A040163 -id:A040163 - cp's OEIS frontend

A040115 Concatenate absolute values of differences between adjacent digits of n.

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, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1

Offset: 0

Felice Russo

Let the decimal expansion of n be abcd...efg, say. Then a(n) has decimal expansion |a-b| |b-c| |c-d| ... |e-f| |f-g|. Leading zeros in a(n) are omitted.
From M. F. Hasler, Nov 09 2019: (Start)
This sequence coincides with A080465 up to a(109) but is thereafter completely different.
Eric Angelini calls a(n) the "ghost" of the number n and considers iterations of n -> n +- a(n) depending on parity of a(n), cf. A329200 and A329201. (End)

			a(371) = 46, for example.
a(110) = 01 = 1, while A080465(110) = 10 - 1 = 9. - _M. F. Hasler_, Nov 09 2019

T. D. Noe, Table of n, a(n) for n = 0..10000
E. Angelini, The Ghost Iteration, Personal blog "Cinquante signes", Nov 2019
E. Angelini, The Ghost Iteration, Personal blog "Cinquante signes", Nov 2019 [Cached copy, pdf file, with permission]

Cf. A010785, A037904, A040114, A040163, A040997.

Cf. A329200, A329201: iterations of n +- a(n).

Table[FromDigits[Abs[Differences[IntegerDigits[n]]]],{n,110}] (* Harvey P. Dale, Dec 16 2021 *)

apply( A040115(n)=fromdigits(abs((n=digits(n+!n))[^-1]-n[^1])), [10..199]) \\ Works for all n >= 0. - M. F. Hasler, Nov 09 2019

a(n) = 0 iff n is a repdigit >= 11 (A010785). - Bernard Schott, May 09 2022

Definition clarified by N. J. A. Sloane, Aug 19 2008
Name edited by M. F. Hasler, Nov 09 2019
Terms a(0) = a(1) = ... = a(9) = 0 prepended by Max Alekseyev, Jul 26 2024

A037904 Greatest digit of n - least digit of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

a(n) = A054055(n)-A054054(n); a(A010785(n)) = 0; for k>0: a(n) = a(n*10^k + A000030(n)) = a(n*10^k + A010879(n)) = a(n*10^k + A054054(n)) = a(n*10^k + A054055(n)) . - Reinhard Zumkeller, Dec 14 2007; corrected by David Wasserman, May 21 2008

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A040163, A064834.

a037904 = f 9 0 where
   f u v 0 = v - u
   f u v z = f (min u d) (max v d) z' where (z', d) = divMod z 10
-- Reinhard Zumkeller, Dec 16 2013

f:= n -> (max-min)(convert(n,base,10)):
map(f, [$1..1000]); # Robert Israel, Jul 07 2016

f[n_] := Block[{d = IntegerDigits[n]}, Max[d] - Min[d]]; Table[ f[n], {n, 1, 15}]

a(n)=my(d=digits(n)); vecmax(d)-vecmin(d) \\ Charles R Greathouse IV, Feb 07 2017

def A037904(n): return int(max(s:=str(n)))-int(min(s)) # Chai Wah Wu, Nov 10 2023

Incorrect comments deleted by Robert Israel, Jul 07 2016

A040997 Absolute value of first digit of n minus sum of other digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 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, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Offset: 1

Felice Russo

This is different from |A055017(n)| = |(x1 + x3 + ...) - (x2 + x4 + ...)|, where x1,...,xk are the digits of n. - M. F. Hasler, Nov 09 2019

			a(371) = |3-7-1| = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A037904, A040114, A040115, A040163, A055017.

Cf. A000030, A007953, A217657.

a040997 n = abs $ a000030 n - a007953 (a217657 n) -- Reinhard Zumkeller, Oct 10 2012

apply( A040997(n)={abs(vecsum(n=digits(n))-n[1]*2)}, [1..199]) \\ M. F. Hasler, Nov 09 2019

If decimal expansion of n is x1 x2 ... xk then a(n) = |x1-x2-x3- ... -xk|.

a(n) = abs(A000030(n) - A007953(A217657(n))). - Reinhard Zumkeller, Oct 10 2012

Name edited and incorrect formula deleted by M. F. Hasler, Nov 09 2019

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

A040114 List of absolute values of differences between digits of 10, 11, 12, ..., listed digit by digit.

Original entry on oeis.org

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, 1, 1, 2, 1, 3, 1, 4, 1, 5

Offset: 10

Felice Russo

Start with the empty sequence. For n = 10, 11, 12, ... do the following. Let the decimal expansion of n be abcd...efg, say. Append the numbers |a-b|, |b-c|, |c-d|, ... |e-f|, |f-g| to the sequence.

The offset is slightly misleading since for n > 99 the index n is in no direct relation with the number whose digits are used to produce a(n), in contrast to A040115 where all digit-differences of n are concatenated, and leading zeros don't appear. For example, a(100) = 1 and a(101) = 0 are the two differences between the digits of 100. Similarly, a(100 + 2k) corresponds to the difference between first and second digit of 100 + k. Therefore, a(120) = 0. - M. F. Hasler, Nov 09 2019

			From _M. F. Hasler_, Nov 09 2019: (Start)
The first term is the difference between digits of 10, which is 1.
The second term is the difference between digits of 11, which is 0.
The 100th term is the difference between the first two digits of 100, 1-0 = 1.
The 101st term is the difference between the last two digits of 100, 0-0 = 0.
The 120th term is the difference between the first two digits of 110, 1-1 = 0: Here "leading zeros" are preserved, in contrast to A040115 where all digit-wise differences of any n are concatenated to one term, and leading zeros disappear.
(End)
When we reach n = 371, for example, we append 4 and 6 to the sequence.

T. D. Noe, Table of n, a(n) for n = 10..1902

Cf. A037904, A040115, A040163, A040997.

Flatten[Table[Abs[Differences[IntegerDigits[n]]],{n,10,200}]] (* Harvey P. Dale, Jun 28 2021 *)

Definition clarified by N. J. A. Sloane, Aug 19 2008.

Name edited by M. F. Hasler, Nov 09 2019

A040164 |First digit - last digit| for n-th prime.

Original entry on oeis.org

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

Offset: 1

Felice Russo

Cf. A000040, A040163.

a(n) = my(pn = prime(n), digs = digits(pn)); abs(digs[1] - digs[#digs]); \\ Michel Marcus, Sep 27 2013

a(n) = A040163(A000040(n)). - Michel Marcus, Sep 27 2013

Extended (and corrected) by Patrick De Geest, Jun 15 1999

A038457 |First digit-last digit| for triangular numbers.

Original entry on oeis.org

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

Offset: 0

Felice Russo

Cf. A000217.

Abs[IntegerDigits[#][[1]]-IntegerDigits[#][[-1]]]&/@Accumulate[ Range[ 0,120]] (* Harvey P. Dale, Mar 22 2020 *)

a(n) = A040163(A000217(n)). - Michel Marcus, Sep 27 2013

cp's OEIS Frontend

A040115 Concatenate absolute values of differences between adjacent digits of n.

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A037904 Greatest digit of n - least digit of n.

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A040997 Absolute value of first digit of n minus sum of other digits of 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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A040114 List of absolute values of differences between digits of 10, 11, 12, ..., listed digit by digit.

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A040164 |First digit - last digit| for n-th prime.

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A038457 |First digit-last digit| for triangular numbers.

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