This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A371091 #23 May 13 2024 09:14:55
%S A371091 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,
%T A371091 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,
%U A371091 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
%N A371091 Number of 1's in the recursive decomposition of primorial base expansion of n.
%C A371091 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.
%H A371091 Antti Karttunen, <a href="/A371091/b371091.txt">Table of n, a(n) for n = 0..60060</a>
%H A371091 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A371091 a(n) = A371090(A276086(n)).
%F A371091 For all n, A267263(n) <= a(n) <= A276150(n).
%e A371091 n A049345(n) recursive a(n) = number of 1's
%e A371091 decomposition in the decomposition
%e A371091 --------------------------------------------------------------------
%e A371091 0 0 () 0
%e A371091 1 1 (1) 1
%e A371091 2 10 (1 0) 1
%e A371091 3 11 (1 1) 2
%e A371091 4 20 ((1 0) 0) 1
%e A371091 5 21 ((1 0) 1) 2
%e A371091 6 100 (1 0 0) 1
%e A371091 7 101 (1 0 1) 2
%e A371091 8 110 (1 1 0) 2
%e A371091 9 111 (1 1 1) 3
%e A371091 10 120 (1 (1 0) 0) 2
%e A371091 11 121 (1 (1 0) 1) 3
%e A371091 12 200 ((1 0) 0 0) 1
%e A371091 ..
%e A371091 21 311 ((1 1) 1 1) 4
%e A371091 ..
%e A371091 24 400 (((1 0) 0) 0 0) 1
%e A371091 ..
%e A371091 29 421 (((1 0) 0) (1 0) 1) 3
%e A371091 30 1000 (1 0 0 0) 1
%e A371091 ..
%e A371091 51 1311 (1 (1 1) 1 1) 5
%e A371091 ..
%e A371091 59 1421 (1 ((1 0) 0) (1 0) 1) 4
%e A371091 60 2000 ((1 0) 0 0 0) 1
%e A371091 ..
%e A371091 111 3311 ((1 1) (1 1) 1 1) 6
%e A371091 ...
%e A371091 360 15000 (1 ((1 0) 1) 0 0 0) 3
%e A371091 ...
%e A371091 2001 93311 ((1 1 1) (1 1) (1 1) 1 1) 9
%e A371091 ....
%e A371091 4311 193311 (1 (1 1 1) (1 1) (1 1) 1 1) 10.
%e A371091 29 is decomposed in piecemeal fashion as: A049345(29) = 421 --> ("20" "10" "1") --> (((1 0) 0) (1 0) 1).
%o A371091 (PARI)
%o A371091 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A371091 A371090(n) = vecsum(apply(e->if(1==e,1,A371091(e)),factor(n)[, 2]));
%o A371091 A371091(n) = A371090(A276086(n));
%Y A371091 Cf. A049345, A267263, A276086, A276150, A371090.
%Y A371091 Cf. A372559 (positions of records and the first occurrence of n).
%Y A371091 Differs from A328482 for the first time at n=360, where a(360) = 3, while A328482(360) = 1.
%K A371091 nonn,base
%O A371091 0,4
%A A371091 _Antti Karttunen_, Mar 31 2024