A134749 First differences of permutational numbers : a(n)=A134640(n+1)-A134640(n).

Original entry on oeis.org

1, 1, 3, 2, 4, 4, 4, 2, 6, 3, 9, 6, 9, 3, 18, 3, 21, 9, 6, 6, 15, 6, 6, 9, 21, 3, 18, 3, 9, 6, 9, 3, -34, 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, 12, 8, 28, 4, 56, 16, 8, 12, 144, 12, 8, 16, 56, 4, 28, 8, 12

Offset: 1

Artur Jasinski, Nov 08 2007

sign

Negative numbers occur at n=!k = A007489(k) and {k=3,4,5,...}

Cf. A007489, A134640, A134703, A134748, A134750.

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}]; d = {}; Do[AppendTo[d, a[[n + 1]] - a[[n]]], {n, 1, Length[a] - 1}] ; d (*Artur Jasinski*)
Differences[Flatten[Table[FromDigits[#,n]&/@Permutations[Range[0,n-1]], {n,5}]]] (* Harvey P. Dale, Dec 09 2014 *)

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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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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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A134749 First differences of permutational numbers : a(n)=A134640(n+1)-A134640(n).

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