A005070 -id:A005070 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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

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.

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

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

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

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 *)

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

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

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)

A005074 Sum of primes = 2 mod 3 dividing n.

Original entry on oeis.org

0, 2, 0, 2, 5, 2, 0, 2, 0, 7, 11, 2, 0, 2, 5, 2, 17, 2, 0, 7, 0, 13, 23, 2, 5, 2, 0, 2, 29, 7, 0, 2, 11, 19, 5, 2, 0, 2, 0, 7, 41, 2, 0, 13, 5, 25, 47, 2, 0, 7, 17, 2, 53, 2, 16, 2, 0, 31, 59, 7, 0, 2, 0, 2, 5, 13, 0, 19, 23, 7, 71, 2, 0, 2, 5, 2, 11, 2, 0, 7, 0, 43, 83, 2, 22, 2, 29, 13, 89, 7, 0, 25, 0, 49, 5

Offset: 1

N. J. A. Sloane

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A005070, A005075, A005076, A005077, A005090, A008472, A079978.

Array[DivisorSum[#, # &, And[PrimeQ@ #, Mod[#, 3] == 2] &] &, 95] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 3] == 2, p, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%3) == 2, p)); \\ Michel Marcus, Jul 11 2017

(define (A005074 n) (if (= 1 n) 0 (+ (if (= 2 (modulo (A020639 n) 3)) (A020639 n) 0) (A005074 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p if p = 2 (mod 3), 0 otherwise.

a(n) = A008472(n) - A005070(n) - 3*A079978(n). - Antti Karttunen, Jul 10 2017

More terms from Antti Karttunen, Jul 10 2017

A005071 Sum of squares of primes = 1 mod 3 dividing n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 169, 49, 0, 0, 0, 0, 361, 0, 49, 0, 0, 0, 0, 169, 0, 49, 0, 0, 961, 0, 0, 0, 49, 0, 1369, 361, 169, 0, 0, 49, 1849, 0, 0, 0, 0, 0, 49, 0, 0, 169, 0, 0, 0, 49, 361, 0, 0, 0, 3721, 961, 49, 0, 169, 0, 4489, 0, 0, 49, 0, 0, 5329, 1369, 0, 361, 49, 169, 6241, 0, 0, 0, 0, 49, 0, 1849, 0, 0, 0, 0, 218

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A000290, A005070, A005072, A005073.

Module[{sp=Select[Prime[Range[100]],Mod[#,3]==1&]},Table[Total[ Select[ sp, Divisible[ n,#]&]^2],{n,70}]] (* Harvey P. Dale, Dec 19 2014 *)
f[p_, e_] := If[Mod[p, 3] == 1, p^2, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

(define (A005071 n) (if (= 1 n) 0 (+ (A000290 (if (= 1 (modulo (A020639 n) 3)) (A020639 n) 0)) (A005071 (A028234 n))))) ;; Antti Karttunen, Jul 09 2017

Additive with a(p^e) = p^2 if p = 1 (mod 3), 0 otherwise.

More terms from Antti Karttunen, Jul 09 2017

A005072 Sum of cubes of primes = 1 mod 3 dividing n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 343, 0, 0, 0, 0, 0, 2197, 343, 0, 0, 0, 0, 6859, 0, 343, 0, 0, 0, 0, 2197, 0, 343, 0, 0, 29791, 0, 0, 0, 343, 0, 50653, 6859, 2197, 0, 0, 343, 79507, 0, 0, 0, 0, 0, 343, 0, 0, 2197, 0, 0, 0, 343, 6859, 0, 0, 0, 226981, 29791, 343, 0, 2197, 0, 300763, 0, 0, 343, 0, 0, 389017, 50653, 0, 6859, 343, 2197, 493039

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A000578, A005070, A005071, A005073.

f[p_, e_] := If[Mod[p, 3] == 1, p^3, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%3) == 1, p^3)); \\ Michel Marcus, Jul 10 2017

(define (A005072 n) (if (= 1 n) 0 (+ (A000578 (if (= 1 (modulo (A020639 n) 3)) (A020639 n) 0)) (A005072 (A028234 n))))) ;; Antti Karttunen, Jul 09 2017

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

More terms from Antti Karttunen, Jul 09 2017

A005073 Sum of 4th powers of primes = 1 mod 3 dividing n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2401, 0, 0, 0, 0, 0, 28561, 2401, 0, 0, 0, 0, 130321, 0, 2401, 0, 0, 0, 0, 28561, 0, 2401, 0, 0, 923521, 0, 0, 0, 2401, 0, 1874161, 130321, 28561, 0, 0, 2401, 3418801, 0, 0, 0, 0, 0, 2401, 0, 0, 28561, 0, 0, 0, 2401, 130321, 0, 0, 0, 13845841, 923521, 2401, 0, 28561, 0, 20151121, 0, 0, 2401, 0, 0, 28398241, 1874161

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A000583, A005070, A005071, A005072.

f[p_, e_] := If[Mod[p, 3] == 1, p^4, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%3) == 1, p^4)); \\ Michel Marcus, Jul 10 2017

(define (A005073 n) (if (= 1 n) 0 (+ (A000583 (if (= 1 (modulo (A020639 n) 3)) (A020639 n) 0)) (A005073 (A028234 n))))) ;; Antti Karttunen, Jul 09 2017

Additive with a(p^e) = p^4 if p = 1 (mod 3), 0 otherwise.

More terms from Antti Karttunen, Jul 09 2017

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A005088 Number of primes = 1 mod 3 dividing n.

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A005074 Sum of primes = 2 mod 3 dividing n.

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A005071 Sum of squares of primes = 1 mod 3 dividing n.

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A005072 Sum of cubes of primes = 1 mod 3 dividing n.

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A005073 Sum of 4th powers of primes = 1 mod 3 dividing n.

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