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

1,7

Links

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

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A020639, A028234.

Sequence in context: A106226 A229658 A306755 * A135435 A339442 A331748

Adjacent sequences: A005067 A005068 A005069 * A005071 A005072 A005073

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms and examples from Antti Karttunen, May 12 2017

Status

approved