A098085 -id:A098085 - cp's OEIS frontend

A099655 a[n]=A098085[n]-A096215[n], difference between next and previous primes to A011974[n], the sum of two consecutive primes.

4, 4, 2, 2, 6, 2, 6, 2, 6, 2, 4, 6, 6, 8, 4, 4, 14, 4, 2, 10, 6, 6, 6, 10, 2, 12, 12, 12, 12, 2, 6, 6, 6, 10, 14, 4, 14, 14, 10, 4, 8, 6, 6, 8, 8, 10, 6, 8, 8, 2, 12, 8, 8, 6, 12, 18, 18, 10, 6, 6, 6, 2, 2, 12, 12, 6, 12, 8, 10, 8, 10, 8, 4, 6, 8, 4, 14, 12, 2, 2, 14, 14, 14, 14, 2, 20, 20, 8, 10, 8

Offset: 1

Labos Elemer, Nov 17 2004

nonn

			n=8, p(8)+p(9)=19+23=42,a[8]=43-41=2=a(8).

Cf. A098085, A096215, A011974.

Mathematica
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<Harvey P. Dale, Mar 07 2017 *)
```

a(n)=NextPrime[p(n)+p(n+1)]-PreviousPrime[p(n)+p(n+1)]

A098084 a(n) satisfies P(n) + P(n+1) + a(n) = least prime >= P(n) + P(n+1), where P(i)=i-th prime.

Original entry on oeis.org

0, 3, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 5, 7, 1, 1, 7, 3, 1, 5, 5, 1, 1, 5, 1, 7, 1, 7, 1, 1, 5, 1, 1, 5, 7, 3, 11, 1, 7, 1, 7, 1, 5, 7, 1, 9, 5, 7, 1, 1, 7, 7, 7, 1, 1, 9, 1, 9, 5, 5, 1, 1, 1, 7, 1, 5, 5, 7, 5, 7, 7, 1, 3, 5, 7, 1, 1, 11, 1, 1, 13, 1, 13, 5, 1, 15, 1, 1, 5, 7, 1, 1, 5, 1, 7, 1, 1, 5, 5, 3, 5, 3, 19

Offset: 1

Pierre CAMI, Sep 13 2004

a(n) = 1 iff prime(n) is in A177017. - Robert Israel, Feb 04 2020

			P(1) + P(2) = 2 + 3 = 5; least prime >= 5 = 5, so a(1)=0.
P(2) + P(3) = 3 + 5 = 8; least prime > 8 = 11, so a(2) = 11 - 8 = 3.
P(3) + P(4) = 5 + 7 = 12; least prime > 12 = 13, so a(3) = 13 - 12 = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

The primes are in A098085.

Cf. A177017.

P:= [seq(ithprime(i),i=1..200)]:
map(t -> nextprime(t-1)-t,P[1..-2]+P[2..-1]); # Robert Israel, Feb 04 2020

f[n_] := Block[{k = 0, p = Prime[n] + Prime[n + 1]}, While[ !PrimeQ[p + k], k++ ]; k]; Table[ f[n], {n, 103}] (* Robert G. Wilson v, Sep 24 2004 *)

More terms from Robert G. Wilson v, Sep 25 2004

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

Original entry on oeis.org

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

Patrick Butler, Oct 05 2023

nonn

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

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

			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.

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

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

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

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

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

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))

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

cp's OEIS Frontend

A099655 a[n]=A098085[n]-A096215[n], difference between next and previous primes to A011974[n], the sum of two consecutive primes.

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A098084 a(n) satisfies P(n) + P(n+1) + a(n) = least prime >= P(n) + P(n+1), where P(i)=i-th prime.

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

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