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

Author

Paolo P. Lava, Mar 13 2019

Keywords

• nonn
• base
• more

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.

Examples

			     | 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 |
		

Links

• Eric Weisstein's World of Mathematics, Permanent
• Eric Weisstein's World of Mathematics, Circulant Matrix

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.

Programs

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);

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

Extensions

a(15) from Vaclav Kotesovec, Aug 19 2021