A344765 -id:A344765 - cp's OEIS frontend

A344768 a(n) = n - A344767(n); Möbius transform of A344765.

0, 0, 2, 0, 2, 7, 2, 0, 3, 12, 2, 11, 2, 15, 15, 0, 2, 19, 2, 13, 22, 23, 2, 25, 5, 28, 9, 32, 2, 19, 2, 0, 33, 36, 30, 41, 2, 39, 40, 48, 2, 28, 2, 27, 47, 47, 2, 47, 7, 50, 51, 29, 2, 53, 58, 23, 58, 60, 2, 72, 2, 63, 48, 0, 55, 66, 2, 72, 69, 63, 2, 24, 2, 76, 76, 43, 71, 77, 2, 47, 27, 84, 2, 57, 70, 87, 87, 96, 2, 73

Offset: 1

Antti Karttunen, May 30 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A008683, A011772, A344765, A344767.

A344767(n) = sumdiv(n,d,moebius(n/d)*A011772(d));
A344768(n) = (n-A344767(n));

a(n) = n - A344767(n) = n - Sum_{d|n} A008683(n/d) * A011772(d).

a(n) = Sum_{d|n} A008683(n/d) * A344765(d).

A344876 a(n) = A344875(n) - A011772(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 0, 8, 0, 6, 0, 11, 3, 0, 0, 16, 0, 13, 6, 19, 0, 15, 0, 24, 0, 35, 0, 9, 0, 0, 9, 32, 10, 48, 0, 35, 12, 45, 0, 16, 0, 38, 23, 43, 0, 30, 0, 48, 15, 45, 0, 51, 30, 42, 18, 56, 0, 41, 0, 59, 21, 0, 23, 49, 0, 96, 21, 52, 0, 57, 0, 72, 24, 70, 39, 60, 0, 60, 0, 80, 0, 36, 30, 83, 27, 118, 0, 61, 59, 131

Offset: 1

Antti Karttunen, Jun 03 2021

Apparently A000961 gives the positions of zeros.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000961, A011772, A344875, A344969, A344973, A344976.

Cf. also A344763, A344765, A344874.

A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
A344875[n_] := Product[{p, e} = pe; If[p == 2, 2^(1+e)-1, p^e-1], {pe, FactorInteger[n]}];
a[n_] := If[n == 1, 0, A344875[n] - A011772[n]];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)

A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344876(n) = (A344875(n)-A011772(n));

a(n) = A344875(n) - A011772(n).

a(n) >= A344976(n).

A344763 a(n) = n - A011772(n).

Original entry on oeis.org

0, -1, 1, -3, 1, 3, 1, -7, 1, 6, 1, 4, 1, 7, 10, -15, 1, 10, 1, 5, 15, 11, 1, 9, 1, 14, 1, 21, 1, 15, 1, -31, 22, 18, 21, 28, 1, 19, 27, 25, 1, 22, 1, 12, 36, 23, 1, 16, 1, 26, 34, 13, 1, 27, 45, 8, 39, 30, 1, 45, 1, 31, 36, -63, 40, 55, 1, 52, 46, 50, 1, 9, 1, 38, 51, 20, 56, 66, 1, 16, 1, 42, 1, 36, 51, 43, 58, 56, 1, 55

Offset: 1

Antti Karttunen, May 30 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001065, A011772, A344764, A344765, A344769.

A011772[n_] := Module[{m = 1}, While[!IntegerQ[(m(m+1))/(2n)], m++]; m];
a[n_] := n - A011772[n];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)

PARI
```
A344763(n) = (n-A011772(n));
```

from sympy.ntheory.modular import crt
from sympy import factorint
from math import prod
from itertools import combinations
def A344763(n):
    plist = tuple(p**q for p, q in factorint(2*n).items())
    return 1-n if len(plist) == 1 else n-int(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)))) # Chai Wah Wu, Jun 15 2022

a(n) = n - A011772(n).
a(n) = A344765(n) - A001065(n).
a(2^k) = 1-2^k. - Chai Wah Wu, Jun 15 2022

A344766 a(n) = gcd(sigma(n), A011772(n)).

Original entry on oeis.org

1, 3, 2, 7, 2, 3, 2, 15, 1, 2, 2, 4, 2, 1, 1, 31, 2, 1, 2, 3, 2, 1, 2, 15, 1, 6, 2, 7, 2, 3, 2, 63, 1, 2, 2, 1, 2, 1, 4, 15, 2, 4, 2, 4, 3, 1, 2, 4, 3, 3, 1, 1, 2, 3, 2, 24, 2, 2, 2, 3, 2, 1, 1, 127, 1, 1, 2, 2, 1, 4, 2, 3, 2, 6, 4, 28, 3, 12, 2, 2, 1, 2, 2, 16, 2, 1, 1, 4, 2, 1, 1, 1, 2, 1, 1, 63, 2, 3, 4, 1, 2, 3, 2, 2, 2

Offset: 1

Antti Karttunen, May 30 2021

nonn

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

Cf. A000203, A011772, A344765.

Cf. also A344764.

PARI
```
A344766(n) = gcd(sigma(n), A011772(n));
```

a(n) = gcd(A000203(n), A011772(n)).

a(n) = gcd(A000203(n), A344765(n)) = gcd(A011772(n), A344765(n)).

A344769 a(n) = A005187(n) - A011772(n).

Original entry on oeis.org

0, 0, 2, 0, 4, 7, 5, 0, 8, 14, 9, 14, 11, 18, 21, 0, 16, 26, 17, 23, 33, 30, 20, 31, 23, 37, 24, 46, 26, 41, 27, 0, 53, 50, 53, 62, 35, 54, 62, 63, 39, 61, 40, 53, 77, 65, 43, 62, 47, 73, 81, 62, 50, 77, 95, 61, 92, 84, 55, 101, 57, 88, 93, 0, 103, 119, 65, 118, 112, 117, 68, 79, 71, 109, 122, 93, 129, 140, 75, 94, 79, 121

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

Cf. A000120, A005187, A011772, A294898, A344763, A344765.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344769(n) = (A005187(n) - A011772(n));

from itertools import combinations
from math import prod
from sympy import factorint, divisors
from sympy.ntheory.modular import crt
def A344769(n):
    c = 2*n-bin(n).count('1')
    plist = [p**q for p, q in factorint(2*n).items()]
    if len(plist) == 1:
        return int(c+1+(plist[0] % 2 - 2)*n)
    return int(c-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)))) # Chai Wah Wu, Jun 03 2021

a(n) = A005187(n) - A011772(n).
a(n) = A344765(n) + A294898(n).
a(2^e) = 0, for e >= 0.
If n is an odd prime power, then a(n) = n+1-A000120(n). - Chai Wah Wu, Jun 03 2021

A344874 a(n) = A047994(n) - A011772(n).

Original entry on oeis.org

0, -2, 0, -4, 0, -1, 0, -8, 0, 0, 0, -2, 0, -1, 3, -16, 0, 0, 0, -3, 6, -1, 0, -1, 0, 0, 0, 11, 0, -7, 0, -32, 9, 0, 10, 16, 0, -1, 12, 13, 0, -8, 0, -2, 23, -1, 0, -2, 0, 0, 15, -3, 0, -1, 30, -6, 18, 0, 0, 9, 0, -1, 21, -64, 23, 9, 0, 32, 21, 4, 0, -7, 0, 0, 24, -2, 39, 12, 0, -4, 0, 0, 0, -12, 30, -1, 27, 38, 0, -3, 59

Offset: 1

Antti Karttunen, Jun 03 2021

sign

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

Cf. A011772, A047994.

Cf. also A344763, A344765, A344876.

A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A047994(n) = { my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); };
A344874(n) = (A047994(n)-A011772(n));

cp's OEIS Frontend

A344768 a(n) = n - A344767(n); Möbius transform of A344765.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A344876 a(n) = A344875(n) - A011772(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A344763 a(n) = n - A011772(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A344766 a(n) = gcd(sigma(n), A011772(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A344769 a(n) = A005187(n) - A011772(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

A344874 a(n) = A047994(n) - A011772(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI