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

1,91

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.

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)

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 *)

Prog

(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

Crossrefs

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

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

Sequence in context: A110515 A355437 A071936 * A076142 A084904 A084928

Adjacent sequences: A005085 A005086 A005087 * A005089 A005090 A005091

Keyword

nonn

Author

N. J. A. Sloane

Status

approved