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

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, 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, 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

Offset

1,2

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

From Antti Karttunen, Mar 23 2017: (Start)

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)).

If n is a prime power p^k (a term of A000961), a(n) = 1 + a(k).

(End)

Other identities. For all n >= 1:

a(n) = A106490(n) + A064372(n).

a(n) = A106494(A106444(n)).

Example

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.

Maple

a:= proc(n) option remember; `if`(n=1, 1,
      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, 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 *)

Prog

(Scheme, with memoization-macro definec)
(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
(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, 1, if(A028234(n)==1, 1 + a(A067029(n)), 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. A000961, A028234, A064372, A067029, A106444, A106490, A106492, A106494.

Sequence in context: A205562 A217984 A196437 * A073184 A377519 A073182

Adjacent sequences: A106488 A106489 A106490 * A106492 A106493 A106494

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