A371091 Number of 1's in the recursive decomposition of primorial base expansion of n.

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

Author

Antti Karttunen, Mar 31 2024

Keywords

• nonn
• base

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.

Examples

			     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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..60060
• Index entries for sequences related to primorial base

Crossrefs

Cf. A049345, A267263, A276086, A276150, A371090.

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.

Programs

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]));
A371091(n) = A371090(A276086(n));

Formula

a(n) = A371090(A276086(n)).

For all n, A267263(n) <= a(n) <= A276150(n).