A265652 Triangle read by rows: T(n,k) is the sum of the union of the divisors of n and k.

1, 3, 3, 4, 6, 4, 7, 7, 10, 7, 6, 8, 9, 12, 6, 12, 12, 12, 16, 17, 12, 8, 10, 11, 14, 13, 19, 8, 15, 15, 18, 15, 20, 24, 22, 15, 13, 15, 13, 19, 18, 21, 20, 27, 13, 18, 18, 21, 22, 18, 27, 25, 30, 30, 18, 12, 14, 15, 18, 17, 23, 19, 26, 24, 29, 12, 28, 28, 28, 28, 33, 28, 35, 36, 37, 43, 39, 28

Offset: 1

Author

Franklin T. Adams-Watters, Dec 11 2015

Keywords

• nonn
• tabl
• look

Comments

Does every positive integer except 2 and 5 occur here? The stronger form of Goldbach's conjecture (every even integer > 6 is the sum of two distinct primes) suffices to show that every odd integer (except 5) is in the sequence, since T(p,q) = p + q + 1.

Examples

			Triangle begins:
   1
   3  3
   4  6  4
   7  7 10  7
   6  8  9 12  6
  12 12 12 16 17 12
  ...
The divisors of 3 are {1, 3}; the divisors of 4 are {1, 2, 4}. The union is {1, 2, 3, 4}, summing to 10; so T(4,3) = 10.
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Crossrefs

Cf. A000203 (first column and main diagonal).

T(2n,n) gives A062731.

Cf. A132442, A245093.

Programs

Haskell

a265652 n k = a265652_tabl !! (n-1) !! (k-1)
a265652_row n = a265652_tabl !! (n-1)
a265652_tabl = zipWith (zipWith (-))
   (zipWith (map . (+)) a000203_list a245093_tabl) a132442_tabl
-- Reinhard Zumkeller, Dec 12 2015

Maple

seq(seq(numtheory:-sigma(n) + numtheory:-sigma(k) - numtheory:-sigma(igcd(n,k)), k=1..n), n=1..10); # Robert Israel, Dec 17 2015

Mathematica

Table[Total@ Union[Divisors@ n, Divisors@ k], {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Dec 18 2015 *)

PARI

T(n,k) = sigma(n) + sigma(k) - sigma(gcd(n,k))

Formula

T(n,k) = sigma(n) + sigma(k) - sigma(gcd(n,k)).

T(n,k) = A000203(n) + A245093(n,k) - A132442(n,k). - Reinhard Zumkeller, Dec 12 2015