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A265652 Triangle read by rows: T(n,k) is the sum of the union of the divisors of n and k. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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 A337019

Adjacent sequences:  A265649 A265650 A265651 * A265653 A265654 A265655

KEYWORD

nonn,tabl,look

AUTHOR

Franklin T. Adams-Watters, Dec 11 2015

STATUS

approved

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Last modified October 17 17:21 EDT 2021. Contains 348065 sequences. (Running on oeis4.)