A284599 Sum of twin prime (A001097) divisors of n.

0, 0, 3, 0, 5, 3, 7, 0, 3, 5, 11, 3, 13, 7, 8, 0, 17, 3, 19, 5, 10, 11, 0, 3, 5, 13, 3, 7, 29, 8, 31, 0, 14, 17, 12, 3, 0, 19, 16, 5, 41, 10, 43, 11, 8, 0, 0, 3, 7, 5, 20, 13, 0, 3, 16, 7, 22, 29, 59, 8, 61, 31, 10, 0, 18, 14, 0, 17, 3, 12, 71, 3, 73, 0, 8, 19, 18, 16, 0, 5, 3, 41, 0, 10, 22, 43, 32, 11, 0, 8

Offset: 1

Author

Ilya Gutkovskiy, Mar 30 2017

Keywords

• nonn

Examples

			a(15) = 8 because 15 has 4 divisors {1, 3, 5, 15} among which 2 are twin primes {3, 5} therefore 3 + 5 = 8.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Twin Primes.
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A001097, A008472, A062729, A284203.

Programs

Maple

N:= 200: # to get a(1)..a(N)
P:= select(isprime, {seq(i,i=3..N+2)}):
TP:= P intersect map(`-`,P,2):
TP:= TP union map(`+`,TP,2):
V:= Vector(N):
for p in TP do
  pm:= [seq(i,i=p..N,p)];
  V[pm]:= map(`+`,V[pm],p);
od:
convert(V,list); # Robert Israel, Mar 30 2017

Mathematica

Table[Total[Select[Divisors[n], PrimeQ[#1] && (PrimeQ[#1 - 2] || PrimeQ[#1 + 2]) &]], {n, 80}]

PARI

a(n) = sumdiv(n, d, d*(isprime(d) && (isprime(d-2) || isprime(d+2)))); \\ Michel Marcus, Apr 02 2017

Python

from sympy import divisors, isprime
def a(n): return sum([i for i in divisors(n) if isprime(i) and (isprime(i - 2) or isprime(i + 2))])
print([a(n) for n in range(1, 91)]) # Indranil Ghosh, Mar 30 2017

Formula

G.f.: Sum_{k>=1} A001097(k)*x^A001097(k)/(1 - x^A001097(k)).

a(n) = Sum_{d|n, d twin prime} d.

a(A062729(n)) = 0.

a(A001097(n)) = A001097(n).

Additive with a(p^e) = p if p is in A001097, and 0 otherwise. - Amiram Eldar, May 15 2025