A371090 -id:A371090 - cp's OEIS frontend

A376886 The number of distinct factors of n of the form p^(k!), where p is a prime and k >= 1, when the factorization is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 3, 3

Offset: 1

Amiram Eldar, Oct 08 2024

See A376885 for details about this factorization.
First differs from A371090 at n = 2^18 = 262144.
Differs from A064547 at n = 64, 128, 192, 256, 320, 384, 448, 512, ... .
Differs from A058061 at n = 128, 384, 512, 640, 896, ... .

			For n = 8 = 2^3, the representation of 3 in factorial base is 11, i.e., 3 = 1! + 2!, so 8 = (2^(1!))^1 * (2^(2!))^1 and a(8) = 1 + 1 = 2.
For n = 16 = 2^4, the representation of 4 in factorial base is 20, i.e., 4 = 2 * 2!, so 16 = (2^(2!))^2 and a(16) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007623, A058061, A060130, A077761, A371090.

Similar sequences: A064547, A318464, A376885.

fdignum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 0, s++]; m++]; s]; f[p_, e_] := fdignum[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

fdignum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, if(r > 0, s ++); m++); s;}
a(n) = {my(e = factor(n)[, 2]); sum(i = 1, #e, fdignum(e[i]));}

Additive with a(p^e) = A060130(e).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.12589120926760155013..., where f(x) = -x + (1-x) * Sum_{k>=1} A060130(k) * x^k.

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

A376886 The number of distinct factors of n of the form p^(k!), where p is a prime and k >= 1, when the factorization is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.

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