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A348733 a(n) = gcd(A003959(n), A034448(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

%I A348733 #13 Nov 05 2021 21:12:25
%S A348733 1,3,4,1,6,12,8,9,2,18,12,4,14,24,24,1,18,6,20,6,32,36,24,36,2,42,4,8,
%T A348733 30,72,32,3,48,54,48,2,38,60,56,54,42,96,44,12,12,72,48,4,2,6,72,14,
%U A348733 54,12,72,72,80,90,60,24,62,96,16,1,84,144,68,18,96,144,72,18,74,114,8,20,96,168,80,6,2,126,84,32
%N A348733 a(n) = gcd(A003959(n), A034448(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.
%C A348733 This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 1444 = 2^2 * 19^2, where a(1444) = 10 != 1*2 = a(4)*a(361). See A348740 for the list of such positions.
%H A348733 Antti Karttunen, <a href="/A348733/b348733.txt">Table of n, a(n) for n = 1..21125</a>
%F A348733 a(n) = gcd(A003959(n), A034448(n)).
%F A348733 a(n) = gcd(A003959(n), A348732(n)) = gcd(A034448(n), A348732(n)).
%F A348733 a(n) = A003959(n) / A348734(n) = A034448(n) / A348735(n).
%t A348733 f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), Times @@ f2 @@@ f]; Array[a, 100] (* _Amiram Eldar_, Nov 05 2021 *)
%o A348733 (PARI)
%o A348733 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A348733 A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
%o A348733 A348733(n) = gcd(A003959(n), A034448(n));
%Y A348733 Cf. A003959, A034448, A348732, A348734, A348735, A348740.
%Y A348733 Cf. also A344695, A348047, A348503, A348946 for similar, almost multiplicative sequences.
%K A348733 nonn
%O A348733 1,2
%A A348733 _Antti Karttunen_, Nov 05 2021