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A323548
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Amicable numbers under the calculation of the determinant of the circulant matrix formed by their decimal digits.
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0
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108, 182, 473, 513, 1139005, 3798233, 142250866, 186519853, 245578912, 387304234, 12410397495, 15303786345, 28309184956, 28670744905
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OFFSET
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1,1
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COMMENTS
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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}.
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)
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LINKS
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EXAMPLE
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| 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 |
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MAPLE
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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):
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PROG
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(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
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CROSSREFS
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KEYWORD
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base,nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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