A331181 -id:A331181 - 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];

A331182 Number of values of k, 1 <= k <= n, with A083254(k) = A083254(n), where A083254(n) = 2*phi(n) - n.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 3, 3, 4, 1, 1, 1, 2, 3, 4, 1, 1, 2, 5, 2, 3, 1, 1, 1, 5, 1, 6, 1, 1, 1, 7, 3, 2, 1, 1, 1, 4, 4, 8, 1, 1, 2, 1, 2, 5, 1, 2, 1, 3, 3, 9, 1, 1, 1, 10, 4, 6, 1, 1, 1, 6, 1, 1, 1, 1, 1, 11, 2, 7, 1, 1, 1, 2, 2, 12, 1, 1, 2, 13, 2, 4, 1, 1, 1, 8, 3, 14, 1, 1, 1, 2, 2, 1, 1, 1, 1, 5, 1

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A083254.

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

Cf. A000010, A083254.

Cf. also A081373, A331175, A331181.

Module[{b}, b[_] = 0;
a[n_] := With[{t = 2 EulerPhi[n] - 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; };
A083254(n) = (2*eulerphi(n)-n);
v331182 = ordinal_transform(vector(up_to,n,A083254(n)));
A331182(n) = v331182[n];

A331184 Number of values of k, 1 <= k <= n, with A318310(k) = A318310(n).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1

Offset: 1

Antti Karttunen, Jan 12 2020

nonn

Ordinal transform of A318310, or equally, of the ordered pair [A000120(n), A033879(n)], i.e., binary weight of n & deficiency of n.

Cf. A000120, A033879, A318310.

Cf. also A331181, A331183.

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; };
A318310aux(n) = [hammingweight(n), (2*n) - sigma(n)];
v331184 = ordinal_transform(vector(up_to,n,A318310aux(n)));
A331184(n) = v331184[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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A331182 Number of values of k, 1 <= k <= n, with A083254(k) = A083254(n), where A083254(n) = 2*phi(n) - n.

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A331184 Number of values of k, 1 <= k <= n, with A318310(k) = A318310(n).

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