A143617 -id:A143617 - cp's OEIS frontend

A010371 Number of segments used to represent n on a 7-segment calculator display; version where '6', '7' and '9' use 6, 4 and 6 segments, respectively.

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

Offset: 0

Olivier.Gagneux(AT)roche.com

Except for 1 and 3 every positive integer occurs; A143616 and A143617 give record values and where they occur. - Reinhard Zumkeller, Aug 27 2008
The difference between this sequence and A006942 lies in the representation chosen for the digit 7,
                    
  | |                 |
    |  (here), vs.    |  in A006942.
If we mark with ' the "sans serif" graphical representation which uses one segment less and with * the "heavier" version, we have the following variants:
  A063720 (6', 7', 9'), A277116 (6*, 7', 9'), A074458 (6*, 7*, 9'),
  ___________________ A006942 (6*, 7', 9*), A010371 (6*, 7*, 9*) = this.
  Sequences A234691, A234692 and variants make precise which segments are lit in each digit. They are related through the Hamming weight A000120, see formula. The sequence could be extended to negative arguments with a(-n) = a(n)+1. - M. F. Hasler, Jun 17 2020

			LCD Display (cf. Casio scientific calculator fx-3600P):
   _     _  _       _   _   _   _   _
  | | |  _| _| |_| |_  |_  | | |_| |_|
  |_| | |_  _|   |  _| |_|   | |_|  _|

R. Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to calculator display

Segment variations: A006942, A063720, A074458, A277116.

Cf. A000120, A234691, A234692.

a010371 n = a010371_list !! n
a010371_list = [6,2,5,5,4,5,6,4,7,6] ++ f 10 where
   f x = (a010371 x' + a010371 d) : f (x + 1)
         where (x',d) = divMod x 10
-- Reinhard Zumkeller, Mar 15 2013

MapIndexed[(f[#2[[1]]-1] = #1)&, {6, 2, 5, 5, 4, 5, 6, 4, 7, 6}]; a[n_] := Total[f /@ IntegerDigits[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2017 *)

apply( {A010371(n)=digits(6255456476)[n%10+1]+if(n>9, self()(n\10))}, [0..99]) \\ M. F. Hasler, Jun 17 2020

For n > 9, a(n) = a(floor(n/10)) + a(n mod 10). - Reinhard Zumkeller, Aug 27 2008
a(n) = A000120(A234691(n)) = A000120(A234692(n))
  = A006942(n) + A102679(n) - A102681(n) (add number of digits 7)
= A074458(n) + A102683(n) (add number of digits 9). - M. F. Hasler, Jun 17 2020

Corrected and extended by Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 27 1999

Edited name, comments, cross-references. - M. F. Hasler, Jun 17 2020

A216261 Smallest positive number using exactly n segments on a calculator display (when '6' and '7' are represented using 6 resp. 3 segments).

Original entry on oeis.org

1, 7, 4, 2, 0, 8, 10, 18, 22, 20, 28, 68, 88, 108, 188, 200, 208, 288, 688, 888, 1088, 1888, 2008, 2088, 2888, 6888, 8888, 10888, 18888, 20088, 20888, 28888, 68888, 88888, 108888, 188888, 200888, 208888, 288888, 688888, 888888, 1088888, 1888888, 2008888, 2088888, 2888888

Offset: 2

Reinhard Zumkeller, Mar 15 2013

Essentially the same as A038619 and A143617. One could argue that a(3) should rather be -1 (prior to adding "positive" in the definition), which does use 3 segments on typical 7-segment displays, and is smaller than 7. Also, most pocket calculators and the Unicode standard (cf. links) use 4 rather than 3 segments to represent a '7' (as in A074458 and A010371, rather than A063720, A277116 or A006942), in which case a(3) is undefined if negative numbers are not allowed. No digit '9' will ever occur here, whether it would be represented with 6 or only 5 segments. However, digit '6' does occur, as the second smallest digit using 6 segments as does '0', which cannot occur as leading digit. If '6' is represented with 5 segments, any prefix 68 would be replaced with 80. - M. F. Hasler and Kevin Ryde, Jun 17 2020

Unicode, Symbols for Legacy Computing, The Unicode Standard, Version 13.0, 2020.
Index entries for sequences related to calculator display
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,10,-10).

Cf. A006942, A010371, A063720, A074458, A277116; A234691, A234692.

Cf. A038619 and A143617 (identical up to initial terms).

import Data.Maybe (fromJust)
import Data.List (elemIndex)
a216261 = fromJust . (`elemIndex` a006942_list)
-- Reinhard Zumkeller, Mar 15 2013

Drop[#, 2] &@ CoefficientList[Series[(x^2 + 6 x^3 - 3 x^4 - 2 x^5 - 2 x^6 + 8 x^7 + 2 x^8 - 2 x^9 - 56 x^10 + 28 x^11 + 28 x^12 + 60 x^13 - 60 x^14 - 28 x^17 + 28 x^18)/((1 - x) (1 - 10 x^7)), {x, 0, 50}], x] (* Michael De Vlieger, Jan 29 2016 *)

Vec((x^2 +6*x^3 -3*x^4 -2*x^5 -2*x^6 +8*x^7 +2*x^8 -2*x^9 -56*x^10 +28*x^11 +28*x^12 +60*x^13 -60*x^14 -28*x^17 +28*x^18)/((1-x)*(1-10*x^7)) + O(x^50)) \\ Michel Marcus, Jan 29 2016

A006942(a(n)) = n and A006942(m) <> n for m < a(n).
a(n+7) = 10*a(n) + 8 for n > 10. This can be deduced from a(n) = min{10*a(n-A006942(r))+r, r=0..9} via strong induction. - David Radcliffe, Jan 29 2016
G.f.: (x^2 +6*x^3 -3*x^4 -2*x^5 -2*x^6 +8*x^7 +2*x^8 -2*x^9 -56*x^10 +28*x^11 +28*x^12 +60*x^13 -60*x^14 -28*x^17 +28*x^18)/((1-x)*(1-10*x^7)). - David Radcliffe, Jan 29 2016

Name and cross-references edited by M. F. Hasler, Jun 17 2020

A038619 Smallest positive number that needs more lines when shown on a 7-segment display (digital clock) than any previous term.

Original entry on oeis.org

1, 2, 6, 8, 10, 18, 20, 28, 68, 88, 108, 188, 200, 208, 288, 688, 888, 1088, 1888, 2008, 2088, 2888, 6888, 8888, 10888, 18888, 20088, 20888, 28888, 68888, 88888, 108888, 188888, 200888, 208888, 288888, 688888, 888888, 1088888, 1888888, 2008888, 2088888, 2888888, 6888888, 8888888

Offset: 1

Jan Kristian Haugland

For n > 1, a(n) uses n + 3 segments to be displayed, when a digit '6' uses 6 segments (as in A234691, A234692 and A277116, A074458, A006942, A010371, but not in A063720). Sequence A143617 is the same but starts with 0, 8, ... and A216261 has additional terms 7 & 4 before 2 and 22 before 20. - M. F. Hasler, Jun 23 2020

			Digits 0, 1, 2, ..., 9 use 6, 2, 5, 5, 4, 5, 6, 3, 7, 6 lines / segments.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,10,-10).

Cf. A143617, A216261; A234691, A234692; A277116, A074458, A006942, A010371, A063720.

Block[{f, s}, MapIndexed[(f[#2[[1]] - 1] = #1) &, {6, 2, 5, 5, 4, 5, 6, 3, 7, 6}]; s = Array[Total[f /@ IntegerDigits[#]] &, 10^7]; Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* or *)
Nest[Append[#1, If[#2 > 13, 10 #1[[-7]] + 8, 10 #1[[-6]] + Boole[#2 != 13] 8]] & @@ {#, Length@ # + 1} &, {1, 2, 6, 8, 10, 18, 20}, 36] (* Michael De Vlieger, Jun 23 2020 *)
LinearRecurrence[{1,0,0,0,0,0,10,-10},{1,2,6,8,10,18,20,28,68,88,108,188,200,208},50] (* Harvey P. Dale, Aug 11 2025 *)

apply( {A038619(n)=if(n>7, self()(n-6-(n>13))*10+(n!=13)*8, [1,2,6, 8,10,18,20][n])}, [1..33]) \\ M. F. Hasler, Jun 23 2020

For n >= 3, the terms with n digits are given by: 108*A + B, 188*A + B, 200*A + B, 208*A + B, 288*A + B, 688*A + B, 888*A + B where A = 10^(n-3), B = 8*(A - 1)/9.
From M. F. Hasler, Jun 23 2020: (Start)
a(n) = 10*a(n-7) + 8 for n > 13 (and with a(n-6) for 7 < n < 13).
G.f.: (1 + x + 4*x^2 + 2*x^3 + 2*x^4 + 8*x^5 + 2*x^6 - 2*x^7 + 30*x^8 - 20*x^9 + 60*x^11 - 68*x^12 - 12*x^13)/((1 - x)*(1 - x^10)).
(End)
a(n) = a(n-1) + 10*a(n-7) - 10*a(n-8), for n >= 15. - Wesley Ivan Hurt, Jun 29 2020

Edited and offset corrected to 1 by M. F. Hasler, Jun 23 2020
More terms from Michael De Vlieger, Jun 23 2020
More terms from M. F. Hasler, Jun 23 2020

A143616 Record values in A010371.

Original entry on oeis.org

6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Reinhard Zumkeller, Aug 27 2008

a(n)=A010371(A143617(n)) and A010371(m)A143617(n);

{1,2,3,4,5,10} = complement(range of this sequence).

Index entries for linear recurrences with constant coefficients, signature (2,-1).

a(n)=n+5+(n>4) \\ Charles R Greathouse IV, Oct 26 2011

a(n) = n + 6 for n > 4. [Charles R Greathouse IV, Oct 26 2011]

cp's OEIS Frontend

A010371 Number of segments used to represent n on a 7-segment calculator display; version where '6', '7' and '9' use 6, 4 and 6 segments, respectively.

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A216261 Smallest positive number using exactly n segments on a calculator display (when '6' and '7' are represented using 6 resp. 3 segments).

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A038619 Smallest positive number that needs more lines when shown on a 7-segment display (digital clock) than any previous term.

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A143616 Record values in A010371.

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