A005088 Number of primes = 1 mod 3 dividing n.

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The first instance of a(n)=2 is for n=91; the first instance of a(n)=3 is for n=1729. 1729 is famously Ramanujan's taxi cab number -- see A001235. - Harvey P. Dale, Jun 25 2013

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.

Crossrefs

Cf. A001221, A005070, A005090, A020639, A028234, A079978.

Cf. A121940 (first number having n such factors).

Programs

Maple

A005088 := proc(n)
    local a,pe;
    a := 0 ;
    for pe in ifactors(n)[2] do
        if modp(op(1,pe),3)= 1 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, May 19 2020

Mathematica

Join[{0},Table[Count[Transpose[FactorInteger[n]][[1]],?(Mod[#-1,3] == 0&)],{n,2,100}]] (* _Harvey P. Dale, Sep 22 2021 *)
Array[DivisorSum[#, 1 &, And[PrimeQ@ #, Mod[#, 3] == 1] &] &, 91] (* Michael De Vlieger, Jul 11 2017 *)

PARI

a(n)=my(f=factor(n)[,1]); sum(i=1,#f,f[i]%3==1) \\ Charles R Greathouse IV, Jan 16 2017

Scheme

(define (A005088 n) (if (= 1 n) 0 (+ (modulo (modulo (A020639 n) 3) 2) (A005088 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Formula

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

From Antti Karttunen, Jul 10 2017: (Start)

a(1) = 0; for n > 1, ((A020639(n) mod 3) mod 2) + a(A028234(n)).

a(n) = A001221(n) - A005090(n) - A079978(n).

(End)