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.

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, 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, 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

Offset: 1

Author

Matthew Campbell, Aug 29 2015

Keywords

• nonn
• tabf
• cofr

Examples

			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.
From _Michael De Vlieger_, Aug 29 2015: (Start)
Triangle begins:
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,   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, 42,  2,  2,  7,  1,  1,  4
...
(End)
		

Links

• Matthew Campbell, Table of n, a(n) for n = 1..116505 The first 225 rows are in the b-file.

Crossrefs

Cf. A000040.

For the continued fractions of the harmonic numbers, see A100398.

For the numerator of the sum, see A024451.

For the denominator of the sum, see A002110.

Programs

Maple

seq(op(numtheory:-cfrac(s,'quotients')),s=ListTools:-PartialSums(map2(`/`,1,[seq(ithprime(i),i=1..20)]))); # Robert Israel, Sep 06 2015

Mathematica

Table[ContinuedFraction[Sum[1/Prime@k, {k, n}]], {n, 11}] // Flatten (* Michael De Vlieger, Aug 29 2015 *)

PARI

row(n) = contfrac(sum(k=1, n, 1/prime(k)));
tabf(nn) = for(n=1, nn, print(row(n))); \\ Michel Marcus, Sep 18 2015