A331180 - OEIS

%I #9 Jan 12 2022 03:21:10

%S 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,

%T 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,

%U 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

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

%C Ordinal transform of A323910.

%H Antti Karttunen, <a href="/A331180/b331180.txt">Table of n, a(n) for n = 1..65537</a>

%t f[n_] := 2 n - DivisorSigma[1, n];

%t A323910[n_] := A323910[n] = If[n == 1, 1, -Sum[f[n/d] A323910[d], {d, Most@Divisors[n]}]];

%t Module[{b}, b[_] = 0; a[n_] := With[{t = A323910[n]}, b[t] = b[t] + 1]];

%t Array[a, 105] (* _Jean-François Alcover_, Jan 12 2022 *)

%o (PARI)

%o up_to = 65537;

%o 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; };

%o DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(d<n, v[n/d]*u[d], 0))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.

%o A033879(n) = (2*n-sigma(n));

%o v331180 = ordinal_transform(DirInverse(vector(up_to,n,A033879(n))));

%o A331180(n) = v331180[n];

%Y Cf. A033879, A323910.

%Y Cf. also A331178, A331181.

%K nonn

%O 1,8

%A _Antti Karttunen_, Jan 11 2020