A072926 a(n) = Sum_{k=1..n} A051699(k).

1, 1, 1, 2, 2, 3, 3, 4, 6, 7, 7, 8, 8, 9, 11, 12, 12, 13, 13, 14, 16, 17, 17, 18, 20, 23, 25, 26, 26, 27, 27, 28, 30, 33, 35, 36, 36, 37, 39, 40, 40, 41, 41, 42, 44, 45, 45, 46, 48, 51, 53, 54, 54, 55, 57, 60, 62, 63, 63, 64, 64, 65, 67, 70, 72, 73, 73, 74, 76, 77, 77, 78, 78, 79

Offset: 1

Author

Benoit Cloitre, Aug 11 2002

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A051699.

Programs

Mathematica

f[n_] := If[PrimeQ[n], 0, Min[NextPrime[n] - n, n - NextPrime[n, -1]]]; Accumulate[Table[f[n], {n, 1, 100}]] (* Amiram Eldar, May 05 2022 *)

PARI

a(n)=sum(k=1,n,vecmin(vector(k,i,abs(k-prime(i)))))

Python

from sympy import nextprime; p = a = 1
while p < 71:
    q = nextprime(p); h = (q - p)//2
    for i in range(q-p): a += h - abs(h-i); print(a, end = ', ')
    p = q # Ya-Ping Lu, Apr 06 2025

Formula

Conjecture: a(n) is asymptotic to C*n*log(n) with C = 0.29... .

From Ya-Ping Lu, Apr 06 2025: (Start)

C = lim_{n->oo} a(n)/(n*log(n)) = 0.44 approximately.

a(prime(m)) = 1 + Sum_(i=3..m) (1/4)*(prime(i)-prime(i-1))^2. (End)