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A237273
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Triangle read by rows: T(n,k) = k+m, if k < m and k*m = n, or T(n,k) = k, if k^2 = n. Otherwise T(n,k) = 0. With n>=1 and 1<=k<=A000196(n).
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5
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1, 3, 4, 5, 2, 6, 0, 7, 5, 8, 0, 9, 6, 10, 0, 3, 11, 7, 0, 12, 0, 0, 13, 8, 7, 14, 0, 0, 15, 9, 0, 16, 0, 8, 17, 10, 0, 4, 18, 0, 0, 0, 19, 11, 9, 0, 20, 0, 0, 0, 21, 12, 0, 9, 22, 0, 10, 0, 23, 13, 0, 0, 24, 0, 0, 0, 25, 14, 11, 10, 26, 0, 0, 0, 5
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OFFSET
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1,2
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COMMENTS
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The first element of column k is in row k^2.
Column k lists k, k-1 zeros, and the positive integers but starting from 2*k+1 interleaved with k-1 zeros.
Row n has only one positive term iff n is a noncomposite number (A008578).
It appears that there are only eight rows that do not contain zeros. The indices of these rows are in A018253 (the divisors of 24).
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LINKS
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Table of n, a(n) for n=1..75.
Index entries for sequences related to sigma(n)
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EXAMPLE
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Triangle begins:
1;
3;
4;
5, 2;
6, 0;
7, 5;
8, 0;
9, 6;
10, 0, 3;
11, 7, 0;
12, 0, 0;
13, 8, 7;
14, 0, 0;
15, 9, 0;
16, 0, 8;
17, 10, 0, 4;
18, 0, 0, 0;
19, 11, 9, 0;
20, 0, 0, 0;
21, 12, 0, 9;
22, 0, 10, 0;
23, 13, 0, 0;
24, 0, 0, 0;
25, 14, 11, 10;
26, 0, 0, 0, 5;
27, 15, 0, 0, 0;
28, 0, 12, 0, 0;
29, 16, 0, 11, 0;
30, 0, 0, 0, 0;
31, 17, 13, 0, 11;
...
For n = 9 the divisors of n are 1, 3, 9, so row 9 is 10, 0, 3, because 1*9 = 9 and 3^2 = 9. The sum of row 9 is A000203(9) = 13.
For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12, so row 12 is 13, 8, 7, because 1*12 = 12, 2*6 = 12 and 3*4 = 12. The sum of row 12 is A000203(12) = 28.
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PROG
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(PARI) T(n, k) = if (n % k, 0, if (k^2==n, k, k + n/k));
tabf(nn) = {for (n = 1, nn, v = vector(sqrtint(n), k, T(n, k)); print(v); ); } \\ Michel Marcus, Jun 19 2019
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CROSSREFS
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Row sums give A000203.
Row n has length A000196(n).
Column 1 is A065475.
Cf. A000290, A008578, A018253, A027750, A196020, A210959, A212119, A212120, A228812-A228814, A231347, A236104, A236631, A237519, A237593.
Sequence in context: A121845 A239639 A099816 * A272025 A262411 A280488
Adjacent sequences: A237270 A237271 A237272 * A237274 A237275 A237276
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KEYWORD
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nonn,tabf
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AUTHOR
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Omar E. Pol, Feb 08 2014
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STATUS
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approved
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