A191251 -id:A191251 - cp's OEIS frontend

A191250 Fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 001.

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

Offset: 1

Clark Kimberling, May 28 2011

nonn

A few 3-substitution sequences:
A191250:  0 -> 01, 1 -> 02, 2 -> 001
A191254:  0 -> 01, 1 -> 02, 2 -> 01
A191255:  0 -> 01, 1 -> 02, 2 -> 03, 3 -> 01
A191258:  0 -> 01, 1 -> 02, 2 -> 03, 3 -> 001
A191261:  0 -> 01, 1 -> 002, 2 -> 01
A191265:  0 -> 001, 1 -> 002, 2 -> 01
A191269:  0 -> 001, 1 -> 02, 2 -> 01
See also A189576 and A189628.

Cf. A191251, A191252, A191253.

t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 0, 1}}] &, {0},8]  (* A191250 *)
Flatten[Position[t, 0]]  (* A191251 *)
Flatten[Position[t, 1]]  (* A191252 *)
Flatten[Position[t, 2]]  (* A191253 *)

A043306 Sum of all digits in all base-b representations for n, for 2 <= b <= n.

Original entry on oeis.org

1, 3, 4, 8, 10, 16, 17, 21, 25, 35, 34, 46, 52, 60, 58, 74, 73, 91, 92, 104, 114, 136, 128, 144, 156, 168, 171, 199, 193, 223, 221, 241, 257, 281, 261, 297, 315, 339, 333, 373, 367, 409, 416, 430, 452, 498, 472, 508, 515, 547, 556, 608, 598, 638, 634, 670, 698, 756, 717, 777

Offset: 2

Clark Kimberling

			5 = 101_2 = 12_3 = 11_4 = 10_5. Thus a(5) = 2 + 3 + 2 + 1 = 8.

Alois P. Heinz, Table of n, a(n) for n = 2..1000
Robin Fissum, Digit sums and the number of prime factors of the factorial n!=1.2...n, Bachelor's project in BMAT, Norwegian University of Science and Technology, 2020.
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme, Math. Semesterber., Vol. 49, No. 2 (2002), pp. 209-226; alternative link.
Vladimir Shevelev, Compact integers and factorials, Acta Arith., Vol. 126, No. 3 (2007), pp. 195-236 (see p. 205).

Cf. A014837, A043000, A068953, A191251, A191322, A191350, A309531.

Row sums of A240236.

Table[Sum[Total[First[RealDigits[n, i]]], {i, 2, n}], {n, 2, 80}] (* Carl Najafi, Aug 16 2011 *)

a(n) = sum(i=2, n, vecsum(digits(n, i))); \\ Michel Marcus, Jan 03 2017

a(n) = sum(b=2, n, sumdigits(n, b)); \\ Michel Marcus, Aug 18 2017

from sympy.ntheory.digits import digits
def a(n): return sum(sum(digits(n, b)[1:]) for b in range(2, n+1))
print([a(n) for n in range(2, 62)]) # Michael S. Branicky, Apr 04 2022

From Vladimir Shevelev, Jun 03 2011: (Start)
a(n) = (n-1)*n - Sum_{i=2..n} (i-1)*Sum_{r>=1} floor(n/i^r).
a(n) <= (n-1)^2*log(n+1)/log(n).
Problem: find a better upper estimate. (End)
From Amiram Eldar, Apr 16 2021: (Start)
a(n) = A014837(n) + 1.
a(n) ~ (1-Pi^2/12)*n^2 + O(n^(3/2)) (Fissum, 2020). (End)

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A191250 Fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 001.

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A043306 Sum of all digits in all base-b representations for n, for 2 <= b <= n.

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