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A371091
Number of 1's in the recursive decomposition of primorial base expansion of n.
4
0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 3, 4, 4, 5, 4, 5, 2, 3, 3, 4, 3, 4, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 3, 4, 4, 5, 4, 5, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 3, 4, 4, 5, 4, 5, 3, 4, 4, 5, 4, 5, 4, 5, 5, 6, 5, 6, 3, 4, 4, 5, 4, 5, 1
OFFSET
0,4
COMMENTS
Take the primorial base expansion of n (A049345), and then replace any digit larger than 1 with its own primorial base expansion, and do this recursively until no digits larger than 1 remain. a(n) is then the number of 1's in the completed decomposition. (See the examples). This decomposition offers a way to design a natural primorial based numeral system that does not require an infinite number of arbitrary glyphs for its digits, but instead suffices with just two graphically distinct subfigures whose exact positions in the whole hierarchically organized composite glyph determines the numerical value of that glyph, a bit like in Maya numerals or Babylonian cuneiform digits, but based on a primorial number system instead of vigesimal or sexagesimal.
FORMULA
a(n) = A371090(A276086(n)).
For all n, A267263(n) <= a(n) <= A276150(n).
EXAMPLE
n A049345(n) recursive a(n) = number of 1's
decomposition in the decomposition
--------------------------------------------------------------------
0 0 () 0
1 1 (1) 1
2 10 (1 0) 1
3 11 (1 1) 2
4 20 ((1 0) 0) 1
5 21 ((1 0) 1) 2
6 100 (1 0 0) 1
7 101 (1 0 1) 2
8 110 (1 1 0) 2
9 111 (1 1 1) 3
10 120 (1 (1 0) 0) 2
11 121 (1 (1 0) 1) 3
12 200 ((1 0) 0 0) 1
..
21 311 ((1 1) 1 1) 4
..
24 400 (((1 0) 0) 0 0) 1
..
29 421 (((1 0) 0) (1 0) 1) 3
30 1000 (1 0 0 0) 1
..
51 1311 (1 (1 1) 1 1) 5
..
59 1421 (1 ((1 0) 0) (1 0) 1) 4
60 2000 ((1 0) 0 0 0) 1
..
111 3311 ((1 1) (1 1) 1 1) 6
...
360 15000 (1 ((1 0) 1) 0 0 0) 3
...
2001 93311 ((1 1 1) (1 1) (1 1) 1 1) 9
....
4311 193311 (1 (1 1 1) (1 1) (1 1) 1 1) 10.
29 is decomposed in piecemeal fashion as: A049345(29) = 421 --> ("20" "10" "1") --> (((1 0) 0) (1 0) 1).
PROG
(PARI)
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A371090(n) = vecsum(apply(e->if(1==e, 1, A371091(e)), factor(n)[, 2]));
CROSSREFS
Cf. A372559 (positions of records and the first occurrence of n).
Differs from A328482 for the first time at n=360, where a(360) = 3, while A328482(360) = 1.
Sequence in context: A267263 A060130 A328482 * A257695 A257694 A281543
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Mar 31 2024
STATUS
approved