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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.
8
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
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
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