

A208239


Triangle read by rows: T(n,m) = n + k  n/k, where k is the mth divisor of n; 1 <= m <= tau(n).


2



1, 1, 3, 1, 5, 1, 4, 7, 1, 9, 1, 5, 7, 11, 1, 13, 1, 6, 10, 15, 1, 9, 17, 1, 7, 13, 19, 1, 21, 1, 8, 11, 13, 16, 23, 1, 25, 1, 9, 19, 27, 1, 13, 17, 29, 1, 10, 16, 22, 31, 1, 33, 1, 11, 15, 21, 25, 35, 1, 37, 1, 12, 19, 21, 28, 39, 1, 17, 25, 41, 1, 13, 31, 43, 1, 45, 1, 14, 19, 22, 26
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OFFSET

1,3


COMMENTS

nth row sum is equal to A038040(n) = d(n)*n, where d = A000005.
Numbers n such that n + k  n/k is noncomposite number for all divisors k of n: 1, 2, 3, 6, 7, 10, 15, 19, 22, 30, 31, 37, 42, 57, 70, 79, 87, 97,...
Numbers n such that n + k  n/k is nonprime number for all divisor k of n: 1, 5, 8, 11, 13, 17, 23, 25, 29, 32, 38, 41, 43, 47, 53, 56, 59, 61, 62, 67, 68, 71, 73, 80, 81, 83, 88, 89, 93, 98, 101, 103, 107, 109, 111, 113, 121, 123, 125, 127,...
Smallest m such that n = m + k  m/k for all k is divisor of n, or 0 if no such m exists : 1, 0, 2, 4, 3, 8, 4, 12, 5, 8, 6, 20, 7, 24, 8, 12, 9, 32, 10, 36, 11, 16, 12, 44, 13, 24, 14, 20, 15, 56, 16, 60, 17, 24,..
Number of ways to write n as (p  q)/(1  1/q), where p is prime and q is a prime divisor of n: 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 0, 1, 0, 3, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 3, 0, 0, 1, 1, 0,...
Numbers n of the form (s  r)(1/s  1) where s is divisor of n and r is antidivisor of n: 10, 12, 14,...
The nth row starts with 1 and ends with 2n1; the first differences are symmetric w.r.t. reversal of the row (which corresponds to exchange of k and n/k). The second term in even lines is n/2+2.  M. F. Hasler, Jan 26 2013
If n is prime then nth row is 1, 2n1.  Zak Seidov, Feb 22 2013
T(n,A000005(n)) = A005408(n1).  Reinhard Zumkeller, Feb 25 2013


LINKS

Zak Seidov, Rows n = 1..200 of irregular triangle, flattened


FORMULA

T(n,k) = n + A027750(n,k) + A027750(n,A000005(n)+1k), 1<=k<=A000005(n).  Reinhard Zumkeller, Feb 25 2013


EXAMPLE

Triangle begins:
1,
1, 3,
1, 5,
1, 4, 7,
1, 9,
1, 5, 7, 11,
1, 13,
1, 6, 10, 15,
1, 9, 17,
1, 7, 13, 19,
1, 21,
1, 8, 11, 13, 16, 23.
In this last, 12th line (ending with 2*121), the first differences are (7,3,2,3,7).


MATHEMATICA

row[n_] := Table[n + k  n/k, {k, Divisors[n]}]; Table[row[n], {n, 1, 24}] // Flatten (* JeanFrançois Alcover, Jan 21 2013 *)


PROG

(Haskell)
a208239 n k = a208239_row n !! k
a208239_row n = map (+ n) $ zipWith () divs $ reverse divs
where divs = a027750_row n
a208239_tabl = map a208239_row [1..]
 Reinhard Zumkeller, Feb 25 2013


CROSSREFS

Row lengths are A000005.
Cf. A027750, A038040, A087909.
Sequence in context: A254765 A300893 A325249 * A114567 A001051 A214737
Adjacent sequences: A208236 A208237 A208238 * A208240 A208241 A208242


KEYWORD

nonn,tabf


AUTHOR

Gerasimov Sergey, Jan 11 2013


STATUS

approved



