cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A306593 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 334, 65, 42, 76, 455, 41, 40, 98, 123, 667, 64, 52, 778, 788, 51, 50, 899, 63, 86, 7787, 2025885, 8788, 62, 74, 46996, 61, 60, 66898, 67997, 85, 73, 78998, 88899, 88999, 335, 72, 4579975, 878888, 71, 70, 10243, 5354, 355, 989999, 114
Offset: 0

Views

Author

Paolo P. Lava, Feb 27 2019

Keywords

Comments

Here only the least numbers are listed: e.g., a(75) = 1031, even if 10002110 also produces 75.
The sequence is infinite because any number of the form (91*10^n - 10) / 90 for n > 0 (A267623 or A283508) has the determinant of the circulant matrix equal to n but, in general, it is not the least possible term. - Giovanni Resta, Mar 06 2019

Examples

			                        | 3 3 4 |
a(10) = 334 because det | 4 3 3 | = 10
                        | 3 4 3 |
.
and 334 is the least number to have this property.
.
                          | 4 6 9 9 6 |
                          | 6 4 6 9 9 |
a(34) = 46996 because det | 9 6 4 6 9 | = 34
                          | 9 9 6 4 6 |
                          | 6 9 9 6 4 |
.
and 46996 is the least number to have this property.

Links

Crossrefs

Cf. A177894, A219324, A267623, A283508, A306662, A323485, A323486.

Programs

  • Maple
    with(linalg): P:=proc(q) local a,b,c,d,j,k,i,n,t;
print(0); for i from 1 to q do for n from 1 to q do
d:=ilog10(n)+1; a:=convert(n, 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;
if i=det(t) then print(n); break; fi; od; od; end: P(10^7);
  • PARI
    md(n) = my(d = if (n, digits(n), [0])); matdet(matrix(#d, #d, i, j, d[1+lift(Mod(j-i, #d))]));
a(n) = my(k=0); while(md(k) != n, k++); k; \\ Michel Marcus, Mar 20 2019

Formula

A177894(a(n)) = n when a(n) >= 0. - Rémy Sigrist, Feb 27 2019

A323486 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to n*k.

Original entry on oeis.org

1, 10168, 119700, 196, 1973082, 63980523693, 167037139360, 1350720096, 1543479071, 17239680, 4000206089, 219566358180, 104171259465, 2380649994, 113323907385, 14059155927, 19925280
Offset: 1

Views

Author

Paolo P. Lava, Jan 17 2019

Keywords

Examples

			      det | 1 | = 1 = 1*1.
.
          | 1 0 1 6 8|
          | 8 1 0 1 6|
      det | 6 8 1 0 1| = 20336 = 2*10168.
          | 1 6 8 1 0|
          | 0 1 6 8 1|

Links

Crossrefs

Cf. A219324, A323485.

Programs

  • Maple
    with(linalg): P:=proc(q) local a,b,c,d,i,j,k,n,t;
for i from 1 to q do for n from 1 to q do
d:=ilog10(n)+1; a:=convert(n,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; if i*n=det(t) then
print(n); break; fi; od; od; end: P(10^7);

Extensions

a(6)-a(17) from Giovanni Resta, Jan 21 2019

A306662 Least number k such that the determinant of the circulant matrix of its representation in base 2 is equal to n.

Original entry on oeis.org

0, 1, 5, 11, 23, 47, 95, 191, 43, 38, 1535, 3071, 571, 12287, 24575, 137, 269, 196607, 393215, 786431, 295, 687, 6291455, 12582911, 69, 155, 100663295, 134, 293, 805306367, 1610612735, 3221225471, 75, 518, 25769803775, 301, 8874
Offset: 0

Views

Author

Paolo P. Lava, Mar 04 2019

Keywords

Comments

Here only the least numbers are listed: e.g., a(10) = 1531, even if 1791, 1919, 1983, 2015, 2031, 2039, 2043, etc. also produce 10.
The sequence is infinite because any number of the form 3*2^(n-1) - 1 (A083329) has the determinant of the circulant matrix of its representation in base 2 equal to n but, in general, it is not the least possible term.
It would be nice to characterize the values of n where k < A083329(n).

Examples

			                                      | 1 0 1 1 |
a(3) = 11 because 11 = 1011_2 and det | 1 1 0 1 | = 3
                                      | 1 1 1 0 |
                                      | 0 1 1 1 |
.
and 11 is the least number to have this property.
.
                                       | 1 0 1 1 1 |
                                       | 1 1 0 1 1 |
a(4) = 23 because 23 = 10111_2 and det | 1 1 1 0 1 | = 4
                                       | 1 1 1 1 0 |
                                       | 0 1 1 1 1 |
.
and 23 is the least number to have this property.

Crossrefs

Cf. A083329, A219324, A306593, A323485, A323486.

Programs

  • Maple
    with(linalg): P:=proc(q) local a, b, c, d, j, k, i, n, t;
print(0); for i from 1 to q do for n from 1 to q do
a:=convert(n, base, 2); d:=nops(a); 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;
if i=det(t) then print(n); break; fi; od; od; end: P(10^7);

Extensions

a(31)-a(36) from Giovanni Resta, Mar 05 2019

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

Original entry on oeis.org

108, 182, 473, 513, 1139005, 3798233, 142250866, 186519853, 245578912, 387304234, 12410397495, 15303786345, 28309184956, 28670744905
Offset: 1

Views

Author

Paolo P. Lava, Jan 18 2019

Keywords

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

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
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