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A106491 Total number of bases and exponents in Quetian Superfactorization of n, including the unity-exponents at the tips of branches.

%I A106491 #30 Jul 09 2025 04:25:05
%S A106491 1,2,2,3,2,4,2,3,3,4,2,5,2,4,4,4,2,5,2,5,4,4,2,5,3,4,3,5,2,6,2,3,4,4,
%T A106491 4,6,2,4,4,5,2,6,2,5,5,4,2,6,3,5,4,5,2,5,4,5,4,4,2,7,2,4,5,5,4,6,2,5,
%U A106491 4,6,2,6,2,4,5,5,4,6,2,6,4,4,2,7,4,4,4,5,2,7,4,5,4,4,4,5,2,5,5,6,2,6
%N A106491 Total number of bases and exponents in Quetian Superfactorization of n, including the unity-exponents at the tips of branches.
%H A106491 Alois P. Heinz, <a href="/A106491/b106491.txt">Table of n, a(n) for n = 1..10000</a>
%H A106491 A. Karttunen, <a href="/A091247/a091247.scm.txt">Scheme-program for computing this sequence.</a>
%H A106491 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A106491 From _Antti Karttunen_, Mar 23 2017: (Start)
%F A106491 a(1) = 1, and for n > 1, if A028234(n) = 1, a(n) = 1 + a(A067029(n)), otherwise a(n) = 1 + a(A067029(n)) + a(A028234(n)).
%F A106491 If n is a prime power p^k (a term of A000961), a(n) = 1 + a(k).
%F A106491 (End)
%F A106491 Other identities. For all n >= 1:
%F A106491 a(n) = A106490(n) + A064372(n).
%F A106491 a(n) = A106494(A106444(n)).
%e A106491 a(64) = 5, as 64 = 2^6 = 2^(2^1*3^1) and there are 5 nodes in that superfactorization. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 8. See comments at A106490.
%p A106491 a:= proc(n) option remember; `if`(n=1, 1,
%p A106491       add(1+a(i[2]), i=ifactors(n)[2]))
%p A106491     end:
%p A106491 seq(a(n), n=1..100);  # _Alois P. Heinz_, Nov 07 2014
%t A106491 a[n_] := a[n] = If[n == 1, 1, 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 A106491 (Scheme, with memoization-macro definec)
%o A106491 (definec (A106491 n) (cond ((= 1 n) n) ((= 1 (A028234 n)) (+ 1 (A106491 (A067029 n)))) (else (+ 1 (A106491 (A067029 n)) (A106491 (A028234 n)))))) ;; _Antti Karttunen_, Mar 23 2017
%o A106491 (PARI)
%o A106491 A067029(n) = if(n<2, 0, factor(n)[1,2]);
%o A106491 A028234(n) = my(f = factor(n)); if (#f~, f[1, 1] = 1); factorback(f); /* after _Michel Marcus_ */
%o A106491 a(n) = if(n<2, 1, if(A028234(n)==1, 1 + a(A067029(n)), 1 + a(A067029(n)) + a(A028234(n))));
%o A106491 for(n=1, 150, print1(a(n),", ")) \\ _Indranil Ghosh_, Mar 23 2017, after formula by _Antti Karttunen_
%Y A106491 Cf. A000961, A028234, A064372, A067029, A106444, A106490, A106492, A106494.
%K A106491 nonn
%O A106491 1,2
%A A106491 _Antti Karttunen_, May 09 2005 based on _Leroy Quet_'s message ('Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003