A308485 a(n) is the sum of the integer logs of all integers strictly between prime(n) and prime(n+1).

0, 4, 5, 19, 7, 25, 8, 32, 54, 10, 65, 48, 12, 51, 74, 93, 12, 92, 61, 12, 111, 68, 130, 180, 47, 22, 89, 13, 73, 397, 80, 150, 28, 344, 15, 124, 196, 149, 138, 223, 15, 314, 15, 138, 19, 497, 620, 148, 26, 86, 244, 16, 341, 221, 282, 269, 16, 238, 196, 52, 439, 775, 166, 22, 260, 731, 389, 515

Offset

1,2

Links

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

Formula

a(n) = Sum_{prime(n) < j < prime(n+1)} A001414(j).

Example

For n = 4, the numbers between prime(4)=7 and prime(5)=11 are 8, 9, 10
so a(4) = A001414(8) + A001414(9) + A001414(10) = 6 + 6 + 7 = 19.

Maple

A001414:= proc(n) local t;
  add(t[1]*t[2], t=ifactors(n)[2])
end proc:
p:= 2: Res:= NULL:
for i from 1 to 100 do
q:= nextprime(p);
Res:= Res, add(A001414(j), j=p+1..q-1);
p:= q;
od:
Res;

Prog

(PARI) sopfr(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]);
a(n) = sum(k=prime(n)+1, prime(n+1)-1, sopfr(k)); \\ Michel Marcus, May 31 2019
(Python)
from sympy import prime, factorint
def A308485(n): return sum(p*e for m in range(prime(n)+1, prime(n+1)) for p, e in factorint(m).items()) # Chai Wah Wu, Apr 01 2022

Crossrefs

Cf. A000040, A001414.

Sequence in context: A243120 A317378 A338866 * A234143 A217686 A042885

Adjacent sequences: A308482 A308483 A308484 * A308486 A308487 A308488

Keyword

nonn

Author

J. M. Bergot and Robert Israel, May 30 2019

Status

approved