A318320 -id:A318320 - cp's OEIS frontend

A318325 a(n) = Sum_{d|n} [moebius(n/d) > 0]*(sigma(d)-d).

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 17, 1, 10, 9, 15, 1, 22, 1, 23, 11, 14, 1, 39, 6, 16, 13, 29, 1, 45, 1, 31, 15, 20, 13, 61, 1, 22, 17, 53, 1, 57, 1, 41, 34, 26, 1, 83, 8, 44, 21, 47, 1, 70, 17, 67, 23, 32, 1, 125, 1, 34, 42, 63, 19, 81, 1, 59, 27, 77, 1, 139, 1, 40, 50, 65, 19, 93, 1, 113, 40, 44, 1, 159, 23, 46, 33, 95, 1, 163, 21

Offset: 1

Antti Karttunen, Aug 26 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001065, A051953, A291784, A318320, A318326, A318441.

A318325(n) = sumdiv(n,d,(1==moebius(n/d))*(sigma(d)-d));

a(n) = Sum_{d|n} [A008683(n/d) == 1]*A001065(d).
a(n) = A051953(n) + A318326(n).
a(n) = A291784(n) - A318441(n).

A318326 a(n) = Sum_{d|n} [moebius(n/d) < 0]*(sigma(d)-d).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 9, 0, 2, 2, 7, 0, 10, 0, 11, 2, 2, 0, 23, 1, 2, 4, 13, 0, 23, 0, 15, 2, 2, 2, 37, 0, 2, 2, 29, 0, 27, 0, 17, 13, 2, 0, 51, 1, 14, 2, 19, 0, 34, 2, 35, 2, 2, 0, 81, 0, 2, 15, 31, 2, 35, 0, 23, 2, 31, 0, 91, 0, 2, 15, 25, 2, 39, 0, 65, 13, 2, 0, 99, 2, 2, 2, 47, 0, 97, 2, 29, 2, 2, 2, 107, 0, 18, 19, 65, 0, 47

Offset: 1

Antti Karttunen, Aug 26 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001065, A051953, A291784, A318320, A318325, A318442.

A318326(n) = sumdiv(n,d,(-1==moebius(n/d))*(sigma(d)-d));

a(n) = Sum_{d|n} [A008683(n/d) == -1]*A001065(d).
a(n) = A318325(n) - A051953(n).
a(n) = A318320(n) - A318442(n).

A318442 a(n) = Sum_{d|n} [moebius(n/d) < 0]*A033879(d).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 2, 5, 1, 1, 1, 7, 6, 1, 1, 5, 1, 3, 8, 11, 1, -3, 4, 13, 5, 5, 1, 9, 1, 1, 12, 17, 10, -7, 1, 19, 14, -1, 1, 15, 1, 9, 11, 23, 1, -11, 6, 21, 18, 11, 1, 11, 14, 1, 20, 29, 1, -17, 1, 31, 15, 1, 16, 27, 1, 15, 24, 29, 1, -31, 1, 37, 25, 17, 16, 33, 1, -9, 14, 41, 1, -15, 20, 43, 30, 5, 1, -1, 18, 21, 32, 47, 22, -27, 1

Offset: 1

Antti Karttunen, Aug 26 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A033879, A083254, A318320, A318325, A318441.

A318442(n) = sumdiv(n,d,(-1==moebius(n/d))*(d+d-sigma(d)));

a(n) = Sum_{d|n} [A008683(n/d) == 1]*A033879(d).
a(n) = A318320(n) - A318326(n).
A083254(n) = A318441(n) - a(n).

A319682 Restricted growth sequence transform of A300832.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 12, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 7, 34, 35, 25, 2, 36, 27, 37, 38, 39, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 53, 54, 2, 55, 56, 57, 2, 58, 59, 60, 61, 62, 2, 63, 64, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 49

Offset: 1

Antti Karttunen, Sep 29 2018

nonn

For all i, j: a(i) = a(j) => A318320(i) = A318320(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A300832, A319681.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300832(n) = { my(m=1); fordiv(n,d,if(-1==moebius(n/d), m *= A019565(d))); m; };
v319682 = rgs_transform(vector(up_to,n,A300832(n)));
A319682(n) = v319682[n];

A318675 Sum of squarefree divisors of n that have an odd number of prime factors.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 9, 8, 2, 17, 5, 19, 7, 10, 13, 23, 5, 5, 15, 3, 9, 29, 40, 31, 2, 14, 19, 12, 5, 37, 21, 16, 7, 41, 54, 43, 13, 8, 25, 47, 5, 7, 7, 20, 15, 53, 5, 16, 9, 22, 31, 59, 40, 61, 33, 10, 2, 18, 82, 67, 19, 26, 84, 71, 5, 73, 39, 8, 21, 18, 96, 79, 7, 3, 43, 83, 54, 22, 45, 32, 13, 89, 40, 20, 25, 34, 49, 24, 5

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A008472, A023900, A048250, A318320, A318674, A318677.

Array[DivisorSum[#, # &, MoebiusMu@ # == -1 &] &, 96] (* Michael De Vlieger, Sep 04 2018 *)

A318675(n) = sumdiv(n,d,(-1==moebius(d))*d);

For all n >= 1, a(n) <= A318677(n).
a(n) = Sum_{d|n} [A008683(d) < 0]*d.
a(n) = A048250(n) - A318674(n).
a(n) = (A048250(n) - A023900(n))/2. - Amiram Eldar, Jun 06 2025

cp's OEIS Frontend

A318325 a(n) = Sum_{d|n} [moebius(n/d) > 0]*(sigma(d)-d).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A318326 a(n) = Sum_{d|n} [moebius(n/d) < 0]*(sigma(d)-d).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A318442 a(n) = Sum_{d|n} [moebius(n/d) < 0]*A033879(d).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A319682 Restricted growth sequence transform of A300832.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A318675 Sum of squarefree divisors of n that have an odd number of prime factors.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula