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A285010 a(n+1) = a(n) + p, where p is the largest prime less than a(n); a(1) = 3.

%I A285010 #28 May 16 2024 03:14:00
%S A285010 3,5,8,15,28,51,98,195,388,771,1540,3071,6138,12271,24540,49073,98142,
%T A285010 196271,392518,785021,1570038,3140045,6280086,12560153,25120300,
%U A285010 50240589,100481176,200962343,401924670,803849309,1607698612,3215397195,6430794388,12861588761,25723177510
%N A285010 a(n+1) = a(n) + p, where p is the largest prime less than a(n); a(1) = 3.
%C A285010 After a(1) the sequence alternates between odd and even numbers (obviously).
%C A285010 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.
%C A285010 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).
%C A285010 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.
%H A285010 Giovanni Resta, <a href="/A285010/b285010.txt">Table of n, a(n) for n = 1..3000</a>
%F A285010 a(n) ~ c * 2^n, where c = 0.748642996358317338.... - _Bill McEachen_, May 09 2024
%e A285010 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.
%t A285010 a[1]=3; a[n_] := a[n] = a[n-1] + NextPrime[a[n-1], -1]; Array[a, 35] (* _Giovanni Resta_, Apr 10 2017 *)
%t A285010 NestList[#+NextPrime[#,-1]&,3,40] (* _Harvey P. Dale_, Aug 22 2020 *)
%o A285010 (PARI) lista(nn) = { print1(a=3, ", "); for (n=2, nn, a += precprime(a-1); print1(a, ", ");); } \\ _Michel Marcus_, Apr 08 2017
%Y A285010 Cf. A123196, A075058, A068524.
%K A285010 easy,nonn
%O A285010 1,1
%A A285010 _David James Sycamore_, Apr 07 2017
%E A285010 More terms from _Michel Marcus_, Apr 08 2017