

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
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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 !! (n1) !! (k1)
a265652_row n = a265652_tabl !! (n1)
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. AdamsWatters, Dec 11 2015


STATUS

approved



