A005070 Sum of primes == 1 (mod 3) dividing n.

0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 13, 7, 0, 0, 0, 0, 19, 0, 7, 0, 0, 0, 0, 13, 0, 7, 0, 0, 31, 0, 0, 0, 7, 0, 37, 19, 13, 0, 0, 7, 43, 0, 0, 0, 0, 0, 7, 0, 0, 13, 0, 0, 0, 7, 19, 0, 0, 0, 61, 31, 7, 0, 13, 0, 67, 0, 0, 7, 0, 0, 73, 37, 0, 19, 7, 13, 79, 0, 0, 0, 0, 7, 0, 43, 0, 0, 0, 0, 20, 0, 31, 0, 19, 0, 97, 7, 0, 0, 0, 0, 103, 13, 7, 0, 0, 0, 109, 0, 37

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			For n = 5, a(5) = 0 as 5 modulo 3 = 2.
For n = 49 = 7*7, a(49) = 7 as 7 modulo 3 = 1, and each such prime is counted only once.
For n = 91 = 7*13, a(91) = 7+13 = 20, as both primes are of the form 3k+1.
For n = 10001 = 73*137, only 73 is of the form 3k+1, thus a(10001) = 73.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10001

Crossrefs

Cf. A020639, A028234.

Programs

Mathematica

Table[DivisorSum[n, # &, And[PrimeQ@ #, Mod[#, 3] == 1] &], {n, 111}] (* Michael De Vlieger, May 12 2017 *)
f[p_, e_] := If[Mod[p, 3] == 1, p, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

Python

from sympy import factorint, primefactors
def a028234(n):
    f = factorint(n)
    m = min(f)
    return 1 if n==1 else n/(m**f[m])
def a020639(n): return min(primefactors(n)) if n>1 else 1
def a(n): return 0 if n==1 else a(a028234(n)) + a020639(n)*(1*(a020639(n)%3==1)) # Indranil Ghosh, May 12 2017

Scheme

(define (A005070 n) (if (= 1 n) 0 (+ (if (= 1 (modulo (A020639 n) 3)) (A020639 n) 0) (A005070 (A028234 n)))))

Formula

Additive with a(p^e) = p if p == 1 (mod 3), 0 otherwise.

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + A020639(n)*[A020639(n) == 1 (mod 3)]. (Here [] is the Iverson bracket, giving in this case 1 whenever the smallest prime is of the form 3k+1, and 0 otherwise.) - Antti Karttunen, May 12 2017

Extensions

More terms and examples from Antti Karttunen, May 12 2017