A139337 -id:A139337 - cp's OEIS frontend

A108571 Any digit d in the sequence says: "I am part of an integer in which you'll find d digits d".

1, 22, 122, 212, 221, 333, 1333, 3133, 3313, 3331, 4444, 14444, 22333, 23233, 23323, 23332, 32233, 32323, 32332, 33223, 33232, 33322, 41444, 44144, 44414, 44441, 55555, 122333, 123233, 123323, 123332, 132233, 132323, 132332, 133223, 133232, 133322, 155555

Offset: 1

Eric Angelini, Jul 05 2005

The sequence is finite. Last term: 999999999888888887777777666666555554444333221.
Number of terms is 66712890763701234740813164553708284. - Zak Seidov, Jan 02 2007
Fixed points of A139337. - Reinhard Zumkeller, Apr 14 2008
Sequence contains squares (A181392) and cubes (A225886^3) but no higher powers, see Comments in A181392. - Giovanni Resta, May 19 2013

			23323 is in the sequence because it has two 2's and three 3's.
23332 is in the sequence because it has two 2's and three 3's.
23333 is not in the sequence because it has only one 2 and four 3's.

T. D. Noe, Table of n, a(n) for n=1..21056 (terms < 10^10)
Michael S. Branicky, Python program

Cf. A127007, A139337, A078348 (subsequence of primes), A181392, A225886.

is(n)={ vecmin(n=vecsort(digits(n))) && #n==normlp(Set(n),1) && !for(i=1,#n, n[i+n[i]-1]==n[i] || return; i+n[i]>#n || n[i+n[i]]>n[i] || return; n[i]>1 && i+=n[i]-1)} \\ M. F. Hasler, Sep 22 2014

# see link for a function that directly generates terms
def ok(n): s = str(n); return all(s.count(d) == int(d) for d in set(s))
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(155555)) # Michael S. Branicky, Jan 22 2021

A339023 Replace each digit d in the decimal representation of n with the digital root of n*d.

Original entry on oeis.org

0, 1, 4, 9, 7, 7, 9, 4, 1, 9, 10, 22, 36, 43, 52, 63, 76, 82, 99, 19, 40, 63, 88, 16, 36, 58, 73, 99, 28, 49, 90, 34, 61, 99, 31, 64, 99, 37, 67, 99, 70, 25, 63, 13, 55, 99, 46, 85, 36, 79, 70, 36, 85, 46, 99, 55, 13, 63, 25, 79, 90, 67, 37, 99, 64, 31, 99, 61

Offset: 0

Sebastian Karlsson, Jan 18 2021

			a(23) = 16 because 2*23 = 46 and 3*23 = 69 and the digital roots of 46 and 69 are 1 and 6.

Sebastian Karlsson, Table of n, a(n) for n = 0..10000
Sebastian Karlsson, The sequence generalized and plotted in arbitrary bases

Cf. A010888, A056992, A059729, A106649, A106746, A139337, A316347, A322131.

dr(n) = if(n, (n-1)%9+1); \\ A010888
a(n) = if (n==0, return(0)); my(d=digits(n), s=""); for (k=1, #d, s=concat(s, dr(n*d[k]))); eval(s); \\ Michel Marcus, Jan 18 2021

def digitalroot(n):
    return 0 if n == 0 else (n-1)%9 + 1
def a(n):
    return int(''.join([str(digitalroot(n*int(d))) for d in str(n)]))
for n in range(0, 68):
    print(a(n), end=', ')

a(9*n + 1) = 9*n + 1.

a(10*n) = 10*a(n). - Sebastian Karlsson, Feb 14 2021

A348179 Replace each decimal digit d of n with the digit that is d steps to the right of d. Interpret the digits of n as a cycle: one step to the right from the last digit is considered to be the first.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 22, 31, 44, 51, 66, 71, 88, 91, 20, 22, 22, 22, 24, 22, 26, 22, 28, 22, 0, 13, 22, 33, 44, 53, 66, 73, 88, 93, 40, 44, 42, 44, 44, 44, 46, 44, 48, 44, 0, 15, 22, 35, 44, 55, 66, 75, 88, 95, 60, 66, 62, 66, 64, 66, 66, 66, 68, 66, 0, 17, 22, 37, 44, 57, 66, 77, 88, 97, 80, 88, 82, 88, 84, 88, 86, 88, 88, 88, 0, 19, 22, 39, 44, 59, 66, 79, 88, 99, 0, 1

Offset: 0

Sebastian Karlsson, Oct 05 2021

First differs from A349422 at a(101). - Sebastian Karlsson, Dec 31 2021

			a(102345) = 004124 = 4124. For example, 4 gets replaced by 2 because moving 4 steps to the right gives: 4 -> 5 -> 1 -> 0 -> 2. Note that from 5 we went to the first digit of the number.

Sebastian Karlsson, Table of n, a(n) for n = 0..10000

Cf. A336668 (fixed points), A349422 (to the left), A349423 (index of first appearance of n).

Cf. A106649, A139337, A322131, A339023, A341767, A341953, A346576, A347323.

import Data.Char (digitToInt)
a n = let s = show n; l = length s in
  read [s !! (mod (i + digitToInt (s !! i)) l) | i <- [0..l-1]] :: Integer

Table[FromDigits@Table[v[[If[(p=Mod[k+v[[k]],t])==0,t,p]]],{k,t=Length[v=IntegerDigits[n]]}],{n,0,67}] (* Giorgos Kalogeropoulos, Oct 08 2021 *)

f(k, d) = d[(k+d[k]-1)%#d + 1];
a(n) = my(d=digits(n), dd=vector(#d, k, f(k, d))); fromdigits(dd); \\ Michel Marcus, Oct 07 2021

def a(n):
    s, l = str(n), len(str(n))
    return int("".join(s[(i + int(s[i])) % l] for i in range(l)))

a(68)-a(101) from Sebastian Karlsson, Dec 31 2021

A349422 Replace each decimal digit d of n with the digit that is d steps to the left of d. Interpret the digits of n as a cycle: one step to the left from the first digit is considered to be the last.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 22, 31, 44, 51, 66, 71, 88, 91, 20, 22, 22, 22, 24, 22, 26, 22, 28, 22, 0, 13, 22, 33, 44, 53, 66, 73, 88, 93, 40, 44, 42, 44, 44, 44, 46, 44, 48, 44, 0, 15, 22, 35, 44, 55, 66, 75, 88, 95, 60, 66, 62, 66, 64, 66, 66, 66, 68, 66, 0, 17, 22, 37, 44, 57, 66, 77, 88, 97, 80, 88, 82, 88, 84, 88, 86, 88, 88, 88, 0, 19, 22, 39, 44, 59, 66, 79, 88, 99, 0, 100

Offset: 0

Sebastian Karlsson, Nov 17 2021

First differs from A348179 at a(101).

			a(3210) = 2020, because:
Moving 3 steps to the left from 3 gives: 3 -> 0 -> 1 -> 2.
Moving 2 steps to the left from 2 gives: 2 -> 3 -> 0.
Moving 1 step to the left from 1 gives: 1 -> 2.
Moving 0 steps to left from 0 gives: 0.

Sebastian Karlsson, Table of n, a(n) for n = 0..10000

Cf. A336668 (fixed points), A348179 (to the right).

Cf. A139337, A346576, A347323.

import Data.Char (digitToInt)
a n = read [s !! mod (i - digitToInt (s !! i)) l | i <- [0..l-1]] :: Integer
    where s = show n; l = length s

a(n) = { my (d=digits(n)); fromdigits(vector(#d, k, d[1+(k-1-d[k])%#d])) } \\ Rémy Sigrist, Nov 17 2021

def a(n):
    s, l = str(n), len(str(n))
    return int("".join(s[(i - int(s[i])) % l] for i in range(l)))

A341767 Replace each digit d in the decimal representation of n with the digital root of n^d.

Original entry on oeis.org

1, 4, 9, 4, 2, 9, 7, 1, 9, 11, 22, 39, 41, 54, 69, 71, 88, 99, 11, 41, 93, 77, 78, 99, 44, 11, 99, 11, 48, 91, 14, 87, 99, 17, 88, 99, 11, 84, 99, 41, 45, 99, 71, 11, 99, 11, 72, 99, 41, 21, 96, 44, 88, 99, 11, 51, 99, 77, 28, 91, 17, 11, 99, 11, 15, 99, 14

Offset: 1

Sebastian Karlsson, Feb 19 2021

If n == 1 (mod 9), then every digit will be replaced by "1". If n == 0 (mod 9), then all nonzero digits will be replaced by "9".

The corresponding n of values a(n)= 1, a(n)= 11, a(n)= 111,... creates a subsequence of A236653. - Davide Rotondo, Mar 04 2024

			a(26) = 11, since 26^2 = 676 and 26^6 = 308915776. 6 + 7 + 6 = 19, 1 + 9 = 10 and 1 + 0 = 1, so the digital root of 676 is 1. 3 + 0 + 8 + 9 + 1 + 5 + 7 + 7 + 6 = 46, 4 + 6 = 10 and 1 + 0 = 1, so the digital root of 308915776 is 1. Thus, for 26, both "2" and "6" will be replaced by "1".

Sebastian Karlsson, Table of n, a(n) for n = 1..10000
Sebastian Karlsson, Generalized and plotted in arbitrary bases

Cf. A010888, A106649, A139337, A322131, A339023.

a[n_] := FromDigits[Mod[n^IntegerDigits[n] - 1, 9] + 1]; Array[a, 100] (* Amiram Eldar, Feb 19 2021 *)

dr(n) = if(n, (n-1)%9+1); \\ A010888
a(n) = my(d=digits(n)); fromdigits(vector(#d, k, dr(n^d[k]))); \\ Michel Marcus, Feb 19 2021

def a(n):
    return int(''.join(str((pow(n, int(d), 9)-1)%9 + 1) for d in str(n)))

a(10*n) = 10*a(n) + 1.

A341953 Replace each digit d in the decimal representation of n with the digital root of d^n.

Original entry on oeis.org

1, 4, 9, 4, 2, 9, 7, 1, 9, 10, 11, 11, 19, 17, 18, 19, 14, 11, 19, 40, 81, 77, 59, 11, 25, 49, 81, 71, 59, 90, 91, 94, 99, 94, 92, 99, 97, 91, 99, 40, 71, 11, 49, 77, 18, 49, 74, 11, 49, 70, 81, 47, 29, 11, 55, 79, 81, 41, 29, 90, 91, 94, 99, 94, 92, 99, 97

Offset: 1

Sebastian Karlsson, Feb 24 2021

The digits 0, 1, 3, 6 and 9 will always be replaced by the same digits: 0 -> 0, 1 -> 1, 3 -> 9, 6 -> 9 and 9 -> 9.

			a(14) = 17, since 1^14 = 1 and 4^14 = 268435456. 2 + 6 + 8 + 4 + 3 + 5 + 4 + 5 + 6 = 43 and 4 + 3 = 7. Thus, the digital root of 268435456 is 7. This means that for 14, "1" gets replaced by "1" and "4" gets replaced by "7".

Robert Israel, Table of n, a(n) for n = 1..10000
Sebastian Karlsson, Generalized and plotted in arbitrary bases

Cf. A010888, A106649, A139337, A322131, A339023, A341767.

digroot[n_] := If[n == 0, 0, Mod[n - 1, 9] + 1]; a[n_] := FromDigits[digroot /@ (IntegerDigits[n]^n)]; Array[a, 100] (* Amiram Eldar, Feb 24 2021 *)

r(n) = if(n, (n-1)%9+1) \\ A010888
a(n) = fromdigits(apply(x->r(x^n), digits(n))); \\ Michel Marcus, Mar 21 2021

def D(d, n):
    return 0 if d == 0 else (pow(d, n, 9)-1)%9 + 1
def a(n):
    return int(''.join(str(D(int(d), n)) for d in str(n)))

A354719 Replace each even digit in n with the digit to its left.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 11, 13, 11, 15, 11, 17, 11, 19, 2, 11, 22, 33, 42, 55, 62, 77, 82, 99, 33, 31, 33, 33, 33, 35, 33, 37, 33, 39, 4, 11, 24, 33, 44, 55, 64, 77, 84, 99, 55, 51, 55, 53, 55, 55, 55, 57, 55, 59, 6, 11, 26, 33, 46, 55, 66, 77

Offset: 0

Gavin Lupo, Jun 03 2022

If the leftmost digit in n is even, wrap around and replace it with the rightmost digit in n (see example).

			n    =  4 2 7 8 1
        ^ ^   ^      even digits,
a(n) =  1 4 7 7 1    to their left in n

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Gavin Lupo, Logarithmic scatterplot of (n, a(n)) for n = 1..10000

Cf. A139337, A348179, A349422.

Array[FromDigits@ Map[If[EvenQ@ #1, #2, #1] & @@ # &, Transpose@ {#, RotateRight[#, 1]}] &@ IntegerDigits[#] &, 67] (* Michael De Vlieger, Jun 19 2022 *)

prec(d, k) = k--; if (! k, k = #d); k;
a(n) = my(d=digits(n), v=d); for (k=1, #d, if (!(d[k] % 2), v[k] = d[prec(d,k)])); fromdigits(v); \\ Michel Marcus, Jun 04 2022

def a(n):
    digits = list(map(int, str(n)))
    out = (d if d%2 else digits[i-1] for i, d in enumerate(digits))
    return int("".join(map(str, out)))
print([a(n) for n in range(68)]) # Michael S. Branicky, Jun 04 2022

cp's OEIS Frontend

A108571 Any digit d in the sequence says: "I am part of an integer in which you'll find d digits d".

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A339023 Replace each digit d in the decimal representation of n with the digital root of n*d.

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A348179 Replace each decimal digit d of n with the digit that is d steps to the right of d. Interpret the digits of n as a cycle: one step to the right from the last digit is considered to be the first.

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A349422 Replace each decimal digit d of n with the digit that is d steps to the left of d. Interpret the digits of n as a cycle: one step to the left from the first digit is considered to be the last.

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A341767 Replace each digit d in the decimal representation of n with the digital root of n^d.

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A341953 Replace each digit d in the decimal representation of n with the digital root of d^n.

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A354719 Replace each even digit in n with the digit to its left.

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