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a(n) = prime(n) + (1 + prime(1 + n))/2.
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%I #19 Jun 21 2019 09:54:43

%S 4,6,9,13,18,22,27,31,38,45,50,58,63,67,74,83,90,95,103,108,113,121,

%T 128,138,148,153,157,162,166,177,193,200,207,214,225,230,239,247,254,

%U 263,270,277,288,292,297,305,323,337,342,346,353,360,367

%N a(n) = prime(n) + (1 + prime(1 + n))/2.

%C Primes occur at n = {4, 8, 14, 16, 19, 21, 27, 31, 37, 40, 42, 48, 51, 53, ...}: {13, 31, 67, 83, 103, 113, 157, 193, 239, 263, 277, 337, 353, 367, 389, 463, 563, 571, ...}.

%C Squares occur at n = {1, 3, 22, 35, 109, 136, 908, 936, ...}: {4, 9, 121, 225, 900, 1156, 10609, 11025, ...}.

%C Semiprimes: 4, 6, 9, 22, 38, 58, 74, 95, 121, 166, ...

%C Other interesting terms(?).

%H Robert Israel, <a href="/A283623/b283623.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = 2 + (1 + 3)/2 = 4,

%e a(2) = 3 + (1 + 5)/2 = 6,

%e a(3) = 5 + (1 + 7)/2 = 9.

%p Primes:= [seq(ithprime(i),i=1..101)]:

%p map(`+`,Primes[1..-2] + Primes[2..-1]/2,1/2); # _Robert Israel_, Jun 20 2019

%t Table[Prime[k] + (1 + Prime[1 + k])/2, {k, 120}]

%o (PARI) for(n=1, 53, print1(prime(n) + (1 + prime(1 + n))/2, ", ")) \\ _Indranil Ghosh_, Mar 12 2017

%Y Cf. A000040 (primes).

%K nonn

%O 1,1

%A _Zak Seidov_, Mar 12 2017