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Search: a130207 -id:a130207
Displaying 1-7 of 7 results found. page 1
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A143230 Triangle read by rows, A130207 * A000012 * A130207. +20
2
1, 1, 1, 2, 2, 4, 2, 2, 4, 4, 4, 4, 8, 8, 16, 2, 2, 4, 4, 8, 4, 6, 6, 12, 12, 24, 12, 36, 4, 4, 8, 8, 16, 8, 24, 16, 6, 6, 12, 12, 24, 12, 36, 24, 36, 4, 4, 8, 8, 16, 8, 24, 16, 24, 16, 10, 10, 20, 20, 40, 20, 60, 40, 60, 40, 100, 4, 4, 8, 8, 16, 8, 24, 16, 24, 16, 40, 16 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Left border = A000010.
Row sums = A143231: (1, 2, 8, 12, 40, 24, ...).
T(n,k) is the number of pairs (a,b), where 0 <= a < n, 0 <= b < k, gcd(a,n) != 1, and gcd(b,k) != 1. - Joerg Arndt, Jun 26 2011
LINKS
Nathaniel Johnston, Rows 1..100, flattened
FORMULA
Triangle read by rows, A130207 * A000012 * A130207, where A130207 = A000010 * 0^(n-k), 1 <= k <= n.
T(n,k) = phi(n) * phi(k), where phi(n) & phi(k) = Euler's totient function.
EXAMPLE
First few rows of the triangle:
1;
1, 1;
2, 2, 4;
2, 2, 4, 4;
4, 4, 8, 8, 16;
2, 2, 4, 4, 8, 4;
6, 6, 12, 12, 24, 12, 36;
4, 4, 8, 8, 16, 8, 24, 16;
6, 6, 12, 12, 24, 12, 36, 24, 36;
...
T(7,5) = 24 = phi(7) * phi(5) = 6 * 4.
MAPLE
with(numtheory): T := proc(n, k) return phi(n)*phi(k): end: seq(seq(T(n, k), k=1..n), n=1..12); # Nathaniel Johnston, Jun 26 2011
CROSSREFS
Cf. A000010, A130207, A143231.
KEYWORD
nonn,easy,tabl
AUTHOR
Gary W. Adamson, Jul 31 2008
STATUS
approved
A143267 Triangle read by rows, A130207 * A000012 * A127648. +20
1
1, 1, 2, 2, 4, 6, 2, 4, 6, 8, 4, 8, 12, 16, 20, 2, 4, 6, 8, 10, 12, 6, 12, 18, 24, 30, 36, 42, 4, 8, 12, 16, 20, 24, 28, 32, 6, 12, 18, 24, 30, 36, 42, 48, 54, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Left border = phi(n), A000010.
Row sums = A143268, phi(n)*T(n): (1, 3, 12, 20, 60, 42,...)
LINKS
Table of n, a(n) for n=1..65.
FORMULA
Triangle read by rows, A130207 * A000012 * A127648; 1<=k<=n. T(n,k) = phi(n)*k.
EXAMPLE
First few rows of the triangle =
1;
1, 2;
2, 4, 6;
2, 4, 6, 8;
4, 8, 12, 16, 20;
2, 4, 6, 8, 10, 12;
6, 12, 18, 24, 30, 36, 42;
4, 8, 12, 16, 20, 24, 28, 32;
...
Row 5 = (4, 8, 12, 16, 20) since the first terms of phi(5) = 4; so we perform (4*1, 4*2, 4*3, 4*4, 4*5).
CROSSREFS
Cf. A130207, A127648, A000010.
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Aug 03 2008
STATUS
approved
A143269 Triangle read by rows, A127648 * A000012 * A130207, 1<=k<=n. +20
1
1, 2, 2, 3, 3, 6, 4, 4, 8, 8, 5, 5, 10, 10, 20, 6, 6, 12, 12, 24, 12, 7, 7, 14, 14, 28, 14, 42, 8, 8, 16, 16, 32, 16, 48, 32, 9, 9, 18, 18, 36, 18, 54, 36, 54, 10, 10, 20, 20, 40, 20, 60, 40, 60, 40, 11, 11, 22, 22, 44, 22, 66, 44, 66, 44, 110 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Row sums = A143270: (1, 4, 12, 24, 50, 72, 126, 176,...).
LINKS
Table of n, a(n) for n=1..66.
FORMULA
Triangle read by rows, A127648 * A000012 * A130207. T(n,k) = n*phi(k)
EXAMPLE
First few rows of the triangle =
1;
2, 2;
3, 3, 6;
4, 4, 8, 8;
5, 5, 10, 10, 20;
6, 6, 12, 12, 24, 12;
7, 7, 14, 1428, 14, 42;
...
Row 5 = (5, 5, 10, 10, 20) = (5*1, 5*1, 5*2, 5*2, 5*4); where phi(k) = (1, 1, 2, 2, 4,...).
CROSSREFS
Cf. A000010, A143270.
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Aug 03 2008
STATUS
approved
A156836 Triangle read by rows, A156348 * A130207 +20
0
1, 1, 1, 1, 0, 2, 1, 2, 0, 2, 1, 0, 0, 0, 4, 1, 3, 6, 0, 0, 2, 1, 0, 0, 0, 0, 0, 6, 1, 4, 0, 8, 0, 0, 0, 4, 1, 0, 12, 0, 0, 0, 0, 0, 6, 1, 5, 0, 0, 20, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 6, 20, 20, 0, 12, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
Row sums = A156834: (1, 2, 3, 5, 5, 12, 7, 17, 19, 30, 11,...).
LINKS
Table of n, a(n) for n=1..91.
FORMULA
Triangle read by rows, A156348 * A130207, where A130207 = an infinite lower
triangular matrix with A000010 as the main diagonal and the rest zeros.
EXAMPLE
First few rows of the triangle =
1;
1, 1;
1, 0, 2;
1, 2, 0, 2;
1, 0, 0, 0, 4;
1, 3, 6, 0, 0, 2;
1, 0, 0, 0, 0, 0, 6;
1, 4, 0, 8, 0, 0, 0, 4;
1, 0, 12, 0, 0, 0, 0, 0, 6;
1, 5, 0, 0, 20, 0, 0, 0, 0, 4;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10;
1, 6, 20, 20, 0, 12, 0, 0, 0, 0, 0, 4;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12;
1, 7, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 6;
...
CROSSREFS
Cf. A156348, A130207, A156834, A000010
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson & Mats Granvik, Feb 16 2009
STATUS
approved
A130209 Diagonalized matrix of d(n), A000005. +10
8
1, 0, 2, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..93.
FORMULA
T(n,n) = A000005(n),
T(n,k) = 0 if n <> k.
EXAMPLE
First few rows of the triangle are:
1;
0, 2;
0, 0, 2;
0, 0, 0, 3;
0, 0, 0, 0, 2;
0, 0, 0, 0, 0, 4;
...
MAPLE
A130209 := proc(n, k)
if k = n then
numtheory[tau](n);
else
0;
end if;
end proc: # R. J. Mathar, Aug 06 2016
CROSSREFS
Cf. A000005, A130207, A130209.
KEYWORD
nonn,tabl,easy
AUTHOR
Gary W. Adamson, May 16 2007
STATUS
approved
A130208 Diagonalized matrix of A000203, Sigma(n). +10
5
1, 0, 3, 0, 0, 4, 0, 0, 0, 7, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 15 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
A130207 replaces Sigma(n) with phi(n), A000010. A130209 replaces Sigma(n) with d(n), A000005.
LINKS
Table of n, a(n) for n=1..36.
FORMULA
Infinite lower triangular matrix with A000203, Sigma(n), in the main diagonal and the rest zeros.
EXAMPLE
First few rows of the triangle are:
1;
0, 3;
0, 0, 4;
0, 0, 0, 7;
0, 0, 0, 0, 6;
0, 0, 0, 0, 0, 12;
...
PROG
(PARI) for(n=1, 9, for(k=2, n, print1("0, ")); print1(sigma(n)", ")) \\ Charles R Greathouse IV, Feb 14 2013
CROSSREFS
Cf. A000203, A130207, A130209.
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, May 16 2007
STATUS
approved
A127505 Triangle T(n,k) = mobius(n/k)*phi(k) if k|n, otherwise T(n,k)=0; 1<=k<=n. +10
0
1, -1, 1, -1, 0, 2, 0, -1, 0, 2, -1, 0, 0, 0, 4, 1, -1, -2, 0, 0, 2, -1, 0, 0, 0, 0, 0, 6, 0, 0, 0, -2, 0, 0, 0, 4, 0, 0, -2, 0, 0, 0, 0, 0, 6, 1, -1, 0, 0, -4, 0, 0, 0, 0, 4, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 1, 0, -2, 0, -2, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,6
LINKS
Table of n, a(n) for n=1..74.
FORMULA
T(n,k) = sum_{j=k..n} A054525(n,j)*A130207(j,k), 1<=k<=n.
EXAMPLE
First few rows of the triangle are:
1;
-1, 1;
-1, 0, 2;
0, -1, 0, 2;
-1, 0, 0, 0, 4;
1, -1, -2, 0, 0, 2;
...
CROSSREFS
Cf. A051731, A000010 (diagonal n=k), A007431 (row sums), A008683 (column k=1).
KEYWORD
easy,tabl,sign
AUTHOR
Gary W. Adamson, Jan 17 2007
STATUS
approved
page 1

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Last modified August 28 01:15 EDT 2024. Contains 375477 sequences. (Running on oeis4.)