A306853 Positive integers equal to the permanent of the circulant matrix formed by their decimal digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 261, 370, 407, 52036, 724212, 223123410

Offset

1,2

Comments

1, 2, 3, 4, 5, 6, 7, 8, 9, 370 and 407 are also equal to the determinant of the circulant matrix formed by their decimal digits.

Links

Table of n, a(n) for n=1..15.

Eric Weisstein's World of Mathematics, Permanent

Eric Weisstein's World of Mathematics, Circulant Matrix

Example

     | 2 6 1 |
perm | 1 2 6 | = 2*2*2 + 6*6*6 + 1*1*1 + 1*2*6 + 6*1*2 + 2*6*1 = 261.
     | 6 1 2 |
.
     | 2 2 3 1 2 3 4 1 0 |
     | 0 2 2 3 1 2 3 4 1 |
     | 1 0 2 2 3 1 2 3 4 |
     | 4 1 0 2 2 3 1 2 3 |
perm | 3 4 1 0 2 2 3 1 2 | = 223123410
     | 2 3 4 1 0 2 2 3 1 |
     | 1 2 3 4 1 0 2 2 3 |
     | 3 1 2 3 4 1 0 2 2 |
     | 2 3 1 2 3 4 1 0 2 |

Maple

with(linalg): P:=proc(q) local a, b, c, d, i, j, k, n, t;
for n from 1 to q do d:=ilog10(n)+1; a:=convert(n, base, 10); c:=[];
for k from 1 to nops(a) do c:=[op(c), a[-k]]; od; t:=[op([]), c];
for k from 2 to d do b:=[op([]), c[nops(c)]];
for j from 1 to nops(c)-1 do b:=[op(b), c[j]]; od;
c:=b; t:=[op(t), c]; od; if n=permanent(t)
then print(n); fi; od; end: P(10^7);

Prog

(PARI) mpd(n) = {my(d = digits(n)); matpermanent(matrix(#d, #d, i, j, d[1+lift(Mod(j-i, #d))])); }
isok(n) = mpd(n) == n; \\ Michel Marcus, Mar 14 2019
(Python)
from sympy import Matrix
A306853_list = []
for n in range(1, 10**6):
    s = [int(d) for d in str(n)]
    m = len(s)
    if n == Matrix(m, m, lambda i, j: s[(i-j) % m]).per():
        A306853_list.append(n) # Chai Wah Wu, Oct 18 2021

Crossrefs

Cf. A219324, A219327, A306662, A306593, A306714.

Up to n=110 the permanent of the circulant matrix of the digits of n is equal to A101337 but from n=111 on it can differ.

Sequence in context: A085134 A229761 A004882 * A308110 A306593 A046469

Adjacent sequences: A306850 A306851 A306852 * A306854 A306855 A306856

Keyword

nonn,base,more

Author

Paolo P. Lava, Mar 13 2019

Extensions

a(15) from Vaclav Kotesovec, Aug 19 2021

Status

approved