A306264 a(n) = 1 + d*a(n/d); a(1)=0. If n has only one prime divisor, then d=n, otherwise d is the greatest proper unitary divisor of n.

0, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 5, 1, 8, 6, 1, 1, 10, 1, 6, 8, 12, 1, 9, 1, 14, 1, 8, 1, 16, 1, 1, 12, 18, 8, 10, 1, 20, 14, 9, 1, 22, 1, 12, 10, 24, 1, 17, 1, 26, 18, 14, 1, 28, 12, 9, 20, 30, 1, 21, 1, 32, 10, 1, 14, 34, 1, 18, 24, 36, 1, 10, 1, 38, 26, 20, 12, 40, 1, 17

Offset: 1

Author

David James Sycamore, Feb 01 2019

Keywords

• nonn

Comments

Name related to recursive formula of A006022.

a(n) = 1 if and only if n is a prime power; p^t; t >= 1.

The sequence of indices k on which a(k) is a record (1,2,6,10,14,18,22,26,30,...), appears to be A111284.

Examples

			a(8) = a(25) = 1 because 8 and 25 are prime powers.
a(30) = 16 because 15 is the greatest proper unitary divisor of 30, so a(30) = 1 + 15*a(2) = 1 + 15 = 16.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A006022, A077610, A111284, A324388.

Programs

Mathematica

f[n_] := If[PrimePowerQ[n], n,
  SelectFirst[Transpose@
    {Reverse@ #[[-Ceiling[Length[#]/2] ;; -2]],
      #[[2 ;; Ceiling[Length[#]/2]]]} &@ Divisors[n],
    CoprimeQ @@ # &][[1]] ]; f[1] = 1;
a[n_] := 1 + #*a[n/#] &[f[n]]; a[1] = 0;
Array[a, 120] (* Michael De Vlieger, Jun 24 2025 *)

PARI

d(n) = if (omega(n) == 1, n, my(v=select(x->(gcd(x, n/x)==1), divisors(n))); v[#v-1]);
lista(nn) = {va = vector(nn); va[1] = 0; for (n=2, nn, dn = d(n); va[n] = 1 + dn*va[n/dn];); va;} \\ Michel Marcus, Feb 10 2019

PARI

A324388(n) = if(1>=omega(n),n,fordiv(n,d,if((d>1)&&(1==gcd(d,n/d)),return(n/d))));
A306264(n) = if(1==n,0,my(d=A324388(n)); 1+(d*A306264(n/d))); \\ Antti Karttunen, Feb 28 2019

Formula

a(1) = 0; for n > 1, a(n) = 1 + (A324388(n) * a(n/A324388(n))). - Antti Karttunen, Feb 28 2019