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A349685
Irregular triangle read by rows: the n-th row contains the elements in the continued fraction of the abundancy index of n.
7
1, 1, 2, 1, 3, 1, 1, 3, 1, 5, 2, 1, 7, 1, 1, 7, 1, 2, 4, 1, 1, 4, 1, 11, 2, 3, 1, 13, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 15, 1, 17, 2, 6, 1, 19, 2, 10, 1, 1, 1, 10, 1, 1, 1, 1, 3, 1, 23, 2, 2, 1, 4, 6, 1, 1, 1, 1, 1, 2, 1, 2, 13, 2, 1, 29, 2, 2, 2, 1, 31, 1, 1, 31
OFFSET
1,3
COMMENTS
The abundancy index of n is sigma(n)/n = A000203(n)/n = A017665(n)/A017666(n).
For a prime p, the p-th row has a length 2 with a(p, 1) = 1 and a(p, 2) = p.
For multiply-perfect numbers m (A007691), the m-th row has a length 1, since their abundancy index is an integer. In particular, for a perfect number m (A000396), the m-th row has a length 1 with a(m, 1) = 2.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10489 (rows 1..2000)
EXAMPLE
The first ten rows of the triangle are:
1,
1, 2,
1, 3,
1, 1, 3,
1, 5,
2,
1, 7,
1, 1, 7,
1, 2, 4,
1, 1, 4,
...
MATHEMATICA
row[n_] := ContinuedFraction[DivisorSigma[1, n]/n]; Table[row[k], {k, 1, 32}] // Flatten
PROG
(PARI) row(n) = contfrac(sigma(n)/n); \\ Michel Marcus, Nov 25 2021
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Amiram Eldar, Nov 25 2021
STATUS
approved