A361806 Sum of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

0, 2, 5, 10, 5, 17, 5, 28, 30, 10, 45, 42, 12, 44, 47, 76, 10, 72, 57, 5, 97, 51, 117, 150, 28, 22, 83, 5, 65, 321, 66, 131, 28, 298, 10, 108, 172, 145, 109, 205, 10, 276, 5, 127, 16, 441, 582, 130, 24, 80, 232, 10, 276, 195, 270, 256, 10, 218, 187, 52, 388, 701, 162

Offset: 1

Author

Karl-Heinz Hofmann, Mar 26 2023

Keywords

• nonn
• easy

Examples

			a(6): 6th prime = 13 and the (6+1)th prime = 17; the composites between are {14,15,16} and the distinct prime factors of this set are {2,7,3,5} (no duplicates allowed); so a(6) = 2 + 7 + 3 + 5 = 17.
		

Links

• Winston de Greef, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A052297, A077218, A008472, A061214.

Programs

Mathematica

a[n_] := Plus @@ Union@ (Join @@ (FactorInteger[#][[;; , 1]] & /@ Range[Prime[n] + 1, Prime[n + 1] - 1])); Array[a, 65] (* Amiram Eldar, Mar 27 2023 *)

PARI

a(n) = my(list=List()); for(i=prime(n)+1, prime(n+1)-1, my(f=factor(i)[,1]); for (k=1, #f, listput(list, f[k]))); vecsum(Set(list)); \\ Michel Marcus, Mar 27 2023

Python

from sympy import primefactors, sieve
def A361806(n):
    primeset = []
    for composites in range (sieve[n]+1, sieve[n+1]):
        for p in primefactors(composites): primeset.append(p)
    return(sum(set(primeset)))

Formula

a(n) = A008472(A061214(n)).