A110592 -id:A110592 - cp's OEIS frontend

A110591 Number of digits in base-4 representation of n.

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Jonathan Vos Post, Jul 29 2005

Number of digits in A007090(n).

In terms of the repetition convolution operator #, where (sequence A) # (sequence B) = the sequence consisting of A(n) copies of B(n), this sequence is the repetition convolution A110594 # n. Over the set of positive infinite integer sequences, # gives a nonassociative noncommutative groupoid (magma) with a left identity (A000012) but no right identity, where the left identity is also a right nullifier and idempotent. For any positive integer constant c, the sequence c*A000012 = (c,c,c,c,...) is also a right nullifier; for c = 1, this is A000012; for c = 3 this is A010701.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000012, A007090, A010701, A049804, A081604, A110594.

import Data.List (unfoldr)
a110591 0 = 1
a110591 n = length $
   unfoldr (\x -> if x == 0 then Nothing else Just (x, x `div` 4)) n
-- Reinhard Zumkeller, Apr 22 2011

A110592 := proc(n)
    if n = 0 then
        1;
    else
        1+floor(log[4](n)) ;
    end if;
end proc:
seq(A110592(n),n=0..50) ; # R. J. Mathar, Sep 02 2020

a[n_] := If[n == 0, 1, Floor[Log[4, n]] + 1];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 24 2020 *)

G.f.: 1 + (1/(1 - x))*Sum_{k>=0} x^(4^k). - Ilya Gutkovskiy, Jan 08 2017

a(n) = floor(log_4(n)) + 1 for n >= 1. - Petros Hadjicostas, Dec 12 2019

A049803 a(n) = n mod 3 + n mod 9 + ... + n mod 3^k, where 3^k <= n < 3^(k+1).

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 1, 2, 0, 2, 4, 3, 5, 7, 6, 8, 10, 0, 2, 4, 3, 5, 7, 6, 8, 10, 0, 3, 6, 6, 9, 12, 12, 15, 18, 9, 12, 15, 15, 18, 21, 21, 24, 27, 18, 21, 24, 24, 27, 30, 30, 33, 36, 0, 3, 6, 6, 9, 12, 12, 15, 18, 9, 12, 15, 15, 18, 21, 21, 24, 27, 18

Offset: 1

Clark Kimberling

nonn

From Petros Hadjicostas, Dec 11 2019: (Start)
Conjecture: For b >= 2, consider the function s(n,b) = Sum_{1 <= b^j <= n} (n mod b^j) from p. 8 in Dearden et al. (2011). Then s(b*n + r, b) = b*s(n,b) + r*N(n,b) for 0 <= r <= b-1, where N(n,b) = floor(log_b(n)) + 1 is the number of digits in the base-b representation of n. As initial conditions, we have s(n,b) = 0 for 1 <= n <= b. (This is a generalization of a result by Robert Israel in A049802.)
Here b = 3 and a(n) = s(n,3).
We have N(n,2) = A070939(n), N(n,3) = A081604(n), N(n,4) = A110591(n), and N(n,5) = A110592(n).
If A_b(x) = Sum_{n >= 1} s(n,b)*x^n is the g.f. of the sequence (s(n,b): n >= 1) and the above conjecture is correct, then it can be proved that A_b(x) = b * A_b(x^b) * (1-x^b)/(1-x) + x * ((b-1)*x^b - b*x^(b-1) + 1)/((1-x)^2 * (1-x^b)) * Sum_{k >= 1} x^(b^k). (End)

Metin Sariyar, Table of n, a(n) for n = 1..32000
B. Dearden, J. Iiams, and J. Metzger, A Function Related to the Rumor Sequence Conjecture, J. Int. Seq. 14 (2011), #11.2.3, Example 7.

Cf. A049802, A049804, A070939, A081604, A110591, A110592, A330358.

a:= n-> add(irem(n, 3^j), j=1..ilog[3](n)):
seq(a(n), n=1..105);  # Alois P. Heinz, Dec 13 2019

Table[n * Floor@Log[3, n] - Sum[Floor[n*3^-k]*3^k, {k, Log[3, n]}], {n, 100}] (* after Federico Provvedi in A049802*) (* Metin Sariyar, Dec 12 2019 *)

a(n) = sum(k=1, logint(n, 3), n % 3^k); \\ Michel Marcus, Dec 12 2019

From Petros Hadjicostas, Dec 11 2019: (Start)
Conjecture: a(3*n+r) = 3*a(n) + r*A081604(n) = 3*a(n) + r*(floor(log_3(n)) + 1) for n >= 1 and r = 0, 1, 2.
If the conjecture above is true, the g.f. A(x) satisfies A(x) = 3*(1+x+x^2)*A(x^3) + x*(2*x+1)/(1-x^3) * Sum_{k >= 1} x^(3^k). (End)

A049804 a(n) = n mod 4 + n mod 16 + ... + n mod 4^k, where 4^k <= n < 4^(k+1).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 2, 4, 6, 4, 6, 8, 10, 8, 10, 12, 14, 12, 14, 16, 18, 0, 2, 4, 6, 4, 6, 8, 10, 8, 10, 12, 14, 12, 14, 16, 18, 0, 2, 4, 6, 4, 6, 8, 10, 8, 10, 12, 14, 12, 14, 16, 18, 0, 3, 6, 9, 8, 11, 14, 17, 16, 19, 22

Offset: 1

Clark Kimberling

nonn

From Petros Hadjicostas, Dec 11 2019: (Start)
Conjecture: For b >= 2, consider the function s(n,b) = Sum_{1 <= b^j <= n} (n mod b^j) from p. 8 in Dearden et al. (2011). Then s(b*n + r, b) = b*s(n,b) + r*N(n,b) for 0 <= r <= b-1, where N(n,b) = floor(log_b(n)) + 1 is the number of digits in the base-b representation of n. As initial conditions, we have s(n,b) = 0 for 1 <= n <= b. (This is a generalization of a result by Robert Israel in A049802.)
Here b = 4 and a(n) = s(n,4).
We have N(n,2) = A070939(n), N(n,3) = A081604(n), N(n,4) = A110591(n), and N(n,5) = A110592(n).
If A_b(x) = Sum_{n >= 1} s(n,b)*x^n is the g.f. of the sequence (s(n,b): n >= 1) and the above conjecture is correct, then it can be proved that A_b(x) = b * A_b(x^b) * (1-x^b)/(1-x) + x * ((b-1)*x^b - b*x^(b-1) + 1)/((1-x)^2 * (1-x^b)) * Sum_{k >= 1} x^(b^k). (End)

Metin Sariyar, Table of n, a(n) for n = 1..32000
B. Dearden, J. Iiams, and J. Metzger, A Function Related to the Rumor Sequence Conjecture, J. Int. Seq. 14 (2011), #11.2.3, Example 7.

Cf. A049802, A049803, A070939, A081604, A110591, A110592, A330358.

a:= n-> add(irem(n, 4^j), j=1..ilog[4](n)):
seq(a(n), n=1..105);  # Petros Hadjicostas, Dec 13 2019 (after Alois P. Heinz's program for A330358)

Table[n * Floor@Log[4, n] - Sum[Floor[n*4^-k]*4^k, {k, Log[4, n]}], {n, 100}] (* Metin Sariyar, Dec 12 2019 *)
a[n_] := Sum[Mod[n, 4^j], {j, 1, Length[IntegerDigits[n, 4]] - 1}];
Array[a, 105] (* Jean-François Alcover, Dec 31 2021 *)

a(n) = sum(k=1, logint(n, 4), n % 4^k); \\ Michel Marcus, Dec 12 2019

From Petros Hadjicostas, Dec 11 2019: (Start)
Conjecture: a(4*n+r) = 4*a(n) + r*A110591(n) = 4*a(n) + r*(floor(log_4(n)) + 1) for n >= 1 and r = 0, 1, 2, 3.
If the conjecture above is true, the g.f. A(x) satisfies A(x) = 4*(1 + x + x^2 + x^3)*A(x^4) + x*(1 + 2*x + 3*x^2)/(1 - x^4) * Sum_{k >= 1} x^(4^k). (End)

A330358 a(n) = n mod 5 + n mod 25 + ... + n mod 5^k, where 5^k <= n < 5^(k+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 2, 4, 6, 8, 5, 7, 9, 11, 13, 10, 12, 14, 16, 18, 15, 17, 19, 21, 23, 20, 22, 24, 26, 28, 0, 2, 4, 6, 8, 5, 7, 9, 11, 13, 10, 12, 14, 16, 18, 15, 17, 19, 21, 23, 20, 22, 24, 26, 28, 0, 2, 4, 6, 8, 5, 7, 9, 11, 13, 10

Offset: 1

Petros Hadjicostas, Dec 12 2019

Conjecture: For b >= 2, consider the function s(n,b) = Sum_{1 <= b^j <= n} (n mod b^j) from p. 8 in Dearden et al. (2011). Then s(b*n + r, b) = b*s(n,b) + r*N(n,b) for 0 <= r <= b-1, where N(n,b) = floor(log_b(n)) + 1 is the number of digits in the base-b representation of n. As initial conditions, we have s(n,b) = 0 for 1 <= n <= b. (This is a generalization of a result by Robert Israel in A049802.)
Here b = 5 and a(n) = s(n,5).
We have N(n,2) = A070939(n), N(n,3) = A081604(n), N(n,4) = A110591(n), and N(n,5) = A110592(n).
If A_b(x) = Sum_{n >= 1} s(n,b)*x^n is the g.f. of the sequence (s(n,b): n >= 1) and the above conjecture is correct, then it can be proved that A_b(x) = b * A_b(x^b) * (1-x^b)/(1-x) + x * ((b-1)*x^b - b*x^(b-1) + 1)/((1-x)^2 * (1-x^b)) * Sum_{k >= 1} x^(b^k).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A049802, A049803, A049804, A070939, A081604, A110591, A110592.

a:= n-> add(irem(n, 5^j), j=1..ilog[5](n)):
seq(a(n), n=1..105);  # Alois P. Heinz, Dec 13 2019

a[n_] := Sum[Mod[n, 5^j], {j, 1, Length[IntegerDigits[n, 5]] - 1}];
Array[a, 105] (* Jean-François Alcover, Dec 31 2021 *)

a(n) = sum(k=1, logint(n, 5), n % 5^k);
for(n=1, 100, print1(a(n), ", ")); \\ (after Michel Marcus's program in A049804)

Conjecture: a(5*n+r) = 5*a(n) + r*A110592(n) = 5*a(n) + r*(floor(log_5(n)) + 1) for n >= 1 and r = 0, 1, 2, 3, 4.

If the conjecture above is true, the g.f. A(x) satisfies A(x) = 5*(1 + x + x^2 + x^3 + x^4)*A(x^5) + x*(1 + 2*x + 3*x^2 + 4*x^3)/(1 - x^5) * Sum_{k >= 1} x^(5^k).

A110595 a(1)=5. For n > 1, a(n) = 4*5^(n-1) = A005054(n).

Original entry on oeis.org

5, 20, 100, 500, 2500, 12500, 62500, 312500, 1562500, 7812500, 39062500, 195312500, 976562500, 4882812500, 24414062500, 122070312500, 610351562500, 3051757812500, 15258789062500, 76293945312500, 381469726562500

Offset: 1

Jonathan Vos Post, Jul 29 2005

a(n) is the number of n-digit integers that contain only even digits (A014263). - Bernard Schott, Nov 11 2022

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A000351, A007091, A014263, A081604, A110591, A110592, A110593, A110594.

Join[{5},NestList[5#&,20,20]] (* Harvey P. Dale, Jun 19 2013 *)
Rest[CoefficientList[Series[5 x (1 - x)/(1 - 5 x), {x,0,50}], x]] (* G. C. Greubel, Sep 01 2017 *)

my(x='x+O('x^50)); Vec(5*x*(1-x)/(1-5*x)) \\ G. C. Greubel, Sep 01 2017

O.g.f.: 5*x*(1-x)/(1-5*x). - Better definition from R. J. Mathar, May 13 2008

Sum_{n>=1} 1/a(n) = 21/80. - Bernard Schott, Nov 11 2022

Better definition from R. J. Mathar, May 13 2008

Incorrect comment removed by Michel Marcus, Nov 11 2022

A357143 a(n) is sum of the base-5 digits of n each raised to the number of digits of n in base 5.

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 5, 10, 17, 4, 5, 8, 13, 20, 9, 10, 13, 18, 25, 16, 17, 20, 25, 32, 1, 2, 9, 28, 65, 2, 3, 10, 29, 66, 9, 10, 17, 36, 73, 28, 29, 36, 55, 92, 65, 66, 73, 92, 129, 8, 9, 16, 35, 72, 9, 10, 17, 36, 73, 16, 17, 24, 43, 80, 35, 36, 43, 62, 99, 72, 73, 80, 99, 136, 27

Offset: 1

Francesco A. Catalanotti, Oct 26 2022

			For n = 13_10 = 23_5 (2 digits in base 5): a(13) = 2^2 + 3^2 = 13.
For n = 73_10 = 243_5 (3 digits in base 5): a(73) = 2^3 + 4^3 + 3^3 = 99.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A357954, A010346, A110592.

Cf. in base 10: A157714, A101337, A151544.

f:= proc(n) local L,d,i;
  L:= convert(n,base,5);
  d:= nops(L);
  add(L[i]^d,i=1..d)
end proc:
map(f,[$1..100]); # Robert Israel, Oct 26 2023

a[n_] := Total[IntegerDigits[n, 5]^IntegerLength[n, 5]]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)

a(n) = my(d=digits(n, 5)); sum(k=1, #d, d[k]^#d); \\ Michel Marcus, Oct 29 2022

from sympy.ntheory.factor_ import digits
def A357143(n):
    t = len(s:=digits(n,5)[1:])
    return sum(d**t for d in s) # Chai Wah Wu, Oct 31 2022

a(n) = Sum_{i=1..A110592(n)} d(i)^A110592(n), where d(i) is the i-th digit of n in base 5.

Corrected and extended by Michel Marcus, Oct 29 2022

A037864 a(n)=(number of digits <=2)-(number of digits >2) in base 5 representation of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Cf. A110592.

A037864 := proc(n)
    a := 0 ;
    dgs := convert(n,base,5);
    for i from 1 to nops(dgs) do
        if op(i,dgs)<=2 then
            a := a+1 ;
        else
            a := a-1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 16 2015

nd5[n_]:=Module[{idn5=IntegerDigits[n,5],a},a=Count[idn5,?(#<=2&)];2a-Length[idn5]]; Array[nd5,90] (* _Harvey P. Dale, Oct 13 2011 *)

A217115 Greatest number (in decimal representation) with n nonprime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

67, 88, 442, 567, 2213, 2837, 3067, 11068, 14713, 15337, 15338, 57943, 73568, 77213, 76697, 289717, 280338, 370443, 386068, 386587, 389713, 1852217, 1524067, 1898442, 1930342, 1932943, 1948568, 7242943, 9261088, 9664717, 9586567, 9654712, 9710942, 9742849, 46305443

Offset: 0

Hieronymus Fischer, Dec 20 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 5^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n-(k(k+1)/2). For n=0,1,2,3,... the m(n) in base-5 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s. Thus, the number of nonprime substrings of m(n) is ((k+1)(k+2)/2)-k-1+i=(k(k+1)/2)+i=n. This proves the statement of existence. Proof of finiteness: Each 4-digit base-5 number has at least 2 nonprime substrings. Hence, each m := 4*floor((n+2)/2)-digit number has at least 2*(m/4) = 2*floor((n+2)/2) >= n+1 nonprime substrings. Consequently, there is a boundary b < 5^(m-1) such that all numbers > b have more than n nonprime substrings. It follows, that the set of numbers with n nonprime substrings is finite.

			a(0) = 67, since 67 = 232_5 (base-5) is the greatest number with zero nonprime substrings in base-5 representation.
a(1) = 88 = 323_5 has 6 substrings in base-5 representation (2, 2, 3, 23, 32, 323), the only nonprime substring is 323_5. 88 is the greatest number with 1 nonprime substring.
a(2) = 442 = 3232_5 has 10 substrings in base-5 representation (2, 2, 3, 3, 23, 32, 32, 232, 323 and 3232), exactly 2 of them are nonprime substrings (323_5=88 and 3232_5=442), and there is no greater number with 2 nonprime substrings in base-5 representation.
a(5) = 2837 = 42322_5 has 5 nonprime substrings in base-5 representation, these are 4, 22, 42, 322 and 4232, all the other substrings are prime. There is no greater number with 5 nonprime substrings in base-5 representation.

Hieronymus Fischer, Table of n, a(n) for n = 0..70

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300 - A213321.
Cf. A217102 - A217109, A217112 - A217119.
Cf. A217302 - A217309.

a(n) >= A217105(n).
a(n) >= A217305(A000217(A110592(a(n)))-n).
a(n) <= 5^(n+3).
a(n) <= 5^(4*floor(n/2)), n>1.
a(n) <= 5^min((n + 6)/2, 9*floor((n+20)/21)).
a(n) <= 125*5^(n/2).
With m := floor(log_5(a(n))) + 1:
a(n+m+1) >= 5*a(n), if a(n)!=1 (mod 5).
a(n+m) >= 5*a(n), if a(n)=1 (mod 5).

A357954 Integers k that are periodic points for some iterations of k->A357143(k).

Original entry on oeis.org

1, 2, 3, 4, 13, 18, 28, 118, 194, 289, 338, 353, 354, 419, 489, 528, 609, 769, 1269, 1299, 2081, 4890, 4891, 9113, 18575, 18702, 20759, 35084, 1874374, 338749352, 2415951874

Offset: 1

Francesco A. Catalanotti, Oct 22 2022

Given the function A357143(k), a number k is a term of the sequence if there exists a j such that A357143^j(k) = k, where j is the number of iterations applied.
The sequence is finite.
Proof: A357143(k) < k for all big enough k. g(k) = A110592(k)*4^A110592(k) is clearly an upper bound of A357143(k). Hence k > g(k) -> k > A357143(k), therefore every periodic point must be in an interval [s;t] such that for every k in [s;t] k <= g(k). Limit_{k->oo} g(k)/k = 0; now using the little-o definition we can show that there always exists a certain k_0 such that, for every k > k_0, k > g(k). The conclusion is that there must exist a finite number of intervals [s;t] and, consequently, a finite number of periodic points.
Every term k of the sequence is a periodic point (either a perfect digital invariant or a sociable digital invariant) for the function A357143(k).
The longest cycle needs 6 iterations to end: [489, 609, 769, 1269, 1299, 2081].

			k=9113 is a fixed point (perfect digital invariant) for the reiterated function A357143(k):
    9113_10 = 242423_5 (a 6-digit number in base 5);
    A357143(9113) = 2^6 + 4^6 + 2^6 + 4^6 + 2^6 + 3^6 = 9113.
k=18702 is a sociable digital invariant for the reiterated function A357143(k), requiring 2 iterations:
  1st iteration:
    18702_10 = 1044302_5 (a 7-digit number in base 5);
    A357143(18702) = 1^7 + 0^7 + 4^7 + 4^7 + 3^7 + 0^7 + 2^7 = 35084;
  2nd iteration:
    35084_10 = 2110314_5 (also a 7-digit number in base 5);
    A357143(35084) = 2^7 + 1^7 + 1^7 + 0^7 + 3^7 + 1^7 + 4^7 = 18702.

Wikipedia, Perfect digital invariant

Cf. A357143 , A010346 (fixed points), A110592 (exponents p(k)).

Cf. A157714 (base-10 sociable digital invariants), A101337 (A357143(k) base 10), A151544 (base-10 periodic points).

cp's OEIS Frontend

A110591 Number of digits in base-4 representation of n.

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A049803 a(n) = n mod 3 + n mod 9 + ... + n mod 3^k, where 3^k <= n < 3^(k+1).

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A049804 a(n) = n mod 4 + n mod 16 + ... + n mod 4^k, where 4^k <= n < 4^(k+1).

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A330358 a(n) = n mod 5 + n mod 25 + ... + n mod 5^k, where 5^k <= n < 5^(k+1).

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A110595 a(1)=5. For n > 1, a(n) = 4*5^(n-1) = A005054(n).

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A357143 a(n) is sum of the base-5 digits of n each raised to the number of digits of n in base 5.

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A037864 a(n)=(number of digits <=2)-(number of digits >2) in base 5 representation of n.

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