|
%I #14 Nov 03 2021 22:32:28
%S 1,1,1,5,1,1,1,3,10,1,1,5,1,1,1,17,1,10,1,5,1,1,1,3,26,1,7,5,1,1,1,11,
%T 1,1,1,50,1,1,1,3,1,1,1,5,10,1,1,17,50,26,1,5,1,7,1,3,1,1,1,5,1,1,10,
%U 65,1,1,1,5,1,1,1,6,1,1,26,5,1,1,1,17,82,1,1,5,1,1,1,3,1,10,1,5,1,1,1,11,1,50,10,130
%N a(n) = usigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.
%C This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 72 = 8*9, where a(72) = 6 != 3*10 = a(8) * a(9).
%H Antti Karttunen, <a href="/A348505/b348505.txt">Table of n, a(n) for n = 1..16384</a>
%H Antti Karttunen, <a href="/A348505/a348505.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F a(n) = A034448(n) / A348503(n) = A034448(n) / gcd(A000203(n), A034448(n)).
%t f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := (usigma = Times @@ f1 @@@ (fct = FactorInteger[n])) / GCD[usigma, Times @@ f2 @@@ fct]; Array[a, 100] (* _Amiram Eldar_, Oct 29 2021 *)
%o (PARI)
%o A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
%o A348505(n) = { my(u=A034448(n)); (u/gcd(u, sigma(n))); };
%Y Cf. A000203, A005117, A034448, A048146, A063880, A348503, A348504, A348506 (positions of ones).
%Y Cf. also A344697, A348049.
%K nonn
%O 1,4
%A _Antti Karttunen_, Oct 29 2021
|
