A323548 Amicable numbers under the calculation of the determinant of the circulant matrix formed by their decimal digits.

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

Table of n, a(n) for n=1..14.

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

Cf. A219324, A323485, A323486.

Sequence in context: A330851 A245032 A208088 * A338382 A338384 A344702

Adjacent sequences: A323545 A323546 A323547 * A323549 A323550 A323551

Keyword

base,nonn,more

Author

Paolo P. Lava, Jan 18 2019

Extensions

a(7)-a(14) from Giovanni Resta, Jan 21 2019

Status

approved