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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).

%I A208239 #32 Mar 13 2015 23:49:58
%S A208239 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,
%T A208239 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,
%U A208239 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
%N 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).
%C A208239 n-th row sum is equal to A038040(n) = d(n)*n, where d = A000005.
%C A208239 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,...
%C A208239 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,...
%C A208239 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,..
%C A208239 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,...
%C A208239 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,...
%C A208239 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
%C A208239 If n is prime then n-th row is 1, 2n-1. - _Zak Seidov_, Feb 22 2013
%C A208239 T(n,A000005(n)) = A005408(n-1). - _Reinhard Zumkeller_, Feb 25 2013
%H A208239 Zak Seidov, <a href="/A208239/b208239.txt">Rows n = 1..200 of irregular triangle, flattened</a>
%F A208239 T(n,k) = n + A027750(n,k) + A027750(n,A000005(n)+1-k), 1<=k<=A000005(n). - _Reinhard Zumkeller_, Feb 25 2013
%e A208239 Triangle begins:
%e A208239 1,
%e A208239 1, 3,
%e A208239 1, 5,
%e A208239 1, 4, 7,
%e A208239 1, 9,
%e A208239 1, 5, 7, 11,
%e A208239 1, 13,
%e A208239 1, 6, 10, 15,
%e A208239 1, 9, 17,
%e A208239 1, 7, 13, 19,
%e A208239 1, 21,
%e A208239 1, 8, 11, 13, 16, 23.
%e A208239 In this last, 12th line (ending with 2*12-1), the first differences are (7,3,2,3,7).
%t A208239 row[n_] := Table[n + k - n/k, {k, Divisors[n]}]; Table[row[n], {n, 1, 24}] // Flatten (* _Jean-François Alcover_, Jan 21 2013 *)
%o A208239 (Haskell)
%o A208239 a208239 n k = a208239_row n !! k
%o A208239 a208239_row n = map (+ n) $ zipWith (-) divs $ reverse divs
%o A208239                 where divs = a027750_row n
%o A208239 a208239_tabl = map a208239_row [1..]
%o A208239 -- _Reinhard Zumkeller_, Feb 25 2013
%Y A208239 Row lengths are A000005.
%Y A208239 Cf. A027750, A038040, A087909.
%K A208239 nonn,tabf
%O A208239 1,3
%A A208239 _Gerasimov Sergey_, Jan 11 2013