A193348 -id:A193348 - cp's OEIS frontend

A193349 Sum of odd divisors of tau(n).

1, 1, 1, 4, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 6, 1, 4, 1, 4, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 6, 4, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 8, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 4, 4, 1, 1, 1, 6, 6, 1, 1, 4, 1, 1

Offset: 1

Michel Lagneau, Jul 23 2011

nonn

			a(36) = 13 because tau(36) = 9 and the sum of the 3 odd divisors  {1, 3, 9} is 13.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000593, A036537, A062069, A193348.

Table[Total[Select[Divisors[DivisorSigma[0,n]], OddQ[ # ]&]], {n, 80}]

PARI
```
a(n)=sumdiv(sigma(n,0),d,(d%2)*d);
```

a(n) = A000593(A000005(n)). - Reinhard Zumkeller, Jul 25 2011
From Amiram Eldar, Aug 12 2024: (Start)
a(n) = 1 if and only if n is in A036537.
a(n) = A062069(n) if and only if n is a square. (End)

A193347 Number of even divisors of tau(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 3, 0, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 0, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 0, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 0, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 3, 1, 2, 0, 2, 1, 4, 2, 2, 2, 3, 1, 4, 2, 2, 2, 2, 2, 4, 1, 2, 2, 0, 1, 3, 1, 3, 3, 2, 1, 4, 1, 3, 2, 2, 1, 3

Offset: 1

Michel Lagneau, Jul 23 2011

nonn

			a(24) = 3 because tau(24) = 8 and the 3 even divisors are {2, 4, 8}.

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

Cf. A000005, A010553, A183063, A193348, A193350.

f[n_] := Block[{d = Divisors[DivisorSigma[0,n]]}, Count[EvenQ[d], True]]; Table[f[n], {n, 80}]

PARI
```
a(n)=sumdiv(sigma(n,0),d,(1-d%2));
```

a(n) = A183063(A000005(n)). - Antti Karttunen, May 28 2017
From Amiram Eldar, Jan 27 2025: (Start)
a(n) = 0 if and only if n is a square.
a(n) = A010553(n) - A193348(n). (End)

cp's OEIS Frontend

A193349 Sum of odd divisors of tau(n).

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