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

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

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.

Index entries for sequences computed from exponents in factorization of n

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[2]), i=ifactors(n)[2]))
    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[[2]]], {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 * A349281 A345222 A327399

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