cp's OEIS Frontend

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

%I A323548 #27 Jan 25 2024 14:30:53
%S A323548 108,182,473,513,1139005,3798233,142250866,186519853,245578912,
%T A323548 387304234,12410397495,15303786345,28309184956,28670744905
%N A323548 Amicable numbers under the calculation of the determinant of the circulant matrix formed by their decimal digits.
%C A323548 Terms of A219324 are not in the list because they are perfect under the same rule.
%C A323548 The pairs in the listed terms are {108, 513}, {182, 473}, {1139005, 3798233}, {142250866, 387304234}, {186519853, 245578912}, {12410397495, 15303786345}, {28309184956, 28670744905}.
%C A323548 From _David A. Corneth_, Jan 21 2019: (Start)
%C A323548 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.
%C A323548 Can permutations be "classified" to narrow the search space when finding terms?
%C A323548 Are there any terms with an even number of digits? (End)
%H A323548 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirculantMatrix.html">Circulant Matrix</a>.
%H A323548 Wikipedia, <a href="https://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix</a>.
%e A323548           | 1 0 8 |               | 5 1 3 |
%e A323548       det | 8 1 0 | = 513 and det | 3 5 1 | = 108.
%e A323548           | 0 8 1 |               | 1 3 5 |
%e A323548 .
%e A323548           | 1 8 2 |               | 4 7 3 |
%e A323548       det | 2 1 8 | = 473 and det | 3 4 7 | = 182.
%e A323548           | 8 2 1 |               | 7 3 4 |
%p A323548 with(linalg): P:=proc(q) local a,b,c,d,j,k,n,p,t,x,y:
%p A323548 for n from 1 to q do x:=n: for p from 1 to 2 do
%p A323548 d:=ilog10(x)+1: a:=convert(x,base,10): c:=[]:
%p A323548 for k from 1 to nops(a) do c:=[op(c),a[-k]]: od: t:=[op([]),c]:
%p A323548 for k from 2 to d do b:=[op([]),c[nops(c)]]:
%p A323548 for j from 1 to nops(c)-1 do b:=[op(b),c[j]]: od:
%p A323548 c:=b: t:=[op(t),c]: od; x:=det(t): if x=0 then break:
%p A323548 else if p=1 then y:=x: fi: fi: od:
%p A323548 if n=x and y<>x then print(n); fi: od: end: P(10^8):
%o A323548 (PARI) is(n) = my(c = amidet(n)); if(c == n, return(0)); amidet(c) == n
%o A323548 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
%Y A323548 Cf. A219324, A323485, A323486.
%K A323548 base,nonn,more
%O A323548 1,1
%A A323548 _Paolo P. Lava_, Jan 18 2019
%E A323548 a(7)-a(14) from _Giovanni Resta_, Jan 21 2019