A305799 -id:A305799 - cp's OEIS frontend

A305798 Dirichlet convolution of A078898 with itself.

1, 2, 2, 5, 2, 8, 2, 12, 5, 12, 2, 22, 2, 16, 8, 28, 2, 28, 2, 34, 10, 24, 2, 56, 5, 28, 14, 46, 2, 52, 2, 64, 14, 36, 8, 83, 2, 40, 16, 88, 2, 70, 2, 70, 26, 48, 2, 136, 5, 64, 20, 82, 2, 94, 10, 120, 22, 60, 2, 164, 2, 64, 34, 144, 12, 106, 2, 106, 26, 100, 2, 220, 2, 76, 36, 118, 8, 124, 2, 216, 42, 84, 2, 224, 14, 88, 32, 184, 2, 192, 10

Offset: 1

Antti Karttunen, Jun 13 2018

nonn

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

Cf. A078898, A305791, A305796, A305797, A305799.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A305798(n) = sumdiv(n,d,A078898(d)*A078898(n/d));

a(n) = Sum_{d|n} A078898(d)*A078898(n/d).

A305791 Dirichlet convolution of A175851 with itself.

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 2, 8, 7, 10, 2, 12, 2, 6, 8, 16, 2, 14, 2, 16, 8, 10, 2, 20, 7, 10, 16, 20, 2, 22, 2, 20, 8, 10, 12, 36, 2, 6, 8, 32, 2, 18, 2, 16, 18, 10, 2, 32, 7, 22, 12, 24, 2, 30, 8, 32, 12, 14, 2, 44, 2, 6, 18, 32, 12, 30, 2, 16, 8, 30, 2, 48, 2, 6, 18, 16, 12, 30, 2, 44, 25, 10, 2, 44, 8, 10, 12, 36, 2, 54, 8, 20, 12, 14, 16, 56, 2

Offset: 1

Antti Karttunen, Jun 13 2018

nonn

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

Cf. A175851.

Cf. also A305796, A305797, A305798, A305799.

A175851(n) = if(1==n,n,1 + n - precprime(n));
A305791(n) = sumdiv(n,d,A175851(d)*A175851(n/d));

a(n) = Sum_{d|n} A175851(d)*A175851(n/d).

A317834 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078899 (the ordinal transform of A006530, the largest prime factor of n).

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 17, 11, 3, 1, 19, 1, 3, 5, 139, 1, 23, 1, 19, 5, 3, 1, 39, 19, 3, 45, 19, 1, 13, 1, 263, 5, 3, 9, 77, 1, 3, 5, 55, 1, 13, 1, 19, 43, 3, 1, 387, 27, 47, 5, 19, 1, 59, 9, 71, 5, 3, 1, 43, 1, 3, 51, 995, 9, 13, 1, 19, 5, 25, 1, 87, 1, 3, 59, 19, 13, 13, 1, 707, 467, 3, 1, 59, 9, 3, 5, 71, 1, 53, 13, 19, 5, 3, 9, 1069, 1

Offset: 1

Antti Karttunen, Aug 12 2018

The first negative term is a(216) = -97.

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

Cf. A078899, A046644 (denominators).

Cf. also A305799, A317833, A317830.

gpf[n_] := If[n == 1, 1, FactorInteger[n][[-1, 1]]];
b[_] = 1;
A078899[n_] := A078899[n] = With[{t = gpf[n]}, b[t]++];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A078899[n] -
     Sum[If[1Jean-François Alcover, Dec 19 2021 *)

up_to = 16384;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
v078899 = ordinal_transform(vector(up_to,n,A006530(n)));
A078899(n) = v078899[n];
A317834aux(n) = if(1==n,n,(A078899(n)-sumdiv(n,d,if((d>1)&&(dA317834aux(d)*A317834aux(n/d),0)))/2);
A317834(n) = numerator(A317834aux(n));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A078899(n) - Sum_{d|n, d>1, d 1.

A305797 Dirichlet convolution of A078898 with A078899.

Original entry on oeis.org

1, 2, 2, 5, 2, 7, 2, 11, 6, 9, 2, 19, 2, 11, 8, 23, 2, 24, 2, 25, 9, 15, 2, 45, 8, 17, 17, 31, 2, 39, 2, 47, 11, 21, 10, 66, 2, 23, 12, 62, 2, 48, 2, 43, 27, 27, 2, 100, 10, 48, 14, 49, 2, 76, 11, 79, 15, 33, 2, 113, 2, 35, 32, 95, 12, 66, 2, 61, 17, 69, 2, 161, 2, 41, 37, 67, 12, 75, 2, 142, 44, 45, 2, 142, 13, 47, 20, 111, 2, 143, 13, 79, 21, 51, 14

Offset: 1

Antti Karttunen, Jun 13 2018

nonn

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

Cf. A078898, A078899, A305798, A305799.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
v078899 = ordinal_transform(vector(up_to,n,A006530(n)));
A078899(n) = v078899[n];
A305797(n) = sumdiv(n,d,A078898(d)*A078899(n/d));

a(n) = Sum_{d|n} A078898(d)*A078899(n/d).

cp's OEIS Frontend

A305798 Dirichlet convolution of A078898 with itself.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A305791 Dirichlet convolution of A175851 with itself.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A317834 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078899 (the ordinal transform of A006530, the largest prime factor of n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A305797 Dirichlet convolution of A078898 with A078899.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula