A208239 Triangle read by rows: T(n,m) = n + k - n/k, where k is the m-th divisor of n; 1 <= m <= tau(n).

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

Offset: 1

Author

Gerasimov Sergey, Jan 11 2013

Keywords

• nonn
• tabf

Comments

n-th 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 anti-divisor of n: 10, 12, 14,...

The n-th row starts with 1 and ends with 2n-1; 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 n-th row is 1, 2n-1. - Zak Seidov, Feb 22 2013

T(n,A000005(n)) = A005408(n-1). - Reinhard Zumkeller, Feb 25 2013

Examples

			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*12-1), the first differences are (7,3,2,3,7).
		

Links

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

Crossrefs

Row lengths are A000005.

Cf. A027750, A038040, A087909.

Programs

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

Mathematica

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

Formula

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