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

1,6

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A006022, A077610, A111284, A324388.

Sequence in context: A344910 A326135 A318305 * A321647 A323410 A182665

Adjacent sequences: A306261 A306262 A306263 * A306265 A306266 A306267

Keyword

nonn

Author

David James Sycamore, Feb 01 2019

Status

approved