A344768 -id:A344768 - cp's OEIS frontend

A344765 a(n) = sigma(n) - A011772(n).

0, 0, 2, 0, 2, 9, 2, 0, 5, 14, 2, 20, 2, 17, 19, 0, 2, 31, 2, 27, 26, 25, 2, 45, 7, 30, 14, 49, 2, 57, 2, 0, 37, 38, 34, 83, 2, 41, 44, 75, 2, 76, 2, 52, 69, 49, 2, 92, 9, 69, 55, 59, 2, 93, 62, 72, 62, 62, 2, 153, 2, 65, 77, 0, 59, 133, 2, 110, 73, 124, 2, 132, 2, 78, 100, 84, 75, 156, 2, 122, 41, 86, 2, 176, 74, 89

Offset: 1

Antti Karttunen, May 30 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A001065, A011772, A294898, A344763, A344766, A344768 (Möbius transform), A344769.

A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
a[n_] := DivisorSigma[1, n] - A011772[n];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)

PARI
```
A344765(n) = (sigma(n) - A011772(n));
```

a(n) = A000203(n) - A011772(n).
a(n) = A001065(n) + A344763(n).
a(n) = Sum_{d|n} A344768(d).
a(n) = A344769(n) - A294898(n).

A344767 Möbius transform of A011772.

Original entry on oeis.org

1, 2, 1, 4, 3, -1, 5, 8, 6, -2, 9, 1, 11, -1, 0, 16, 15, -1, 17, 7, -1, -1, 21, -1, 20, -2, 18, -4, 27, 11, 29, 32, 0, -2, 5, -5, 35, -1, -1, -8, 39, 14, 41, 17, -2, -1, 45, 1, 42, 0, 0, 23, 51, 1, -3, 33, -1, -2, 57, -12, 59, -1, 15, 64, 10, 0, 65, -4, 0, 7, 69, 48, 71, -2, -1, 33, 6, 1, 77, 33, 54, -2, 81, 27, 15, -1, 0

Offset: 1

Antti Karttunen, May 30 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A008683, A011772, A344768.

A344767(n) = sumdiv(n,d,moebius(n/d)*A011772(d));

from itertools import combinations
from math import prod
from sympy import factorint, divisors
from sympy.ntheory.modular import crt
from sympy.ntheory import mobius
def A011772(n):
    plist = [p**q for p, q in factorint(2*n).items()]
    return 2*n-1 if len(plist) == 1 else min(min(crt([m,2*n//m],[0,-1])[0],crt([2*n//m,m],[0,-1])[0]) for m in (prod(d) for l in range(1,len(plist)//2+1) for d in combinations(plist,l)))
def A344767(n): return sum(mobius(n//d)*A011772(d) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 20 2021

a(n) = Sum_{d|n} A008683(n/d) * A011772(d).

cp's OEIS Frontend

A344765 a(n) = sigma(n) - A011772(n).

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