A300239 -id:A300239 - cp's OEIS frontend

A300236 Möbius transform of A032742 (the largest proper divisor of n).

1, 0, 0, 1, 0, 2, 0, 2, 2, 4, 0, 2, 0, 6, 4, 4, 0, 4, 0, 4, 6, 10, 0, 4, 4, 12, 6, 6, 0, 4, 0, 8, 10, 16, 6, 6, 0, 18, 12, 8, 0, 6, 0, 10, 8, 22, 0, 8, 6, 16, 16, 12, 0, 12, 10, 12, 18, 28, 0, 8, 0, 30, 12, 16, 12, 10, 0, 16, 22, 18, 0, 12, 0, 36, 16, 18, 10, 12, 0, 16, 18, 40, 0, 12, 16, 42, 28, 20, 0, 16, 12, 22, 30, 46, 18, 16, 0, 36

Offset: 1

Antti Karttunen, Mar 10 2018

nonn

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

Cf. A008683, A032742, A300239.

Table[DivisorSum[n, # MoebiusMu[n/#]/FactorInteger[#][[1, 1]] &], {n, 98}] (* Michael De Vlieger, Mar 10 2018 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A300236(n) = sumdiv(n,d,moebius(n/d)*A032742(d));

a(n) = Sum_{d|n} A008683(n/d)*A032742(d).

A305807 Dirichlet inverse of A032742 (the largest proper divisor of n).

Original entry on oeis.org

1, -1, -1, -1, -1, -1, -1, -1, -2, -3, -1, 1, -1, -5, -3, -1, -1, 0, -1, 1, -5, -9, -1, 3, -4, -11, -4, 1, -1, 5, -1, -1, -9, -15, -5, 6, -1, -17, -11, 5, -1, 7, -1, 1, -2, -21, -1, 5, -6, -8, -15, 1, -1, 4, -9, 7, -17, -27, -1, 19, -1, -29, -4, -1, -11, 11, -1, 1, -21, -3, -1, 8, -1, -35, -8, 1, -9, 13, -1, 9, -8, -39, -1, 29, -15, -41, -27, 11, -1

Offset: 1

Antti Karttunen, Jun 13 2018

sign

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

Cf. A032742, A300236, A300239, A305808.

b[n_] := If[n == 1, 1, Divisors[n][[-2]]];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A305807(n) = if(1==n,1,-sumdiv(n,d,if(dA032742(n/d)*A305807(d),0)));

a(1) = 1; for n > 1, a(n) = -Sum_{d|n, dA032742(n/d)*a(d).

A305808 Dirichlet convolution of A032742 (the largest proper divisor of n) with itself.

Original entry on oeis.org

1, 2, 2, 5, 2, 8, 2, 12, 7, 12, 2, 22, 2, 16, 12, 28, 2, 30, 2, 34, 16, 24, 2, 56, 11, 28, 24, 46, 2, 56, 2, 64, 24, 36, 16, 87, 2, 40, 28, 88, 2, 76, 2, 70, 46, 48, 2, 136, 15, 70, 36, 82, 2, 108, 24, 120, 40, 60, 2, 172, 2, 64, 62, 144, 28, 116, 2, 106, 48, 108, 2, 228, 2, 76, 70, 118, 24, 136, 2, 216, 81, 84, 2, 236, 36, 88, 60, 184, 2, 228, 28

Offset: 1

Antti Karttunen, Jun 13 2018

nonn

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

Cf. A032742, A300236, A300239, A305807.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A305808(n) = sumdiv(n,d,A032742(n/d)*A032742(d));

a(n) = Sum_{d|n} A032742(n/d)*A032742(d).

cp's OEIS Frontend

A300236 Möbius transform of A032742 (the largest proper divisor of n).

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A305807 Dirichlet inverse of A032742 (the largest proper divisor of n).

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Author

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Links

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Programs

Mathematica

PARI

Formula

A305808 Dirichlet convolution of A032742 (the largest proper divisor of n) with itself.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula