A320111 Number of divisors d of n that are not of the form 4k+2.

1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 4, 2, 2, 4, 4, 2, 3, 2, 4, 4, 2, 2, 6, 3, 2, 4, 4, 2, 4, 2, 5, 4, 2, 4, 6, 2, 2, 4, 6, 2, 4, 2, 4, 6, 2, 2, 8, 3, 3, 4, 4, 2, 4, 4, 6, 4, 2, 2, 8, 2, 2, 6, 6, 4, 4, 2, 4, 4, 4, 2, 9, 2, 2, 6, 4, 4, 4, 2, 8, 5, 2, 2, 8, 4, 2, 4, 6, 2, 6, 4, 4, 4, 2, 4, 10, 2, 3, 6, 6, 2, 4, 2, 6, 8

Offset: 1

Author

Antti Karttunen, Nov 22 2018

Keywords

• nonn
• mult

Comments

Inverse Möbius transform of A152822.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20160
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Matjaž Konvalinka and Vasu Tewari, Some natural extensions of the parking space, arXiv:2003.04134 [math.CO], 2020.

Crossrefs

Cf. A000005, A001620, A152822, A252463.

Cf. also A320107 and A319699, A319700, A319703, A319989.

Programs

Mathematica

Array[DivisorSum[#, 1 &, Mod[#, 4] != 2 &] &, 105] (* Michael De Vlieger, Jun 16 2020 *)
f[2, e_] := e; f[p_, e_] := e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 30 2022 *)

PARI

A320111(n) = sumdiv(n,d,(2!=(d%4)));

PARI

A320111(n) = {my(f); f = factor(n); for(i=1, #f~, if(f[i, 1]>2, f[i, 2]++)); factorback(f[,2])};

PARI

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A320111(n) = numdiv(A252463(n));

PARI

\\ A252463 given above
A001511(n) = 1+valuation(n,2);
A000265(n) = (n>>valuation(n,2));
A320111(n) = if(n<=2, 1, my(p=A252463(n)); A001511(p)*A320111(A000265(p)));

Python

from sympy import divisor_count
def A320111(n): return divisor_count(n if n&1 else n>>1) # Chai Wah Wu, Jul 13 2022

Formula

a(n) = Sum_{d|n, d == 0, 1 or 3 mod 4} 1.

Multiplicative with a(2^e) = e, a(p^e) = (e+1) for odd primes p.

a(2n) = A000005(n), a(2n+1) = A000005(2n+1).

a(n) = A000005(A252463(n)).

a(1) = a(2) = 1; for n > 2, a(n) = A001511(A252463(n)) * a(A000265(A252463(n))).

From Amiram Eldar, Dec 30 2022: (Start)

Dirichlet g.f.: zeta(s)^2*(1 - 1/2^s + 1/4^s).

Sum_{k=1..n} a(k) ~ (3/4)*n*(log(n) + 2*gamma - 1), where gamma is Euler's constant (A001620). (End)