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A106490 Total number of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.

%I A106490 #46 Jul 09 2025 04:24:58
%S A106490 0,1,1,2,1,2,1,2,2,2,1,3,1,2,2,3,1,3,1,3,2,2,1,3,2,2,2,3,1,3,1,2,2,2,
%T A106490 2,4,1,2,2,3,1,3,1,3,3,2,1,4,2,3,2,3,1,3,2,3,2,2,1,4,1,2,3,3,2,3,1,3,
%U A106490 2,3,1,4,1,2,3,3,2,3,1,4,3,2,1,4,2,2,2,3,1,4,2,3,2,2,2,3,1,3,3,4,1,3
%N A106490 Total number of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.
%C A106490 Quetian Superfactorization proceeds by factoring a natural number to its unique prime-exponent factorization (p1^e1 * p2^e2 * ... pj^ej) and then factoring recursively each of the (nonzero) exponents in similar manner, until unity-exponents are finally encountered.
%H A106490 Alois P. Heinz, <a href="/A106490/b106490.txt">Table of n, a(n) for n = 1..10000</a>
%H A106490 A. Karttunen, <a href="/A091247/a091247.scm.txt">Scheme-program for computing this sequence.</a>
%H A106490 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A106490 Additive with a(p^e) = 1 + a(e).
%F A106490 a(1) = 0; for n > 1, a(n) = 1 + a(A067029(n)) + a(A028234(n)). - _Antti Karttunen_, Mar 23 2017
%F A106490 Other identities. For all n >= 1:
%F A106490 a(A276230(n)) = n.
%F A106490 a(n) = A106493(A106444(n)).
%F A106490 a(n) = A106491(n) - A064372(n).
%e A106490 a(64) = 3, as 64 = 2^6 = 2^(2^1*3^1) and there are three non-1 nodes in that superfactorization. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 5. a(65536) = a(2^(2^(2^(2^1)))) = 4.
%p A106490 a:= proc(n) option remember; `if`(n=1, 0,
%p A106490       add(1+a(i[2]), i=ifactors(n)[2]))
%p A106490     end:
%p A106490 seq(a(n), n=1..100);  # _Alois P. Heinz_, Nov 07 2014
%t A106490 a[n_] := a[n] = If[n == 1, 0, Sum[1 + a[i[[2]]], {i,FactorInteger[n]}]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Nov 11 2015, after _Alois P. Heinz_ *)
%o A106490 (Scheme, with memoization-macro definec)
%o A106490 (definec (A106490 n) (if (= 1 n) 0 (+ 1 (A106490 (A067029 n)) (A106490 (A028234 n))))) ;; _Antti Karttunen_, Mar 23 2017
%o A106490 (PARI)
%o A106490 A067029(n) = if(n<2, 0, factor(n)[1,2]);
%o A106490 A028234(n) = my(f = factor(n)); if (#f~, f[1, 1] = 1); factorback(f); /* after _Michel Marcus_ */
%o A106490 a(n) = if(n<2, 0, 1 + a(A067029(n)) + a(A028234(n)));
%o A106490 for(n=1, 150, print1(a(n),", ")) \\ _Indranil Ghosh_, Mar 23 2017, after formula by _Antti Karttunen_
%Y A106490 Cf. A028234, A064372, A067029, A106444, A106491, A106492, A106493.
%Y A106490 Cf. A276230 (gives first k such that a(k) = n, i.e., this sequence is a left inverse of A276230).
%Y A106490 After n=1 differs from A038548 for the first time at n=24, where A038548(24)=4, while a(24)=3.
%K A106490 nonn
%O A106490 1,4
%A A106490 _Antti Karttunen_, May 09 2005 based on _Leroy Quet_'s message ('Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003