A366274 a(n) is the least k such that prime(n+1+k) >= prime(n)+prime(n+1).

1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 8, 9, 10, 10, 10, 13, 13, 13, 14, 14, 15, 15, 16, 18, 20, 20, 19, 19, 18, 22, 24, 24, 25, 27, 27, 27, 29, 28, 29, 30, 31, 31, 33, 33, 32, 34, 37, 39, 38, 39, 40, 40, 41, 42, 42, 43, 42, 43, 43, 43

Offset: 1

Author

Patrick Butler, Oct 05 2023

Keywords

• nonn

Comments

a(n) is the number of primes between prime(n) and prime(n) + prime(n+1).

Conjecture: for n >= 3, a(n) < n.

Examples

			For n = 5 prime(n) = 11. prime(5) + prime(6) = 11+13=24.  The 4th prime after 13 is 29 which is the next prime after 13 greater than or equal to 24. So a(5) = 4.
		

Links

• Patrick Butler, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A001043, A076273, A098084, A098085.

Programs

Maple

R:= 1: pn:= 2: pn1:= 3: p:=5: m:= 4: pp:= 7:
for n from 2 to 100 do
  pn:= pn1; pn1:= nextprime(pn1);
  while pp <= pn + pn1 do m:= m+1; pp:= nextprime(pp); od;
  R:= R, m-n-1;
od:
R; # Robert Israel, Oct 31 2023

Mathematica

A366274[n_]:=PrimePi[Prime[n]+Prime[n+1]-1]-n;Array[A366274,100] (* Paolo Xausa, Dec 09 2023 *)

PARI

a(n) = my(k=1, q=prime(n)+prime(n+1)); while(prime(n+k) < q, k++); k; \\ Michel Marcus, Oct 06 2023

Python

m=0
#list here is a list of prime numbers A000040.
def a(n):
    global list
    sum= list[n]+list[n+1]
    i=n+2
    while True:
        if(list[i]>=sum):
            m=i
            break
        i=i+1
    k = m-(n+1)
    return k
#
#calculate the terms of the sequence a(n).
seq = []
for n in range(0,firstN):
   seq.append(a(n))

Formula

a(n) = A000720(A001043(n)-1)-n = A000720(A076273(n+1))-n. - Paolo Xausa, Dec 09 2023