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A260615 Irregular triangle read by rows: the n-th row is the continued fraction expansion of the sum of the reciprocals of the first n primes.

%I A260615 #32 Oct 02 2015 13:25:40
%S A260615 0,2,0,1,5,1,30,1,5,1,2,12,1,3,1,2,1,9,1,1,7,1,2,1,9,1,2,1,2,12,7,1,2,
%T A260615 2,13,1,1,1,8,13,5,4,1,2,5,8,1,2,6,1,1,4,10,1,2,3,1,3,1,2,238,1,28,1,
%U A260615 42,2,2,7,1,1,4,1,1,1,6,1,41,3,1,1,51,1,9,2,3,2,5,1,2,1,6,1,1,1,3,3,3,1,1,1,3,3,1,2,19,1,13,1,1,3,4,7,1,1,3,2,1,10
%N A260615 Irregular triangle read by rows: the n-th row is the continued fraction expansion of the sum of the reciprocals of the first n primes.
%H A260615 Matthew Campbell, <a href="/A260615/b260615.txt">Table of n, a(n) for n = 1..116505</a> The first 225 rows are in the b-file.
%e A260615 For row 3, the sum of the first three prime reciprocals equals 1/2 + 1/3 + 1/5 = 31/30. The continued fraction expansion of 31/30 is 1 + (1/30). Because of this, the terms in row 3 are 1 and 30.
%e A260615 From _Michael De Vlieger_, Aug 29 2015: (Start)
%e A260615 Triangle begins:
%e A260615 0,  2
%e A260615 0,  1,   5
%e A260615 1, 30
%e A260615 1,  5,   1,  2, 12
%e A260615 1,  3,   1,  2,  1,  9,  1,  1,  7
%e A260615 1,  2,   1,  9,  1,  2,  1,  2, 12,  7
%e A260615 1,  2,   2, 13,  1,  1,  1,  8, 13,  5,  4
%e A260615 1,  2,   5,  8,  1,  2,  6,  1,  1,  4, 10,  1,  2,  3,  1,  3
%e A260615 1,  2, 238,  1, 28,  1, 42,  2,  2,  7,  1,  1,  4
%e A260615 ...
%e A260615 (End)
%p A260615 seq(op(numtheory:-cfrac(s,'quotients')),s=ListTools:-PartialSums(map2(`/`,1,[seq(ithprime(i),i=1..20)]))); # _Robert Israel_, Sep 06 2015
%t A260615 Table[ContinuedFraction[Sum[1/Prime@k, {k, n}]], {n, 11}] // Flatten (* _Michael De Vlieger_, Aug 29 2015 *)
%o A260615 (PARI) row(n) = contfrac(sum(k=1, n, 1/prime(k)));
%o A260615 tabf(nn) = for(n=1, nn, print(row(n))); \\ _Michel Marcus_, Sep 18 2015
%Y A260615 Cf. A000040.
%Y A260615 For the continued fractions of the harmonic numbers, see A100398.
%Y A260615 For the numerator of the sum, see A024451.
%Y A260615 For the denominator of the sum, see A002110.
%K A260615 nonn,tabf,cofr
%O A260615 1,2
%A A260615 _Matthew Campbell_, Aug 29 2015