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A127013
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Triangle read by rows: reversal of A126988.
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6
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1, 1, 2, 1, 0, 3, 1, 0, 2, 4, 1, 0, 0, 0, 5, 1, 0, 0, 2, 3, 6, 1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 2, 0, 4, 8, 1, 0, 0, 0, 0, 0, 3, 0, 9, 1, 0, 0, 0, 0, 2, 0, 0, 5, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 0, 0, 0, 0, 0, 2, 0, 3, 4, 6, 12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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Let j = reversed indices of row terms. Then for any row, j*T(n,k) = n, for nonzero T(n,k). For example, in row 10, we match the terms with their j indices: (1, 0, 0, 0, 0, 2, 0, 0, 5, 10), (dot product) (10, 9, 8, 7, 6, 5, 4, 3, 2, 1); getting 10, 0, 0, 0, 0, 10, 0, 0, 10, 10). The factors of n are found in each row in order, as nonzero terms; e.g., 10 has the factors 1, 2, 5, 10, sum 18. Row sums = sigma(n), A000203.
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REFERENCES
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David Wells, "Prime Numbers, The Most Mysterious Figures in Math", John Wiley & Sons, 2005, Appendix.
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 1..7875
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FORMULA
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Reversed rows of A126988.
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EXAMPLE
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First few rows of the triangle are:
1;
1, 2;
1, 0, 3;
1, 0, 2, 4;
1, 0, 0, 0, 5;
1, 0, 0, 2, 3, 6;
1, 0, 0, 0, 0, 0, 7;
1, 0, 0, 0, 2, 0, 4, 8;
1, 0, 0, 0, 0, 0, 3, 0, 9;
1, 0, 0, 0, 0, 2, 0, 0, 5, 10;
Row 10 = (1, 0, 0, 0, 0, 2, 0, 0, 5, 10), reversal of 10th row of A126988.
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MATHEMATICA
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T[n_, m_]:= If[Mod[n, m]==0, n/m, 0]; Table[T[n, n-m+1], {n, 1, 12}, {m, 1, n}]//Flatten (* G. C. Greubel, Jun 03 2019 *)
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PROG
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(Haskell)
a127013 n k = a127013_tabl !! (n-1) !! (k-1)
a127013_row n = a127013_tabl !! (n-1)
a127013_tabl = map reverse a126988_tabl
-- Reinhard Zumkeller, Jan 20 2014
(PARI) {T(n, k) = if(n%k==0, n/k, 0)};
for(n=1, 12, for(k=1, n, print1(T(n, n-k+1), ", "))) \\ G. C. Greubel, Jun 03 2019
(MAGMA) [[(n mod (n-k+1)) eq 0 select n/(n-k+1) else 0: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 03 2019
(Sage)
def T(n, k):
if (n%k==0): return n/k
else: return 0
[[T(n, n-k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 03 2019
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CROSSREFS
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Cf. A126988, A000203.
Sequence in context: A120111 A130055 A202452 * A117362 A247492 A113214
Adjacent sequences: A127010 A127011 A127012 * A127014 A127015 A127016
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KEYWORD
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nonn,tabl
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AUTHOR
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Gary W. Adamson, Jan 02 2007
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EXTENSIONS
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T(10,10) fixed by Reinhard Zumkeller, Jan 20 2014
More terms added by G. C. Greubel, Jun 03 2019
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STATUS
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approved
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