OFFSET
1,1
COMMENTS
After a(1) the sequence alternates between odd and even numbers (obviously).
There is at least 1 prime between p and 2p (Bertrand), and since there is no prime between p and a(n) there must be at least one prime between a(n) and a(n) + p. Hence the sequence continues indefinitely, and each added prime is added once only.
The underlying sequence of added primes is: 2, 3, 7, 13, 23, 47, 97, ...; namely A075028 or A068524 but without their initial terms (1, 2 respectively).
Four primes occur in the first 24 terms, a(1) = 3, a(2) = 5, a(18) = 196271, and a(24) = 12560153, suggesting a higher density of primes here than in related sequence A123196, in which only three primes arise in the first 500 terms. It would be interesting to examine this further, once more terms become available.
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..3000
FORMULA
a(n) ~ c * 2^n, where c = 0.748642996358317338.... - Bill McEachen, May 09 2024
EXAMPLE
a(1) = 3, the first odd prime. The greatest prime less than 3 is 2, so a(2) = 3 + 2 = 5. Greatest prime less than 5 is 3 so a(3) = 5 + 3 = 8. Likewise a(4) = 8 + 7 = 15; etc.
MATHEMATICA
a[1]=3; a[n_] := a[n] = a[n-1] + NextPrime[a[n-1], -1]; Array[a, 35] (* Giovanni Resta, Apr 10 2017 *)
NestList[#+NextPrime[#, -1]&, 3, 40] (* Harvey P. Dale, Aug 22 2020 *)
PROG
(PARI) lista(nn) = { print1(a=3, ", "); for (n=2, nn, a += precprime(a-1); print1(a, ", "); ); } \\ Michel Marcus, Apr 08 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
David James Sycamore, Apr 07 2017
EXTENSIONS
More terms from Michel Marcus, Apr 08 2017
STATUS
approved