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Search: id:A051731
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| A051731 |
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Triangle read by rows: T(n,k)=1 if k divides n, T(n,k)=0 otherwise. |
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+0
211
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| 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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{T(n,k)*k, k=1..n} setminus {0} = divisors of n; sum(T(n,k)*(k^i),k=1..n) = sigma[i](n) = sum of the i-th power of positive divisors of n; sum(T(n,k),k=1..n)=A000005, sum(T(n,k)*k,k=1..n)=A000203
Row sums are A000005. Diagonal sums are A032741(n+2). Might be called a Mobius matrix. Binomial transform (product by binomial matrix) is A101508. - Paul Barry (pbarry(AT)wit.ie), Dec 05 2004
A054525 = the inverse of this triangle = A129360 * A115369. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007
If the 1 in the lower right corner is moved to the upper right corner then the determinant gives the mobius function. [From Mats Granvik (mats.granvik(AT)abo.fi), Nov 18 2008]
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LINKS
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Mats Granvik, Illustration of A051731
Jeffrey Ventrella, Divisor Plot [From Mats Granvik (mats.granvik(AT)abo.fi), Feb 08 2009]
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FORMULA
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T(n, k)=T(n-k, k) for k<=n/2, T(n, k)=0 for n/2<k<=n-1, T(n, n)=1
Rows given by A074854 converted to binary. Example: A074854(4)= 13(decimal)= 1101(binary); row 4 = 1, 1, 0, 1. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Oct 04 2003
Columns have g.f. x^k/(1-x^(k+1)) (k>=0). - Paul Barry (pbarry(AT)wit.ie), Dec 05 2004
Matrix inverse of triangle A054525, where A054525(n, k) = MoebiusMu(n/k) if k|n, 0 otherwise. - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 09 2006
Equals = A129372 * A115361 as infinite lower triangular matrices. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007
This triangle * [1,2,3,...] = Sigma(n), A000203: (1, 3, 4, 7, 6, 12, 8,...). A051731 * [1/1, 1/2, 1/3,...] = Sigma(n)/n: (1/1, 3/2, 4/3, 7/4, 6/5,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 10 2007
T(n,k) = 0^(n mod k). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 01 2009]
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EXAMPLE
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Triangle begins:
.{1};
.{1,1};
.{1,0,1};
.{1,1,0,1};
.{1,0,0,0,1}; ...
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CROSSREFS
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Cf. A000005, A000203, A074854, A054525, A129372, A115361.
A077049 and A077051 are other presentations of this matrix. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 08 2009]
T(n,k) = A000007(A048158(n,k)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 01 2009]
Sequence in context: A054524 A110471 A103994 this_sequence A135839 A155076 A120529
Adjacent sequences: A051728 A051729 A051730 this_sequence A051732 A051733 A051734
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KEYWORD
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easy,nice,nonn,tabl
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AUTHOR
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Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
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