A036248
Denominator of fraction equal to the continued fraction [ 2, 3, 5, ..., prime(n) ].
Original entry on oeis.org
1, 3, 16, 115, 1281, 16768, 286337, 5457171, 125801270, 3653694001, 113390315301, 4199095360138, 172276300080959, 7412079998841375, 348540036245625584, 18480034001016997327, 1090670546096248467877, 66549383345872173537824, 4459899354719531875502085
Offset: 1
Prime(4) = 7, and 2+1/(3+1/(5+1/7)) = 266/115, so a(4) = 115. - _Jon E. Schoenfield_, Dec 20 2016
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a(n)=if(n<0,0,contfracpnqn(vector(n,i, prime(i)))[2,1])
A083659
Denominator of fraction equal to the continued fraction [p(n); p(n-1),...,5,3,2].
Original entry on oeis.org
1, 2, 7, 37, 266, 2963, 38785, 662308, 12622637, 290982959, 8451128448, 262275964847, 9712661827787, 398481410904114, 17144413330704689, 806185907954024497, 42744997534894003030, 2522761040466700203267, 153931168466003606402317, 10315911048262708329158506
Offset: 1
The 5th term is 266 because 11+1/(7+1/(5+1/(3+1/2))) = 2963/266.
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b:= proc(n) option remember;
`if`(n=1, 0, 1/b(n-1)) + ithprime(n)
end:
a:= n-> denom(b(n)):
seq(a(n), n=1..20); # Alois P. Heinz, Nov 03 2018
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Table[Denominator[FromContinuedFraction[Prime[Range[n, 1, -1]]]], {n, 1, 20}]
Original entry on oeis.org
1, 6, 55, 556, 7839, 118141, 2488800, 54871741, 1374282325, 35786212191, 1182319284628, 40234641889543, 1409394785418633, 53597236487797597, 2091701617809524916, 96271871655725943733, 4719413412748380767833
Offset: 1
a(1) = denominator of 4 = 1.
a(2) = denominator of 4 + 1/6 = denominator of 25/6 = 6.
a(3) = denominator of 4 + 1/(6+1/9) = denominator of 229/55 = 55.
a(10) = denominator of 4+1/(6+1/(9+1/(10+ 1/(14+1/(15+ 1/(21+1/(22+1/(25+1/(26))))))))) = denominator of 149001936472/35786212191 = 35786212191.
A292433
a(0) = 0, a(1) = 1; a(n) = prime(a(n-1))*a(n-1) + a(n-2).
Original entry on oeis.org
0, 1, 2, 7, 121, 79988, 81600798165, 182421074243967704954243
Offset: 0
+---+-------------+--------------------+-------------------+
| n | a(n)/a(n+1) | Continued fraction | Comment |
+---+-------------+--------------------+-------------------+
| 1 | 1/2 | [0; 2] | 2 = prime(a(1)) |
+---+-------------+--------------------+-------------------+
| 2 | 2/7 | [0; 3, 2] | 3 = prime(a(2)) |
+---+-------------+--------------------+-------------------+
| 3 | 7/121 | [0; 17, 3, 2] | 17 = prime(a(3)) |
+---+-------------+--------------------+-------------------+
| 4 | 121/79988 | [0; 661, 17, 3, 2] | 661 = prime(a(4)) |
+---+-------------+--------------------+-------------------+
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RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == Prime[a[n - 1]] a[n - 1] + a[n - 2]}, a[n], {n, 7}]
Showing 1-4 of 4 results.
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