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%I #27 Dec 18 2015 11:27:42
%S 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,
%T 15,15,18,15,20,24,22,15,13,15,13,19,18,21,20,27,13,18,18,21,22,18,27,
%U 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
%N Triangle read by rows: T(n,k) is the sum of the union of the divisors of n and k.
%C 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.
%H Reinhard Zumkeller, <a href="/A265652/b265652.txt">Rows n = 1..125 of triangle, flattened</a>
%F T(n,k) = sigma(n) + sigma(k) - sigma(gcd(n,k)).
%F T(n,k) = A000203(n) + A245093(n,k) - A132442(n,k). - _Reinhard Zumkeller_, Dec 12 2015
%e Triangle begins:
%e 1
%e 3 3
%e 4 6 4
%e 7 7 10 7
%e 6 8 9 12 6
%e 12 12 12 16 17 12
%e ...
%e 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.
%p seq(seq(numtheory:-sigma(n) + numtheory:-sigma(k) - numtheory:-sigma(igcd(n,k)), k=1..n), n=1..10); # _Robert Israel_, Dec 17 2015
%t Table[Total@ Union[Divisors@ n, Divisors@ k], {n, 12}, {k, n}] // Flatten (* _Michael De Vlieger_, Dec 18 2015 *)
%o (PARI) T(n,k) = sigma(n) + sigma(k) - sigma(gcd(n,k))
%o (Haskell)
%o a265652 n k = a265652_tabl !! (n-1) !! (k-1)
%o a265652_row n = a265652_tabl !! (n-1)
%o a265652_tabl = zipWith (zipWith (-))
%o (zipWith (map . (+)) a000203_list a245093_tabl) a132442_tabl
%o -- _Reinhard Zumkeller_, Dec 12 2015
%Y Cf. A000203 (first column and main diagonal).
%Y T(2n,n) gives A062731.
%Y Cf. A132442, A245093.
%K nonn,tabl,look
%O 1,2
%A _Franklin T. Adams-Watters_, Dec 11 2015
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