A031298 -id:A031298 - cp's OEIS frontend

A031007 Triangle T(n,k): Write n in base 7, reverse order of digits, to get row n.

0, 1, 2, 3, 4, 5, 6, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 6, 4, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 6, 5, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6, 6

Offset: 0

Clark Kimberling

Cf. A030308, A030341, A030386, A031235, A030567, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Flatten[Table[Reverse[IntegerDigits[n,7]],{n,0,50}]] (* Harvey P. Dale, Feb 25 2014 *)

A031007(n, k=-1)={k<0&&error("Flattened sequence not yet implemented.");n\7^k%7} \\ Assuming that columns start with k=0 as in A030308, A030341, ... TO DO: implement flattened sequence, such that A030567(n)=a(n). - M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

A064834 If n (in base 10) is d_1 d_2 ... d_k then a(n) = Sum_{i = 1..[k/2] } |d_i - d_{k-i+1}|.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 0, 1, 2, 3

Offset: 0

N. J. A. Sloane, Oct 25 2001

Might be called the Palindromic Deviation (or PD(n)) of n, since it measures how far n is from being a palindrome. - W. W. Kokko, Mar 13 2013

a(A002113(n)) = 0; a(A029742(n)) > 0; A136522(n) = A000007(a(n)). - Reinhard Zumkeller, Sep 18 2013

			a(456) = | 4 - 6 | = 2, a(4567) = | 4 - 7 | + | 5 - 6 | = 4.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Index entries for sequences related to palindromes

Cf. A037904, A040163, A064844.

Cf. A031298, A037888.

a064834 n = sum $ take (length nds `div` 2) $
                  map abs $ zipWith (-) nds $ reverse nds
   where nds = a031298_row n
-- Reinhard Zumkeller, Sep 18 2013

f:=proc(n)
local t1,t2,i;
t1:=convert(n,base,10);
t2:=nops(t1);
add( abs(t1[i]-t1[t2+1-i]),i=1..floor(t2/2) );
end;
[seq(f(n),n=0..120)]; # N. J. A. Sloane, Mar 24 2013

f[n_] := (k = IntegerDigits[n]; l = Length[k]; Sum[ Abs[ k[[i]] - k[[l - i + 1]]], {i, 1, Floor[l/2] } ] ); Table[ f[n], {n, 0, 100} ]

from sympy import floor, ceiling
def A064834(n):
    x, y = str(n), 0
    lx2 = len(x)/2
    for a,b in zip(x[:floor(lx2)],x[:ceiling(lx2)-1:-1]):
        y += abs(int(a)-int(b))
    return y
# Chai Wah Wu, Aug 09 2014

More terms from Vladeta Jovovic, Matthew Conroy and Robert G. Wilson v, Oct 26 2001

A076489 Number of common (distinct) digits of consecutive natural numbers.

Original entry on oeis.org

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

Offset: 0

Labos Elemer, Oct 21 2002

a(A226637(n)) = 0. Reinhard Zumkeller, Sep 01 2013

This is the prefix overlap between the decimal expansions of n and n+1 (cf. A238845). - N. J. A. Sloane, Mar 22 2014

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

Cf. A001477, A031298, A076490, A238845, A239092 (partial sums).

import Data.List (intersect, nub)
a076489 n = a076489_list !! n
a076489_list = map (length . nub) $
               zipWith intersect (tail a031298_tabf) a031298_tabf
-- Reinhard Zumkeller, Sep 01 2013

Table[Length[Intersection[IntegerDigits[w], IntegerDigits[w+1]]], {w, 0, 200}]

Initial zero prepended and offset adjusted by Reinhard Zumkeller, Sep 01 2013

A068505 Decimal representation of n interpreted in base b+1, where b=A054055(n) is the largest digit in decimal representation of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 6, 7, 8, 11, 14, 17, 20, 23, 26, 29, 12, 13, 14, 15, 19, 23, 27, 31, 35, 39, 20, 21, 22, 23, 24, 29, 34, 39, 44, 49, 30, 31, 32, 33, 34, 35, 41, 47, 53, 59, 42, 43, 44, 45, 46, 47, 48, 55, 62, 69, 56, 57, 58, 59, 60, 61

Offset: 1

Reinhard Zumkeller, Mar 11 2002, Feb 23 2008

a(n) = n iff n < 10 OR n is a "9ish number": a(A011539(n)) = A011539(n). - Reinhard Zumkeller, Dec 29 2011

			a(20)=2*3^1+0*1=6, a(21)=2*3^1+1*1=7, a(22)=2*3^1+2*1=8,
a(23)=2*4^1+3*1=11, a(24)=2*5^1+4*1=14, a(25)=2*6^1+5*1=17,
a(26)=2*7^1+6*1=20, a(27)=2*8^1+7*1=23, a(28)=2*9^1+8*1=26,
a(29)=2*10^1+9*1=29, a(30)=3*4^1+0*1=12, a(31)=3*4^1+1*1=13.

R. Zumkeller, Table of n, a(n) for n = 1..11000

Cf. A031298.

a068505 n = foldr (\d v -> v * b + d) 0 dds where
b = maximum dds + 1
dds = a031298_row n
-- Reinhard Zumkeller, Feb 17 2013, Dec 29 2011

f:= proc(n) local b,L,i;
L:= convert(n,base,10);
b:= max(L);
add(L[i]*(b+1)^(i-1),i=1..nops(L));
end proc:
map(f, [$1..100]); # Robert Israel, Feb 02 2016

a[n_] := (id = IntegerDigits[n] // Reverse; b = Max[id]+1; id.b^Range[0, Length[id]-1]); Table[a[n], {n, 1, 75}] (* Jean-François Alcover, May 15 2013 *)
Table[FromDigits[IntegerDigits[n],Max[IntegerDigits[n]+1]],{n,80}] (* Harvey P. Dale, Dec 02 2015 *)

a(n)=my(d = digits(n), b = vecmax(d)); subst(Pol(d), x, b+1); \\ Michel Marcus, Feb 12 2016

Definition clarified and comment corrected by Martin Büttner, Feb 02 2016

A133048 Powerback(n): reverse the decimal expansion of n, drop any leading zeros, then apply the powertrain map of A133500 to the resulting number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 1, 4, 9, 16, 25, 36, 49, 64, 81, 3, 1, 8, 27, 64, 125, 216, 343, 512, 729, 4, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 5, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 6, 1, 64, 729, 4096, 15625, 46656, 117649

Offset: 0

J. H. Conway and N. J. A. Sloane, Dec 31 2007

a(A221221(n)) = A133500(A221221(n)) = A222493(n). - Reinhard Zumkeller, May 27 2013

			E.g. 240 -> (0)42 -> 4^2 = 16; 12345 -> 54321 -> 5^4*3^2*1 = 5625.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Cf. A131571 (fixed points), A133059 and A133134 (records); A133500 (powertrain).
Cf. A133144 (length of trajectory), A031346 and A003001 (persistence).
Cf. A031298.

a133048 0 = 0
a133048 n = train $ dropWhile (== 0) $ a031298_row n where
   train []       = 1
   train [x]      = x
   train (u:v:ws) = u ^ v * (train ws)
-- Reinhard Zumkeller, May 27 2013

powerback:=proc(n) local a,i,j,t1,t2,t3;
if n = 0 then RETURN(0); fi;
t1:=convert(n, base, 10); t2:=nops(t1);
for i from 1 to t2 do if t1[i] > 0 then break; fi; od:
a:=1; t3:=t2-i+1;
for j from 0 to floor(t3/2)-1 do a := a*t1[i+2*j]^t1[i+2*j+1]; od:
if t3 mod 2 = 1 then a:=a*t1[t2]; fi;
RETURN(a); end;

ptm[n_]:=Module[{idn=IntegerDigits[IntegerReverse[n]]},If[ EvenQ[ Length[idn]],Times@@ (#[[1]]^#[[2]]&/@Partition[idn,2]),(Times@@(#[[1]]^#[[2]]&/@Partition[ Most[ idn],2]))Last[idn]]];Array[ptm,70,0] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 05 2020 *)

A218978 Table read by rows: n-th row lists all distinct substrings of decimal representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 10, 1, 11, 1, 2, 12, 1, 3, 13, 1, 4, 14, 1, 5, 15, 1, 6, 16, 1, 7, 17, 1, 8, 18, 1, 9, 19, 0, 2, 20, 1, 2, 21, 2, 22, 2, 3, 23, 2, 4, 24, 2, 5, 25, 2, 6, 26, 2, 7, 27, 2, 8, 28, 2, 9, 29, 0, 3, 30, 1, 3, 31, 2, 3, 32, 3

Offset: 0

Reinhard Zumkeller, May 02 2015, Nov 10 2012

A120004(n) = length of n-th row;

A154771(n) = sum of n-th row.

			Rows 100 .. 112:
.  100:  {0, 1, 10, 100},
.  101:  {0, 1, 10, 101},
.  102:  {0, 1, 2, 10, 102},
.  103:  {0, 1, 3, 10, 103},
.  104:  {0, 1, 4, 10, 104},
.  105:  {0, 1, 5, 10, 105},
.  106:  {0, 1, 6, 10, 106},
.  107:  {0, 1, 7, 10, 107},
.  108:  {0, 1, 8, 10, 108},
.  109:  {0, 1, 9, 10, 109},
.  110:  {0, 1, 10 ,11, 110},
.  111:  {1, 11, 111},
.  112:  {1, 2, 11, 12, 112}.

Reinhard Zumkeller, Rows n=0..1000 of triangle, flattened

Cf. A031298, A219031 (squares in row), A262188 (palindromes in row).

import Data.List (inits, tails, sort, nub, genericIndex)
a218978 n k = a218978_row n !! k
a218978_row n = genericIndex a218978_tabf n
a218978_tabf = map (sort . nub . map (foldr (\d v -> 10 * v + d) 0) .
                   concatMap (tail . inits) . tails) a031298_tabf
-- Reinhard Zumkeller, corrected: Sep 15 2015, May 02 2015, Nov 10 2012

A167831 Largest m<=n such that no carry occurs when adding m to n in decimal arithmetic.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 10, 11, 12, 13, 14, 14, 13, 12, 11, 10, 20, 21, 22, 23, 24, 24, 23, 22, 21, 20, 30, 31, 32, 33, 34, 34, 33, 32, 31, 30, 40, 41, 42, 43, 44, 44, 43, 42, 41, 40, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27

Offset: 0

Reinhard Zumkeller, Nov 14 2009

A167832(n) = a(n) + n.

R. Zumkeller, Table of n, a(n) for n = 0..9999

Cf. A167877, A035327 for the ternary and binary cases.

Cf. A031298.

a167831 n = head [x | let ds = a031298_row n, x <- [n, n-1 ..],
                      all (< 10) $ zipWith (+) ds (a031298_row x)]
-- Reinhard Zumkeller, Mar 15 2014

A262188 Table read by rows: row n contains all distinct palindromes contained as substrings in decimal representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 11, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 0, 2, 1, 2, 2, 22, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 0, 3, 1, 3, 2, 3, 3, 33, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 0, 4, 1, 4, 2, 4, 3, 4, 4, 44, 4, 5, 4, 6, 4

Offset: 0

Reinhard Zumkeller, Sep 14 2015

Length of row n = A262190(n);
T(n,0) = A054054(n);
T(n,A262190(n)-1) = A047813(n).

			.     n |  T(n,*)           n |  T(n,*)              n |  T(n,*)
.  -----+-----------    ------+-------------    -------+--------------
.   100 |  0,1           1000 |  0,1             10000 |  0,1
.   101 |  0,1,101       1001 |  0,1,1001        10001 |  0,1,10001
.   102 |  0,1,2         1002 |  0,1,2           10002 |  0,1,2
.   103 |  0,1,3         1003 |  0,1,3           10003 |  0,1,3
.   104 |  0,1,4         1004 |  0,1,4           10004 |  0,1,4
.   105 |  0,1,5         1005 |  0,1,5           10005 |  0,1,5
.   106 |  0,1,6         1006 |  0,1,6           10006 |  0,1,6
.   107 |  0,1,7         1007 |  0,1,7           10007 |  0,1,7
.   108 |  0,1,8         1008 |  0,1,8           10008 |  0,1,8
.   109 |  0,1,9         1009 |  0,1,9           10009 |  0,1,9
.   110 |  0,1,11        1010 |  0,1,101         10010 |  0,1,1001
.   111 |  1,11,111      1011 |  0,1,11,101      10011 |  0,1,11,1001
.   112 |  1,2,11        1012 |  0,1,2,101       10012 |  0,1,2,1001
.   113 |  1,3,11        1013 |  0,1,3,101       10013 |  0,1,3,1001
.   114 |  1,4,11        1014 |  0,1,4,101       10014 |  0,1,4,1001
.   115 |  1,5,11        1015 |  0,1,5,101       10015 |  0,1,5,1001
.   116 |  1,6,11        1016 |  0,1,6,101       10016 |  0,1,6,1001
.   117 |  1,7,11        1017 |  0,1,7,101       10017 |  0,1,7,1001
.   118 |  1,8,11        1018 |  0,1,8,101       10018 |  0,1,8,1001
.   119 |  1,9,11        1019 |  0,1,9,101       10019 |  0,1,9,1001
.   120 |  0,1,2         1020 |  0,1,2           10020 |  0,1,2
.   121 |  1,2,121       1021 |  0,1,2           10021 |  0,1,2
.   122 |  1,2,22        1022 |  0,1,2,22        10022 |  0,1,2,22
.   123 |  1,2,3         1023 |  0,1,2,3         10023 |  0,1,2,3
.   124 |  1,2,4         1024 |  0,1,2,4         10024 |  0,1,2,4
.   125 |  1,2,5         1025 |  0,1,2,5         10025 |  0,1,2,5  .

Reinhard Zumkeller, Rows n = 0..10000 of triangle, flattened
Index entries for sequences related to palindromes

Cf. A262190 (row lengths), A054054 (left edge), A047813 (right edge), A136522, A002113.

Cf. A031298, A218978.

import Data.List (inits, tails, nub, sort)
a262188 n k = a262188_tabf !! n !! k
a262188_row n = a262188_tabf !! n
a262188_tabf = map (sort . nub . map (foldr (\d v -> 10 * v + d) 0) .
   filter (\xs -> length xs == 1 || last xs > 0 && reverse xs == xs) .
          concatMap (tail . inits) . tails) a031298_tabf

A262188_row(n,b=10)=Set(concat(vector(logint(n+!n,b)+1,m,m=n\=b^(m>1);select(is_A002113,vector(logint(m+!m,b)+1,k,m%b^k))))) \\ M. F. Hasler, Jun 19 2018

A119246 Numbers containing in decimal representation their digital root.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 29, 30, 39, 40, 49, 50, 59, 60, 69, 70, 79, 80, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 109, 118, 127, 128, 136, 138, 145, 148, 154, 158, 163, 168, 172, 178, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 198, 199, 200

Offset: 1

Reinhard Zumkeller, May 10 2006

Complement of A119247.

For terms u: all digital permutations of u form terms; u*10 and all insertions of 0 are terms; if v is another term, then the concatenations uv, vu are also terms, as well as all insertions of v in u; these properties allow the construction of all terms beginning with {d:1<=d<=9}. - Reinhard Zumkeller, May 19 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

Cf. A052018, A031298, A138166.

a119246 n = a119246_list !! (n-1)
a119246_list =
    filter (\x -> a010888 x `elem` a031298_row (fromInteger x)) [0..]
-- Reinhard Zumkeller, Dec 16 2013, Apr 14 2011

d[n_] := IntegerDigits[n]; Select[Range[0, 200], MemberQ[d[#1], NestWhile[Total[d[#]] &, #1, # > 9 &]] &] (* Jayanta Basu, Jul 13 2013 *)

A227876 Write the decimal digits of n and take successive absolute differences; sequence is the sum of all digits at each level of the pyramid.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 2, 4, 6, 8, 10, 12, 14, 16, 18, 4, 4, 4, 6, 8, 10, 12, 14, 16, 18, 6, 6, 6, 6, 8, 10, 12, 14, 16, 18, 8, 8, 8, 8, 8, 10, 12, 14, 16, 18, 10, 10, 10, 10, 10, 10, 12, 14, 16, 18, 12, 12, 12, 12, 12, 12, 12, 14, 16, 18, 14, 14, 14, 14, 14, 14, 14, 14, 16, 18, 16, 16, 16, 16, 16, 16, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 3, 4, 7, 10, 13, 16, 19, 22, 25, 28, 4, 3, 6, 9, 12, 15, 18, 21, 24, 27, 7, 6, 7, 8, 11, 14, 17, 20, 23, 26

Offset: 0

Filipi R. de Oliveira, Oct 25 2013

A given nonnegative integer n is decomposed into its digits and the absolute differences between the digits are taken, then the differences between differences between digits (and so on, until the top of the difference pyramid is reached). The sum of the resulting digits is a(n).

			a(364)=19
.
____1____
__3_:_2_ --> 3+6+4+|3-6|+|6-4|+||3-6|-|6-4||=3+6+4+3+2+1=19
3_:_6_:_4

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A031298.

a227876 n = fst $ until (null . snd) h (0, a031298_row n) where
            h (s, ds) = (s + sum ds, map abs $ zipWith (-) ds $ tail ds)
-- Reinhard Zumkeller, Apr 28 2014

Join[{0},Table[Total[Abs[Flatten[NestList[Differences[Abs[#]]&, IntegerDigits[n], IntegerLength[n]-1]]]],{n,130}]] (* Harvey P. Dale, Mar 02 2015 *)

a(n)=my(d=digits(n),s); while(#d, s+=sum(i=1,#d,d[i]); d=vector(#d-1,i,abs(d[i+1]-d[i]))); s \\ Charles R Greathouse IV, Oct 25 2013

a(n)=n, if 0<=n<=9;
a(n)=n-9*floor(n/10)+|-n+11*floor(n/10)|, if 10<=n<=99;
a(n)=n-9*floor(n/10)-9*floor(n/100)+|-floor(n/10)+11*floor(n/100)|+|-n+11*floor(n/10)-10*floor(n/100)|+||-floor(n/10)+11*floor(n/100)|-|-n+11*floor(n/10)-10*floor(n/100)||, if 100<=n<=999.

cp's OEIS Frontend

A031007 Triangle T(n,k): Write n in base 7, reverse order of digits, to get row n.

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A064834 If n (in base 10) is d_1 d_2 ... d_k then a(n) = Sum_{i = 1..[k/2] } |d_i - d_{k-i+1}|.

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A076489 Number of common (distinct) digits of consecutive natural numbers.

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A068505 Decimal representation of n interpreted in base b+1, where b=A054055(n) is the largest digit in decimal representation of n.

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A133048 Powerback(n): reverse the decimal expansion of n, drop any leading zeros, then apply the powertrain map of A133500 to the resulting number.

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A218978 Table read by rows: n-th row lists all distinct substrings of decimal representation of n.

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A167831 Largest m<=n such that no carry occurs when adding m to n in decimal arithmetic.

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