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A323548
Amicable numbers under the calculation of the determinant of the circulant matrix formed by their decimal digits.
0
108, 182, 473, 513, 1139005, 3798233, 142250866, 186519853, 245578912, 387304234, 12410397495, 15303786345, 28309184956, 28670744905
OFFSET
1,1
COMMENTS
Terms of A219324 are not in the list because they are perfect under the same rule.
The pairs in the listed terms are {108, 513}, {182, 473}, {1139005, 3798233}, {142250866, 387304234}, {186519853, 245578912}, {12410397495, 15303786345}, {28309184956, 28670744905}.
From David A. Corneth, Jan 21 2019: (Start)
For all 3-digit numbers k, the corresponding matrices of permutations of digits (unless perhaps leading 0) have the same determinant. In general, the number of determinants is much less than the number of permutations of digits.
Can permutations be "classified" to narrow the search space when finding terms?
Are there any terms with an even number of digits? (End)
LINKS
Eric Weisstein's World of Mathematics, Circulant Matrix.
Wikipedia, Circulant matrix.
EXAMPLE
| 1 0 8 | | 5 1 3 |
det | 8 1 0 | = 513 and det | 3 5 1 | = 108.
| 0 8 1 | | 1 3 5 |
.
| 1 8 2 | | 4 7 3 |
det | 2 1 8 | = 473 and det | 3 4 7 | = 182.
| 8 2 1 | | 7 3 4 |
MAPLE
with(linalg): P:=proc(q) local a, b, c, d, j, k, n, p, t, x, y:
for n from 1 to q do x:=n: for p from 1 to 2 do
d:=ilog10(x)+1: a:=convert(x, 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; x:=det(t): if x=0 then break:
else if p=1 then y:=x: fi: fi: od:
if n=x and y<>x then print(n); fi: od: end: P(10^8):
PROG
(PARI) is(n) = my(c = amidet(n)); if(c == n, return(0)); amidet(c) == n
amidet(n) = my(d = digits(n), qd = #d, m = matrix(qd, qd)); for(i = 1, qd, for(j = 1, qd, m[i, j] = d[1 + (j - i)%qd])); ami = matdet(m); ami \\ David A. Corneth, Jan 21 2019
CROSSREFS
KEYWORD
base,nonn,more
AUTHOR
Paolo P. Lava, Jan 18 2019
EXTENSIONS
a(7)-a(14) from Giovanni Resta, Jan 21 2019
STATUS
approved