A331178 -id:A331178 - cp's OEIS frontend

A331180 Number of values of k, 1 <= k <= n, with A323910(k) = A323910(n), where A323910 is Dirichlet inverse of deficiency of n.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 4, 1, 1, 1, 3, 1, 1, 2, 4, 1, 1, 1, 4, 5, 1, 1, 2, 1, 3, 1, 5, 1, 5, 1, 6, 1, 1, 1, 6, 1, 1, 2, 6, 1, 2, 1, 7, 1, 3, 1, 2, 1, 2, 8, 9, 1, 1, 1, 3, 2, 2, 1, 3, 1, 1, 1, 7, 1, 10, 1, 11, 2, 1, 2, 1, 1, 2, 2, 7, 1, 1, 1, 8, 2

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A323910.

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

Cf. A033879, A323910.

Cf. also A331178, A331181.

f[n_] := 2 n - DivisorSigma[1, n];
A323910[n_] := A323910[n] = If[n == 1, 1, -Sum[f[n/d] A323910[d], {d, Most@Divisors[n]}]];
Module[{b}, b[] = 0; a[n] := With[{t = A323910[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

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; };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v331180 = ordinal_transform(DirInverse(vector(up_to,n,A033879(n))));
A331180(n) = v331180[n];

A331179 Number of values of k, 1 <= k <= n, with A173557(k) = A173557(n), where A173557(n) = Product_{p-1 | p is prime and divisor of n}.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 5, 1, 5, 1, 3, 2, 2, 1, 6, 4, 3, 7, 3, 1, 2, 1, 6, 1, 2, 1, 8, 1, 2, 2, 5, 1, 4, 1, 3, 3, 2, 1, 9, 4, 6, 1, 5, 1, 10, 2, 5, 2, 2, 1, 4, 1, 2, 6, 7, 1, 2, 1, 3, 1, 3, 1, 11, 1, 3, 5, 3, 2, 4, 1, 7, 12, 3, 1, 7, 1, 2, 1, 4, 1, 6, 2, 3, 3, 2, 3, 13, 1, 6, 3, 8, 1, 2, 1, 8, 2

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A173557.

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

Cf. A173557.

Cf. also A081373, A331175, A331178.

A173557[n_] := If[n == 1, 1, Times @@ (FactorInteger[n][[All, 1]] - 1)];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A173557[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

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; };
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
v331179 = ordinal_transform(vector(up_to, n, A173557(n)));
A331179(n) = v331179[n];

cp's OEIS Frontend

A331180 Number of values of k, 1 <= k <= n, with A323910(k) = A323910(n), where A323910 is Dirichlet inverse of deficiency of n.

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