A207480 a(n) = (3/2)*(1+prime(n)) - prime(n+1).

1, 2, 1, 5, 4, 8, 7, 7, 14, 11, 16, 20, 19, 19, 22, 29, 26, 31, 35, 32, 37, 37, 38, 46, 50, 49, 53, 52, 44, 61, 61, 68, 61, 74, 71, 74, 79, 79, 82, 89, 82, 95, 94, 98, 89, 95, 109, 113, 112, 112, 119, 112, 121, 124, 127, 134, 131, 136, 140, 133, 134, 151

Offset: 2

Author

Zak Seidov, Feb 18 2012

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n (cf. A062234).

Note that a(1) = 3/2 hence offset is 2.

There are many cases of two successive terms of the same value, the first case is a(8)=a(9)=7: p(8)=19, p(9)=23, p(10)=29, (3/2)*(1+19)-23 = (3/2)*(1+23)-29 = 7.

The first case of 3 equal successive terms is a(691..693)=2588 for corresponding 4 consecutive primes primes p(691..694)= 5189, 5197, 5209, 5227.

The first case of 4 equal successive terms is a(12702874..12702878)=15579672 for corresponding 5 consecutive primes primes p(12702874..12702878)= 231159373,231159389,231159413,231159449,231159503.

Also of interest are cases with a(n)>a(n-1), e.g., a(27..29): 53, 52, 44 (the general tendency is, of course, increasing a(n) with n).

Links

• Zak Seidov, Table of n, a(n) for n = 2..1001

Crossrefs

Cf. A062234.

Programs

Maple

a:= n-> 3*(1+ithprime(n))/2-ithprime(n+1):
seq(a(n), n=2..63);  # Alois P. Heinz, Feb 14 2022

Mathematica

(3(#[[1]]+1)/2)-#[[2]]&/@Partition[Prime[Range[2,70]],2,1] (* Harvey P. Dale, Jul 27 2016 *)

PARI

a(n) = my(p=prime(n)); (3/2)*(1+p) - nextprime(p+1); \\ Michel Marcus, Feb 14 2022