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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. 14
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..10000 A. Karttunen, Scheme-program for computing this sequence. FORMULA Additive with a(p^e) = 1 + a(e). a(1) = 0; for n > 1, a(n) = 1 + a(A067029(n)) + a(A028234(n)). - Antti Karttunen, Mar 23 2017 Other identities. For all n >= 1: a(A276230(n)) = n. a(n) = A106493(A106444(n)). a(n) = A106491(n) - A064372(n). EXAMPLE 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. MAPLE a:= proc(n) option remember; `if`(n=1, 0,       add(1+a(i), i=ifactors(n)))     end: seq(a(n), n=1..100);  # Alois P. Heinz, Nov 07 2014 MATHEMATICA a[n_] := a[n] = If[n == 1, 0, Sum[1 + a[i[]], {i, FactorInteger[n]}]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *) PROG (Scheme, with memoization-macro definec) (definec (A106490 n) (if (= 1 n) 0 (+ 1 (A106490 (A067029 n)) (A106490 (A028234 n))))) ;; Antti Karttunen, Mar 23 2017 (PARI) A067029(n) = if(n<2, 0, factor(n)[1, 2]); A028234(n) = my(f = factor(n)); if (#f~, f[1, 1] = 1); factorback(f); /* after Michel Marcus */ a(n) = if(n<2, 0, 1 + a(A067029(n)) + a(A028234(n))); for(n=1, 150, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 23 2017, after formula by Antti Karttunen CROSSREFS Cf. A028234, A064372, A067029, A106444, A106491, A106492, A106493. Cf. A276230 (gives first k such that a(k) = n, i.e., this sequence is a left inverse of A276230). After n=1 differs from A038548 for the first time at n=24, where A038548(24)=4, while a(24)=3. Sequence in context: A238949 A076755 A317751 * A327399 A122375 A038548 Adjacent sequences:  A106487 A106488 A106489 * A106491 A106492 A106493 KEYWORD nonn AUTHOR Antti Karttunen, May 09 2005 based on Leroy Quet's message ('Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003. STATUS approved

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Last modified May 9 18:53 EDT 2021. Contains 343744 sequences. (Running on oeis4.)