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%I #6 Nov 05 2021 18:26:55
%S 1444,3249,3364,4332,4563,6498,7220,7569,9126,10092,10108,12996,13924,
%T 15138,15884,16245,16820,17689,18252,18772,21125,21660,22743,22815,
%U 23104,23548,24548,24964,25992,27436,30276,30324,31329,31684,31941,32490,33212,35378,35739,36100,36504,37004,37845,38988,41209,41772
%N Positions k where A348733(k) is not multiplicative.
%C Numbers k with a factorization into coprime x and k/x with A348733(x) * A348733(k/x) <> A348733(k).
%t f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a1[1] = 1; a1[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), Times @@ f2 @@@ f]; f3[p_, e_] := a1[p^e]; a2[n_] := Times @@ f3 @@@ FactorInteger[n]; Position[Table[a2[n] - a1[n], {n, 1, 42000}], _?(# != 0 &)] // Flatten (* _Amiram Eldar_, Nov 05 2021 *)
%o (PARI)
%o A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
%o A348733(n) = gcd(A003959(n), A034448(n));
%o A348733mult(n) = { my(f = factor(n)); prod(k=1, #f~, A348733(f[k, 1]^f[k, 2])); };
%o isA348740(n) = (A348733(n)!=A348733mult(n));
%Y Cf. A003959, A034448, A348733, A348734, A348735.
%Y Cf. also A344702.
%K nonn
%O 1,1
%A _Antti Karttunen_, Nov 05 2021
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