A005094 Number of distinct primes of the form 4k+1 dividing n minus number of distinct primes of the form 4k+3 dividing n.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign
• easy

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005089, A005091, A078613, A086239, A221264, A221265, A267099.

Cf. A079635 (difference when counted with multiplicity).

Programs

Haskell

a005094 n = a005089 n - a005091 n  -- Reinhard Zumkeller, Jan 07 2013

Mathematica

Join[{0},Table[Total[Which[Mod[#,4]==1,1,Mod[#,4]==3,-1,True,0]&/@ FactorInteger[ n][[All,1]]],{n,2,100}]] (* Harvey P. Dale, Sep 03 2022 *)

Formula

Additive with a(p^e) = 0 if p = 2, 1 if p == 1 (mod 4), -1 if p == 3 (mod 4).

From Reinhard Zumkeller, Jan 07 2013: (Start)

a(n) = A005089(n) - A005091(n).

a(A221264(n)) < 0.

a(A078613(n)) = 0.

a(A221265(n)) > 0. (End)

a(A267099(n)) = -a(n). - Antti Karttunen, Feb 03 2016

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = -A086239 = -0.334981... . - Amiram Eldar, Jan 02 2024

Extensions

More precise definition from Antti Karttunen, Feb 03 2016