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A306853
Positive integers equal to the permanent of the circulant matrix formed by their decimal digits.
2
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
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
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
KEYWORD
nonn,base,more
AUTHOR
Paolo P. Lava, Mar 13 2019
EXTENSIONS
a(15) from Vaclav Kotesovec, Aug 19 2021
STATUS
approved