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

1,2

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.

Links

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

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A000203 (first column and main diagonal).

T(2n,n) gives A062731.

Cf. A132442, A245093.

Sequence in context: A196456 A196485 A196718 * A074883 A338015 A196245

Adjacent sequences: A265649 A265650 A265651 * A265653 A265654 A265655

Keyword

nonn,tabl,look

Author

Franklin T. Adams-Watters, Dec 11 2015

Status

approved