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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.

%I A306264 #29 Jun 24 2025 10:54:04
%S A306264 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,
%T A306264 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,
%U A306264 1,21,1,32,10,1,14,34,1,18,24,36,1,10,1,38,26,20,12,40,1,17
%N 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.
%C A306264 Name related to recursive formula of A006022.
%C A306264 a(n) = 1 if and only if n is a prime power; p^t; t >= 1.
%C A306264 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.
%H A306264 Antti Karttunen, <a href="/A306264/b306264.txt">Table of n, a(n) for n = 1..20000</a>
%H A306264 Antti Karttunen, <a href="/A306264/a306264.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A306264 a(1) = 0; for n > 1, a(n) = 1 + (A324388(n) * a(n/A324388(n))). - _Antti Karttunen_, Feb 28 2019
%e A306264 a(8) = a(25) = 1 because 8 and 25 are prime powers.
%e A306264 a(30) = 16 because 15 is the greatest proper unitary divisor of 30, so a(30) = 1 + 15*a(2) = 1 + 15 = 16.
%t A306264 f[n_] := If[PrimePowerQ[n], n,
%t A306264   SelectFirst[Transpose@
%t A306264     {Reverse@ #[[-Ceiling[Length[#]/2] ;; -2]],
%t A306264       #[[2 ;; Ceiling[Length[#]/2]]]} &@ Divisors[n],
%t A306264     CoprimeQ @@ # &][[1]] ]; f[1] = 1;
%t A306264 a[n_] := 1 + #*a[n/#] &[f[n]]; a[1] = 0;
%t A306264 Array[a, 120] (* _Michael De Vlieger_, Jun 24 2025 *)
%o A306264 (PARI) d(n) = if (omega(n) == 1, n, my(v=select(x->(gcd(x, n/x)==1), divisors(n))); v[#v-1]);
%o A306264 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
%o A306264 (PARI)
%o A306264 A324388(n) = if(1>=omega(n),n,fordiv(n,d,if((d>1)&&(1==gcd(d,n/d)),return(n/d))));
%o A306264 A306264(n) = if(1==n,0,my(d=A324388(n)); 1+(d*A306264(n/d))); \\ _Antti Karttunen_, Feb 28 2019
%Y A306264 Cf. A006022, A077610, A111284, A324388.
%K A306264 nonn
%O A306264 1,6
%A A306264 _David James Sycamore_, Feb 01 2019