A372559 -id:A372559 - cp's OEIS frontend

A343048 a(n) is the least number whose sum of digits in primorial base equals n.

0, 1, 3, 5, 11, 17, 23, 29, 59, 89, 119, 149, 179, 209, 419, 629, 839, 1049, 1259, 1469, 1679, 1889, 2099, 2309, 4619, 6929, 9239, 11549, 13859, 16169, 18479, 20789, 23099, 25409, 27719, 30029, 60059, 90089, 120119, 150149, 180179, 210209, 240239, 270269

Offset: 0

Rémy Sigrist, Apr 05 2021

Equivalently, this sequence gives positions of records in A276150.

			The first terms, alongside their primorial base representation, are:
  n   a(n)  prim(a(n))
  --  ----  ----------
   0     0           0
   1     1           1
   2     3          11
   3     5          21
   4    11         121
   5    17         221
   6    23         321
   7    29         421
   8    59        1421
   9    89        2421
  10   119        3421
  11   149        4421
  12   179        5421
  13   209        6421
  14   419       16421
  15   629       26421

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Index entries for sequences related to primorial base

Cf. A200748, A276150.
One less than A060735.
Cf. also A372559.

a(n) = my (v=0, pp=1); forprime (p=2, oo, if (n==0, return (v), my (d=min(p-1, n)); n-=d; v+=d*pp; pp*=p))

A276150(a(n)) = n.

a(n) = A060735(n)-1. - Antti Karttunen, Nov 14 2024

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

Original entry on oeis.org

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

Antti Karttunen, Mar 31 2024

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.

			     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).

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

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.

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));

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

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

cp's OEIS Frontend

A343048 a(n) is the least number whose sum of digits in primorial base equals n.

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