A087192 -id:A087192

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A155167

(L)-sieve transform of A004767 = {3,7,11,15,...,4n-1,...}.

+10
10

1, 2, 3, 5, 7, 10, 14, 19, 26, 35, 47, 63, 85, 114, 153, 205, 274, 366, 489, 653, 871, 1162, 1550, 2067, 2757, 3677, 4903, 6538, 8718, 11625, 15501, 20669, 27559, 36746, 48995, 65327, 87103, 116138, 154851, 206469

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

See A152009 for the definition of the (L)-sieve transform.

This appears to be the same sequence that is defined in Problem 193 of Popular Computing, Number 55 (see link). - N. J. A. Sloane, Apr 16 2015

LINKS

Table of n, a(n) for n=1..40.

Popular Computing (Calabasas, CA), Coding Fun: Rearranging All The Numbers, Annotated and scanned copy of pages PC55-2, PC55-3, and PC55-1(cover) of Vol. 5 (No. 55, Oct 1977).

FORMULA

All listed terms satisfy the recurrence a(n) = floor((4*a(n-1)+3)/3), with a(1)=1.

MAPLE

# Maple program for Popular Computing Problem 193, which produces terms which appear to match this sequence, from N. J. A. Sloane, Apr 16 2015

with(LinearAlgebra): M:=1000; B:=300;

t1:=Array(1..M, 0); t2:=Array(1..M, 0); t3:=Array(1..M, -1);

for n from 1 to M do t1[n]:=n+2; od:

for n from 1 to B do

i:=t1[1];

if t3[i] = -1 then t3[i]:=n-1; fi;

for j from 1 to i do t2[j]:=t1[j+1]; od:

t2[i+1]:=i;

for p from i+2 to M-2 do t2[p]:=t1[p]; od;

for q from 1 to M-2 do t1[q]:=t2[q]; od:

od:

[seq(t3[n], n=3..B)];

MATHEMATICA

NestList[Floor[(4#+3)/3]&, 1, 40] (* Harvey P. Dale, Oct 04 2021 *)

CROSSREFS

Cf. A004767, A006999, A061419, A087192, A152009.

KEYWORD

nonn

AUTHOR

John W. Layman, Jan 21 2009

STATUS

approved

A087165

a(n)=1 when n == 1 (mod 4), otherwise a(n) = a(n - ceiling(n/4)) + 1. Removing all the 1's results in the original sequence with every term incremented by 1.

+10
3

1, 2, 3, 4, 1, 5, 2, 6, 1, 3, 7, 2, 1, 4, 8, 3, 1, 2, 5, 9, 1, 4, 2, 3, 1, 6, 10, 2, 1, 5, 3, 4, 1, 2, 7, 11, 1, 3, 2, 6, 1, 4, 5, 2, 1, 3, 8, 12, 1, 2, 4, 3, 1, 7, 2, 5, 1, 6, 3, 2, 1, 4, 9, 13, 1, 2, 3, 5, 1, 4, 2, 8, 1, 3, 6, 2, 1, 7, 4, 3, 1, 2, 5, 10, 1, 14, 2, 3, 1, 4, 6, 2, 1, 5, 3, 9, 1, 2, 4, 7, 1, 3, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Indices of records are given by A087192: a(A087192(n))=n, where A087192(n) = ceiling(A087192(n-1)*4/3).

From Benoit Cloitre, Mar 07 2009: (Start)

To construct the sequence:

Step 1: start from a sequence of 1's, leaving 3 undefined places between 1's, giving 1,(),(),(),1,(),(),(),1,(),(),(),1,(),(),(),1,(),(),(),1,...

Step 2: replace the first undefined place with a 2 and leave 3 undefined places between 2's, giving 1,2,(),(),1,(),2,(),1,(),(),2,1,(),(),(),1,2,(),(),1,...

Step 3: replace the first undefined place with a 3 and leave 3 undefined places between 3's, giving 1,2,3,(),1,(),2,(),1,3,(),2,1,(),(),3,1,2,(),(),1,...

Step 4: replace the first undefined place with a 4 and leave 3 undefined places between 4's, giving 1,2,3,4,1,(),2,(),1,3,(),2,1,4,(),3,1,2,(),(),1,...

Iterating the process indefinitely yields the sequence: 1,2,3,4,1,5,2,6,1,3,7,2,1,4,8,3,1,2,5,9,1,... (End)

LINKS

Table of n, a(n) for n=1..103.

FORMULA

a(n) = 4 + A244041(4*(n-1)) - A244041(4*n). - Tom Edgar and James Van Alstine, Aug 05 2014

a(4*n) = a(3*n)+1.

a(4*n+1) = 1.

a(4*n+2) = a(3*n+1)+1.

a(4*n+3) = a(3*n+2)+1. - Robert Israel, Aug 05 2014

a(n) < k*log(n) + 4 for n > 1 where k = 1/log(4/3) < 3.5. - Charles R Greathouse IV, Sep 22 2022

MAPLE

for n from 1 to 100 do

if n mod 4 = 1 then A[n]:= 1

else A[n]:= A[n - ceil(n/4)] + 1

od:

seq(A[n], n=1..100); # Robert Israel, Aug 05 2014

PROG

(PARI) a(n)=my(s); while(n>4, if(n%4==1, return(s+1)); n=(n\4*3)+max(n%4 - 1, 0); s++); s+n \\ Charles R Greathouse IV, Sep 22 2022

CROSSREFS

Cf. A001511, A087088, A087192, A244041.

a(n+1) - a(n) = 4*A018902(n-3), n > 2.

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Aug 24 2003

STATUS

approved

A363758

Maximum sum of digits for any number with n digits in fractional base 4/3.

+10
3

0, 3, 6, 8, 9, 12, 13, 15, 17, 19, 22, 24, 26, 28, 30, 32, 33, 36, 37, 40, 42, 44, 46, 48, 50, 52, 54, 56, 57, 60, 62, 65, 67, 70, 71, 73, 75, 77, 80, 83, 84, 87, 90, 93, 94, 96, 98, 101, 104, 106, 108, 109, 112, 115, 117, 120, 122, 123, 126, 129, 131, 133, 134

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

This sequence is strictly increasing since if a(n) is attained by the sum of digits of k, then the final digit of k is 3 and (k - (k mod 3))*4/3 + 3 is the same digits with a new second-least significant 1, 2 or 3 inserted, and so a(n+1) >= a(n) + 1.

Terms can be derived from A357425 by a(n) = s for the largest s where A357425(s) has n digits in base 4/3.

LINKS

Kevin Ryde, Table of n, a(n) for n = 0..207

Kevin Ryde, Mean Digit Plot (and LaTeX source).

Index entries for sequences related to fractional bases

FORMULA

a(n) = Max_{4*A087192(n-1) <= i < 4*A087192(n)} A244041(i), for n>=2.

EXAMPLE

For n=9, the numbers with 9 digits in base 4/3 are 60 to 79 and among them the maximum sum of digits is A244041(75) = 19 (those digits being 321023323), and so a(9) = 19.

CROSSREFS

Cf. A024631 (base 4/3), A244041 (sum of digits).

Cf. A357425 (smallest with sum s), A087192.

KEYWORD

nonn,base

AUTHOR

Kevin Ryde, Jun 20 2023

STATUS

approved

A364751

Minimum sum of digits for any number of length n digits in fractional base 4/3.

+10
1

0, 3, 5, 6, 6, 8, 8, 9, 10, 10, 11, 11, 11, 11, 13, 14, 16, 17, 17, 17, 18, 19, 21, 22, 22, 23, 24, 26, 26, 26, 27, 28, 29, 29, 29, 29, 29, 29, 31, 33, 34, 35, 36, 37, 38, 38, 38, 39, 39, 41, 41, 42, 42, 43, 43, 45, 45, 46, 46, 48, 50, 50, 52, 52, 52, 52, 53, 55

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

0 is taken to be 1 digit long so a(1) = 0.

Terms can be derived from A364779 by a(n) = s for the smallest s where k = A364779(s) is >= n digits long (noting that stripping trailing 0's from k suffices to show numbers with sum of digits s exist at each length down to where sum s-1 exists).

LINKS

Kevin Ryde, Table of n, a(n) for n = 1..209

Kevin Ryde, Mean Digit Plot

Index entries for sequences related to fractional bases

FORMULA

a(n) = Min_{4*A087192(n-1) <= k < 4*A087192(n)} A244041(k), for n >= 2.

CROSSREFS

Cf. A024631 (base 4/3), A244041 (sum of digits), A364779 (largest with sum).

Cf. A363758 (maximum sum).

KEYWORD

nonn,base

AUTHOR

Kevin Ryde, Sep 07 2023

STATUS

approved

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