A137564 -id:A137564 - cp's OEIS frontend

A337857 a(n) is the smallest positive integer m with no repeated digits such that A137564(n + m) = n, or a(n) = 0 if no m exists.

10, 20, 30, 40, 50, 60, 70, 80, 90, 90, 0, 109, 120, 127, 136, 145, 154, 163, 172, 180, 190, 0, 209, 218, 230, 236, 245, 254, 263, 270, 280, 290, 0, 309, 318, 327, 340, 345, 354, 360, 370, 380, 390, 0, 409, 418, 427, 436, 450, 450, 460, 470, 480, 490, 0, 509, 518, 527, 536, 540

Offset: 1

Rodolfo Kurchan, Sep 26 2020

Terms computed by Claudio Meller.

We set a(n)=0 when n has repeated digits; for example, a(11) = 0, a(22) = 0, a(100) = 0, a(101) = 0, since compact(c) cannot result in such n. Is n=450 the first other number that has no solution?

Michel Marcus, Table of n, a(n) for n = 1..449

Cf. A010784, A137564.

f(n) = {my(d=digits(n)); fromdigits(vecextract(d, vecsort(vecsort(d, , 9))))}; \\ A137564
isokd(m) = my(d=digits(m)); #d == #Set(d); \\ A010784
a(n) = my(d=digits(n)); if (#Set(d) == #d, my(m=1); while (!isokd(m) || (f(n+m) != n), m++); m); \\ Michel Marcus, Jan 13 2022

def has_repeated_digits(n): s = str(n); return len(s) > len(set(s))
def A137564(n):
    seen, out, s = set(), "", str(n)
    for d in s:
        if d not in seen: out += d; seen.add(d)
    return int(out)
def a(n):
    if n == 0 or has_repeated_digits(n): return 0
    m = 1
    while has_repeated_digits(m) or A137564(n+m) != n: m += 1
    return m
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Jul 23 2022

A106612 a(n) = numerator(n/(n+11)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 3, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 4, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 5, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 6, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 0

N. J. A. Sloane, May 15 2005

In general, the numerators of n/(n+p) for prime p and n >= 0, form a sequence with the g.f.: x/(1-x)^2 - (p-1)*x^p/(1-x^p)^2. - Paul D. Hanna, Jul 27 2005

a(n) <> n iff n = 11 * k, in this case, a(n) = k. - Bernard Schott, Feb 19 2019

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A109052, A137564 (differs, e.g., for n=100).

Cf. Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106611 (k = 7 thru 10), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

List([0..80],n->NumeratorRat(n/(n+11))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+11)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

seq(numer(n/(n+11)),n=0..80); # Muniru A Asiru, Feb 19 2019

f[n_]:=Numerator[n/(n+11)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,-1},{0,1,2,3,4,5,6,7,8,9,10,1,12,13,14,15,16,17,18,19,20,21},80] (* Harvey P. Dale, Jul 05 2021 *)

vector(100, n, n--; numerator(n/(n+11))) \\ G. C. Greubel, Feb 19 2019

[lcm(n,11)/11 for n in range(0, 54)] # Zerinvary Lajos, Jun 09 2009

G.f.: x/(1-x)^2 - 10*x^11/(1-x^11)^2. - Paul D. Hanna, Jul 27 2005
a(n) = lcm(n,11)/11.
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109052(n)/11.
Dirichlet g.f.: zeta(s-1)*(1-10/11^s). (End)
a(n) = 2*a(n-11) - a(n-22). - G. C. Greubel, Feb 19 2019
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(11^e) = 11^(e-1), and a(p^e) = p^e if p != 11.
Sum_{k=1..n} a(k) ~ (111/242) * n^2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 21*log(2)/11. - Amiram Eldar, Sep 08 2023

A254143 Products of any two not necessarily distinct terms of A237424.

Original entry on oeis.org

1, 4, 7, 16, 28, 34, 37, 49, 67, 136, 148, 238, 259, 268, 334, 337, 367, 469, 667, 1156, 1258, 1336, 1348, 1369, 1468, 2278, 2338, 2359, 2479, 2569, 2668, 3334, 3337, 3367, 3667, 4489, 4669, 6667, 11356, 11458, 12358, 12469, 12478, 13336, 13348, 13468, 13579

Offset: 1

Reinhard Zumkeller, Jan 28 2015

nonn

Digits are in nondecreasing order for all terms in decimal representation;
a(396) = 1123456789 = 3367 * 333667 is the smallest term containing all nonzero decimal digits: A254323(396) = 123456789;
A254323(n) = A137564(a(n)).

			Initial terms of A237424: 1, 4, 7, 34, 37, 67, 334, 337, 367, 667, 3334 ...
.  n | a(n) = A237424(i) * A237424(j)
. ---+-------------------------------
.  1 |    1 = 1 * 1   = A237424(1)^2
.  2 |    4 = 1 * 4   = A237424(1) * A237424(2)
.  3 |    7 = 1 * 7   = A237424(1) * A237424(3)
.  4 |   16 = 4 * 4   = A237424(2)^2
.  5 |   28 = 4 * 7   = A237424(2) * A237424(3)
.  6 |   34 = 1 * 34  = A237424(1) * A237424(4)
.  7 |   37 = 4 * 37  = A237424(1) * A237424(5)
.  8 |   49 = 7 * 7   = A237424(3)^2
.  9 |   67 = 1 * 67  = A237424(1) * A237424(6)
. 10 |  136 = 4 * 34  = A237424(2) * A237424(4)
. 11 |  148 = 4 * 37  = A237424(2) * A237424(5)
. 12 |  238 = 7 * 34  = A237424(3) * A237424(4)
. 13 |  259 = 7 * 37  = A237424(3) * A237424(5)
. 14 |  268 = 4 * 67  = A237424(2) * A237424(6)
. 15 |  334 = 1 * 334 = A237424(1) * A237424(7)
. 16 |  337 = 1 * 337 = A237424(1) * A237424(8)
. 17 |  367 = 1 * 367 = A237424(1) * A237424(9)
. 18 |  469 = 7 * 67  = A237424(3) * A237424(6)
. 19 |  667 = 1 * 34  = A237424(1) * A237424(10)
. 20 | 1156 = 34 * 34 = A237424(4)^2
see link for more.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Reinhard Zumkeller, First 10000 products of any two terms of A237424

Subsequence of A009994.

Cf. A237424, A254323, A137564, A254338 (initial digits), A254339 (final digits).

import Data.Set (empty, fromList, deleteFindMin, union)
import qualified Data.Set as Set (null)
a254143 n = a254143_list !! (n-1)
a254143_list = f a237424_list [] empty where
   f xs'@(x:xs) zs s
     | Set.null s || x < y = f xs zs' (union s $ fromList $ map (* x) zs')
     | otherwise           = y : f xs' zs s'
     where zs' = x : zs
           (y, s') = deleteFindMin s

listA237424(lim)=my(v=List(),a,t); while(1, for(b=0,a, t=(10^a+10^b+1)/3; if(t>lim, return(Set(v))); listput(v, t)); a++)
list(lim)=my(v=List(),u=listA237424(lim),t); for(i=1,#u, for(j=1,i, t=u[i]*u[j]; if(t>lim,break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, May 13 2015

A254323 Remove in decimal representation of A254143(n) all repeated digits.

Original entry on oeis.org

1, 4, 7, 16, 28, 34, 37, 49, 67, 136, 148, 238, 259, 268, 34, 37, 367, 469, 67, 156, 1258, 136, 1348, 1369, 1468, 278, 238, 2359, 2479, 2569, 268, 34, 37, 367, 367, 489, 469, 67, 1356, 1458, 12358, 12469, 12478, 136, 1348, 13468, 13579, 1468, 2378, 2579

Offset: 1

Reinhard Zumkeller, Jan 28 2015

a(n) <= 123456789 for all n, and a(n) < 123456789 for n < 396;
a(396) = 123456789 = A050289(1);
a(n) = A137564(A254143(n)) = A004086(A227362(A254143(n))).

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

Cf. A254143, A137564, A227362, A004086, A050289.

Cf. A254338 (initial digits), A254339 (final digits).

Haskell
```
a254323 = a137564 . a254143
```

A337864 a(n) is the number formed by removing from n each digit if it is a duplicate of the previous digit, from left to right.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 3, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 4, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 5, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 6, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 0

Rodolfo Kurchan, Sep 27 2020

Please see discussion in A337857.
Similar to A137564 from which first differs at a(101) = 101 here, there a(101) = 10.
Differs from A106612 starting at n=100. - R. J. Mathar, Oct 08 2020

			a(100) = 10. Note that the second zero from the index n = 100 has been removed.
a(101) = 101.
a(1211323171) = 121323171. Note that the third "1" from the index n has been removed.

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A043096 (fixed points a(n)=n).
Cf. A137564, A337857.
Cf. A090079 (in binary).

a(n) = if(n < 10, return(n)); if(n%10 == (n\10)%10, return(a(n\10)), return(a(n\10)*10+n%10)) \\ David A. Corneth, Jul 23 2022

sub a {my($n)=@; $n =~ s/(.)\1+/$1/g; $n} # _Kevin Ryde, Oct 04 2020

from itertools import groupby
def a(n): return int("".join(k for k, g in groupby(str(n))))
print([a(n) for n in range(75)]) # Michael S. Branicky, Jul 23 2022

A370748 a(0)=1; a(n) = sum of all previous terms, eliminating repeated digits (starting from the left).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 653, 6189, 7238, 7961, 875, 8452, 9604, 10658, 176, 17342, 13468, 14852, 16304, 179308, 35861, 3947, 39842, 43826, 48209, 5301, 53602, 589204, 17840, 196248, 139246, 153742, 16854

Offset: 0

Sergio Pimentel, Mar 07 2024

Duplicate digits are eliminated as in A137564.
No term can be greater than 9876543210.
a(260390084) = 9876543210. - Michael S. Branicky, Apr 09 2024

			a(17) = 653 since the sum of a(0) through a(16) is 65536, and eliminating repeated digits gives a(17) = A137564(65536) = 653.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A010784, A137564, A370254.

Cf. A371863 (records), A371864 (indices of records).

seq={1};Do[s=Total[seq];AppendTo[seq,FromDigits[DeleteDuplicates[IntegerDigits[s]]]],43];seq (* James C. McMahon, Apr 09 2024 *)

f(n) = my(d=digits(n)); fromdigits(vecextract(d, vecsort(vecsort(d, , 9)))) \\ A137564
lista(nn) = my(va=vector(nn)); va[1] = 1; for (n=2, nn, va[n] = f(sum(j=1, n-1, va[j]));); va; \\ Michel Marcus, Mar 08 2024

from itertools import islice
def A137564(n):
    seen, out, s = set(), "", str(n)
    for d in s:
        if d not in seen: out += d; seen.add(d)
    return int(out)
def A370748gen(): # generator of terms
    an, s = 1, 0
    while True:
        yield an
        s += an
        an = A137564(s)
print(list(islice(A370748gen(), 45))) # Michael S. Branicky, Apr 09 2024

A373696 a(n) is the least m >= 0 with the same number of digits as n such that for some permutation p of 0..9, applying p to the digits of n yields the digits of m.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10

Offset: 0

Rémy Sigrist, Aug 04 2024

Leading zeros are ignored.
For n > 0, a(n) is the least m > 0 such that A358497(n) = A358497(m).
All positive terms belong to A266946.

			For n = 65507668: the different digits appearing in 65507668 are 6, 5, 0, 7 and 8; so we replace 6's by 1's, 5's by 0's, 0's by 2's, 7's by 3's and 8's by 4's, and a(65507668) = 10023114.

Cf. A137564, A266946, A358497, A373712 (ternary analog).

a(n, base = 10) = { my (d = digits(n, base), m = vector(base, i, -1), u = 1); for (i = 1, #d, if (m[1+d[i]] < 0, m[1+d[i]] = u; u = if (u==1, 0, u==0, 2, u+1);); d[i] = m[1+d[i]];); fromdigits(d, base); }

a(n) <= n with equality iff n = 0 or n belongs to A266946.

a(a(n)) = a(n).

cp's OEIS Frontend

A337857 a(n) is the smallest positive integer m with no repeated digits such that A137564(n + m) = n, or a(n) = 0 if no m exists.

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A106612 a(n) = numerator(n/(n+11)).

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A254143 Products of any two not necessarily distinct terms of A237424.

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A254323 Remove in decimal representation of A254143(n) all repeated digits.

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A337864 a(n) is the number formed by removing from n each digit if it is a duplicate of the previous digit, from left to right.

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A370748 a(0)=1; a(n) = sum of all previous terms, eliminating repeated digits (starting from the left).

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A373696 a(n) is the least m >= 0 with the same number of digits as n such that for some permutation p of 0..9, applying p to the digits of n yields the digits of m.

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