A305791 -id:A305791 - 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).

A317830 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A175851, the ordinal transform of the nextprime function, A151800.

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 9, 11, 7, 1, 3, 1, 3, 5, 171, 1, -1, 1, -5, 5, 7, 1, -1, 11, 7, 29, 35, 1, -7, 1, -41, 5, 7, 9, 93, 1, 3, 5, 11, 1, -3, 1, -5, 3, 7, 1, -61, 11, 7, 9, 27, 1, -29, 5, -1, 9, 11, 1, -29, 1, 3, 3, 771, 9, 9, 1, -5, 5, -3, 1, -73, 1, 3, 3, 19, 9, 9, 1, -141, -45, 7, 1, -53, 5, 7, 9, 43, 1, -63, 5, 11, 9, 11, 13, 1597, 1

Offset: 1

Antti Karttunen, Aug 12 2018

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

Cf. A151800, A175851, A046644 (denominators).

Cf. also A305791, A305805, A305806, A317833, A317834, A317847.

A175851[n_] := If[!CompositeQ[n], 1, n - NextPrime[n, -1] + 1];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A175851[n] - Sum[If[1 < d < n, f[d]* f[n/d], 0], {d, Divisors[n]}])];
a[n_] := Numerator[f[n]];
Array[a, 100] (* Jean-François Alcover, Dec 19 2021 *)

A175851(n) = if(1==n,n,1 + n - precprime(n));
A317830aux(n) = if(1==n,n,(A175851(n)-sumdiv(n,d,if((d>1)&&(dA317830aux(d)*A317830aux(n/d),0)))/2);
A317830(n) = numerator(A317830aux(n));

\\ Memoized implementation:
memo317830 = Map();
A317830aux(n) = if(1==n,n,if(mapisdefined(memo317830,n),mapget(memo317830,n),my(v = (A175851(n)-sumdiv(n,d,if((d>1)&&(dA317830aux(d)*A317830aux(n/d),0)))/2); mapput(memo317830,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A175851(n) - Sum_{d|n, d>1, d 1.

A305796 Dirichlet convolution of A246277 with itself.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 4, 1, 2, 0, 10, 0, 2, 2, 12, 0, 10, 0, 14, 2, 2, 0, 32, 1, 2, 4, 18, 0, 22, 0, 32, 2, 2, 2, 47, 0, 2, 2, 48, 0, 30, 0, 26, 10, 2, 0, 88, 1, 14, 2, 30, 0, 38, 2, 64, 2, 2, 0, 104, 0, 2, 14, 80, 2, 42, 0, 38, 2, 30, 0, 148, 0, 2, 10, 42, 2, 54, 0, 136, 12, 2, 0, 144, 2, 2, 2, 96, 0, 98, 2, 50, 2, 2, 2, 224, 0, 18, 18, 103, 0, 66, 0, 112, 22

Offset: 1

Antti Karttunen, Jun 13 2018

nonn

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

Cf. A246277, A305791, A305798.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
A305796(n) = sumdiv(n,d,A246277(d)*A246277(n/d));

a(n) = Sum_{d|n} A246277(d)*A246277(n/d).

A305805 Dirichlet inverse of A175851.

Original entry on oeis.org

1, -1, -1, -1, -1, 0, -1, 1, -2, -2, -1, 3, -1, 0, -1, -1, -1, 5, -1, 7, -1, -2, -1, 0, -2, -2, 0, -1, -1, 10, -1, 3, -1, -2, -3, -3, -1, 0, -1, 0, -1, 6, -1, 7, 6, -2, -1, -1, -2, 7, -3, 3, -1, 4, -1, 6, -3, -4, -1, -7, -1, 0, 6, -1, -3, 6, -1, 7, -1, 12, -1, -4, -1, 0, 6, 1, -3, 6, -1, -5, 8, -2, -1, 5, -1, -2, -3, -2, -1, 0, -1, 5

Offset: 1

Antti Karttunen, Jun 13 2018

sign

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

Cf. A175851, A305791, A305806.

b[n_] := If[n < 3, 1, n - NextPrime[n + 1, -1] + 1];
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 *)

A175851(n) = if(1==n,n,1 + n - precprime(n));
A305805(n) = if(1==n,1,-sumdiv(n,d,if(dA175851(n/d)*A305805(d),0)));

a(1) = 1; for n > 1, a(n) = -Sum_{d|n, dA175851(n/d)*a(d).

A305806 Möbius transform of A175851.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 2, 3, 0, -1, 0, 1, 2, 2, 0, -2, 0, -3, 2, 3, 0, 0, 2, 3, 2, 3, 0, -5, 0, -2, 2, 3, 4, 4, 0, 1, 2, 2, 0, -3, 0, -3, -2, 3, 0, -2, 2, -2, 4, 1, 0, -2, 2, -2, 4, 5, 0, 3, 0, 1, -2, 2, 4, -1, 0, -3, 2, -5, 0, -4, 0, 1, -2, 1, 4, -1, 0, -4, -2, 3, 0, -3, 2, 3, 4, 4, 0, 2, 2, -1, 4, 5, 6, 8, 0, -2, -2, 2, 0, -7, 0, -4, -6

Offset: 1

Antti Karttunen, Jun 13 2018

sign

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

Cf. A008683, A175851, A305791, A305805.

A175851(n) = if(1==n,n,1 + n - precprime(n));
A305806(n) = sumdiv(n,d,moebius(n/d)*A175851(d));

a(n) = Sum_{d|n} A008683(n/d)*A175851(d).

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A305798 Dirichlet convolution of A078898 with itself.

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A317830 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A175851, the ordinal transform of the nextprime function, A151800.

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A305796 Dirichlet convolution of A246277 with itself.

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A305805 Dirichlet inverse of A175851.

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A305806 Möbius transform of A175851.

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