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

Author

Paolo P. Lava, Jan 18 2019

Keywords

• base
• nonn
• more

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)

Examples

			          | 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 |
		

Links

• Eric Weisstein's World of Mathematics, Circulant Matrix.
• Wikipedia, Circulant matrix.

Crossrefs

Cf. A219324, A323485, A323486.

Programs

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):

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

Extensions

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