A054265 Sum of composite numbers between successive primes.

0, 4, 6, 27, 12, 45, 18, 63, 130, 30, 170, 117, 42, 135, 250, 280, 60, 320, 207, 72, 380, 243, 430, 651, 297, 102, 315, 108, 333, 1560, 387, 670, 138, 1296, 150, 770, 800, 495, 850, 880, 180, 1674, 192, 585, 198, 2255, 2387, 675, 228, 693, 1180, 240, 2214, 1270

Offset: 1

Author

Patrick De Geest, Apr 15 2000

Keywords

• nonn

Examples

			Between 7 and 11 we have 8 + 9 + 10 which is a(4)=27.
		

Links

• James Spahlinger, Table of n, a(n) for n = 1..1000
• Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.

Crossrefs

Cf. A000040, A046933, A054264, A054266, A054267, A054268.

Programs

PARI

a(n) = (prime(n+1) + prime(n))*(prime(n+1) - prime(n) - 1)/2; \\ Michel Marcus, Mar 24 2016

Python

from sympy import nextprime, prime
def A054265(n): return ((p:=prime(n))+(q:=nextprime(p)))*(q-p-1)>>1 # Chai Wah Wu, Jun 01 2024

Formula

a(n) = (prime(n+1) + prime(n))*(prime(n+1) - prime(n) - 1)/2. - Zak Seidov, Sep 12 2002