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A135539
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Triangle read by rows: T(n,k) = number of divisors of n that are >= k.
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21
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1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 4, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Triangle read by rows, partial sums of A051731 starting from the right. A051731 as a lower triangular matrix times an all 1's lower triangular matrix.
G.f. of column k: Sum_{j>=1} x^(k*j)/(1 - x^j).
G.f. of column k: Sum_{j>=k} x^j/(1 - x^j). (End)
Sum_{j=1..n} T(j, k) ~ n * (log(n) + 2*gamma - 1 - H(k-1)), where gamma is Euler's constant (A001620), and H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jan 08 2024
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EXAMPLE
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First few rows of the triangle:
1;
2, 1;
2, 1, 1;
3, 2, 1, 1;
2, 1, 1, 1, 1;
4, 3, 2, 1, 1, 1;
2, 1, 1, 1, 1, 1, 1;
4, 3, 2, 2, 1, 1, 1, 1;
3, 2, 2, 1, 1, 1, 1, 1, 1;
4, 3, 2, 2, 2, 1, 1, 1, 1, 1;
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
6, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1;
...
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MAPLE
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with(numtheory);
f1:=proc(n) local d, s1, t1, t2, i;
d:=tau(n);
s1:=sort(divisors(n));
t1:=Array(1..n, 0);
for i from 1 to d do t1[n-s1[i]+1]:=1; od:
t2:=PSUM(convert(t1, list));
[seq(t2[n+1-i], i=1..n)];
end proc;
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MATHEMATICA
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T[n_, k_] := DivisorSum[n, Boole[# >= k]&];
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PROG
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(PARI) row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ Michel Marcus, Jul 23 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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