A320486 -id:A320486 - cp's OEIS frontend

A321012 Trajectory of 596 under repeated application of the map k -> A320486(k^2).

596, 3216, 103425, 197325, 897162, 652, 2510, 631, 3986, 596, 3216, 103425, 197325, 897162, 652, 2510, 631, 3986, 596, 3216, 103425, 197325, 897162, 652, 2510, 631, 3986, 596, 3216, 103425, 197325, 897162, 652, 2510, 631, 3986, 596, 3216, 103425, 197325

Offset: 1

N. J. A. Sloane, Nov 04 2018

k -> A320486(k) is Eric Angelini's remove-repeated-digits map.
Lars Blomberg has discovered that if we start with any positive integer and repeatedly apply the map k -> A320486(k^2) then we will eventually either:
- reach 0,
- reach one of the four fixed points 1, 1465, 4376, 89476 (see A321010)
- reach the period-10 cycle shown in A321011, or
- reach the period-9 cycle shown in A321012.
Since there are only finitely many possible starting values with all digits distinct, it should not be difficult to check that this is true (and indeed, Lars Blomberg may by now have completed the proof).

			The cycle of length 9 is (596, 3216, 103425, 197325, 897162, 652, 2510, 631, 3986).

Eric Angelini, Postings to Sequence Fans Mailing List, Oct 24 2018 and Oct 26 2018.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A320485, A320486, A321010, A321011.

PadRight[{},80,{596,3216,103425,197325,897162,652,2510,631,3986}] (* Harvey P. Dale, Aug 08 2023 *)

Vec(x*(596 + 3216*x + 103425*x^2 + 197325*x^3 + 897162*x^4 + 652*x^5 + 2510*x^6 + 631*x^7 + 3986*x^8) / ((1 - x)*(1 + x + x^2)*(1 + x^3 + x^6)) + O(x^40)) \\ Colin Barker, Nov 04 2018

From Colin Barker, Nov 04 2018: (Start)
G.f.: x*(596 + 3216*x + 103425*x^2 + 197325*x^3 + 897162*x^4 + 652*x^5 + 2510*x^6 + 631*x^7 + 3986*x^8) / ((1 - x)*(1 + x + x^2)*(1 + x^3 + x^6)).
a(n) = a(n-9) for n>9.
(End)

A321008 a(1)=1; thereafter a(n) is obtained by applying Eric Angelini's remove-repeated-digits map, x->A320486(x), to n*a(n-1), stopping when 0 is reached.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 54, 432, 3, 30, 0

Offset: 1

N. J. A. Sloane, Nov 03 2018

			a(6)=720, so for a(7) we compute 7*720 = 5040 which becomes 54 = a(7).

Eric Angelini, Posting to Sequence Fans Mailing List, Oct 24 2018

N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A320485, A320486, A321009.

A321011 Trajectory of 86 under repeated application of the map k -> A320486(k^2).

Original entry on oeis.org

86, 7396, 547816, 12985, 805, 648025, 1325, 1762, 3106, 94723, 86, 7396, 547816, 12985, 805, 648025, 1325, 1762, 3106, 94723, 86, 7396, 547816, 12985, 805, 648025, 1325, 1762, 3106, 94723, 86, 7396, 547816, 12985, 805, 648025, 1325, 1762, 3106, 94723

Offset: 1

N. J. A. Sloane, Nov 04 2018

k -> A320486(k) is Eric Angelini's remove-repeated-digits map.
Lars Blomberg has discovered that if we start with any positive integer and repeatedly apply the map k -> A320486(k^2) then we will eventually either:
- reach 0,
- reach one of the four fixed points 1, 1465, 4376, 89476 (see A321010)
- reach the period-10 cycle shown in A321011, or
- reach the period-9 cycle shown in A321012.
Since there are only finitely many possible starting values with all digits distinct, it should not be difficult to check that this is true (and indeed, Lars Blomberg may by now have completed the proof).

			The cycle of length 10 is (86, 7396, 547816, 12985, 805, 648025, 1325, 1762, 3106, 94723).

Eric Angelini, Postings to Sequence Fans Mailing List, Oct 24 2018 and Oct 26 2018.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A320485, A320486, A321010, A321012.

LinearRecurrence[{0,0,0,0,0,0,0,0,0,1},{86,7396,547816,12985,805,648025,1325,1762,3106,94723},40] (* or *) PadRight[ {},40,{86,7396,547816,12985,805,648025,1325,1762,3106,94723}] (* Harvey P. Dale, Nov 05 2020 *)

Vec(x*(86 + 7396*x + 547816*x^2 + 12985*x^3 + 805*x^4 + 648025*x^5 + 1325*x^6 + 1762*x^7 + 3106*x^8 + 94723*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^40)) \\ Colin Barker, Nov 04 2018

From Colin Barker, Nov 04 2018: (Start)
G.f.: x*(86 + 7396*x + 547816*x^2 + 12985*x^3 + 805*x^4 + 648025*x^5 + 1325*x^6 + 1762*x^7 + 3106*x^8 + 94723*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-10) for n>10.
(End)

A321010 Numbers k such that f(k^2) = k, where f is Eric Angelini's remove-repeated-digits map x->A320486(x).

Original entry on oeis.org

0, 1, 1465, 4376, 89476

Offset: 1

N. J. A. Sloane, Nov 03 2018

Lars Blomberg has discovered that if we start with any positive integer and repeatedly apply the map m -> A320486(m^2) then we will eventually either:
- reach 0,
- reach one of the four fixed points 1, 1465, 4376, 89476 (this sequence),
- reach the period-10 cycle shown in A321011, or
- reach the period-9 cycle shown in A321012.
From Lars Blomberg, Nov 17 2018: (Start)
Verified by testing all possible 8877690 start values that these are the only fixed points and cycles.
Detailed counts are:
- 561354 reach 0,
- 963738 reach one of the four fixed points 1, 1465, 4376, 89476 (counts 946109, 434, 17065, 130),
- 7271337 reach the period-10 cycle, and
- 81261 reach the period-9 cycle. (End)

Eric Angelini, Postings to Sequence Fans Mailing List, Oct 24 2018 and Oct 26 2018.

N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A320485, A320486, A321011, A321012.

A321149 a(1) = 102735, a(n) = prime(n-1)*a(n-1) but products that are not in A010784 are first reduced as in A320486 (see comments); continue until zero is reached.

Original entry on oeis.org

102735, 2547, 7641, 38205, 267435, 2941785, 8405, 1425, 205, 4715, 1675, 192, 7104, 9164, 394052, 18520, 981560, 579124, 24, 1608, 468, 316, 296, 24568, 186, 18042, 184, 18952, 7864, 8516, 962308, 36

Offset: 1

Hans Havermann, Oct 28 2018

At each step, integers that contain duplicated digits are reduced to terms of A010784 by erasing all digits that appear more than once and bunching up the digits that remain. Leading zeros are ignored and any number that disappears entirely becomes 0. See A320486.

102735 is the smallest of 785 A010784 terms that result in a 362-term sequence, the longest possible.

			2 * 102735 = [205470] => 2547
3 * 2547 = 7641
5 * 7641 = 38205
7 * 38205 = 267435
11 * 267435 = 2941785
13 * 2941785 = [38243205] => 8405
17 * 8405 = [142885] => 1425
19 * 1425 = [27075] => 205
...
2417 * 40 = [96680] => 980
2423 * 980 = [2374540] => 23750
2437 * 23750 = [57878750] => 0

Hans Havermann, Table of n, a(n) for n = 1..362
Hans Havermann, What's so special about 102735?

Cf. A010784, A321148, A320486.

A321009 a(1)=2; thereafter a(n) is obtained by applying Eric Angelini's remove-repeated-digits map, x->A320486(x), to n*a(n-1), stopping when 0 is reached.

Original entry on oeis.org

2, 4, 12, 48, 240, 10, 70, 560, 54, 540, 5940, 71280, 9240, 129360, 19, 304, 5168, 93024, 145, 29, 609, 198, 0

Offset: 1

N. J. A. Sloane, Nov 03 2018

			a(5)=240, so for a(6) we compute 6*240 = 1440 which becomes 10 = a(6).

Eric Angelini, Posting to Sequence Fans Mailing List, Oct 24 2018

Cf. A320485, A320486, A321008.

A321148 a(1) = 24603, a(n) = n*a(n-1) but products that are not in A010784 are first reduced as in A320486 (see comments); continue until zero is reached.

Original entry on oeis.org

24603, 49206, 4768, 19072, 95360, 572160, 4512, 309, 2781, 27810, 3591, 43092, 5019, 702, 153, 28, 476, 56, 1064, 180, 3780, 83160, 92680, 430, 175, 40, 18, 504, 4, 120, 3720, 94, 3102, 105468, 69180, 298, 26, 9, 351, 1, 41, 17, 731, 32164, 17380, 7480, 3160, 5680, 7830, 3915, 15, 780, 130, 72, 3960, 1760, 132, 75, 25, 15, 915, 56730, 570, 36480, 371, 286, 962, 541, 729, 513, 642, 6, 438, 341, 27, 5, 385, 0

Offset: 1

Hans Havermann, Oct 28 2018

At each step, integers that contain duplicated digits are reduced to terms of A010784 by erasing all digits that appear more than once and bunching up the digits that remain. Leading zeros are ignored and any number that disappears entirely becomes 0. See A320486.

24603 is the smallest of 1746 A010784 terms that result in a 78-term sequence, the longest possible.

			2 * 24603 = 49206
3 * 49206 = [147618] => 4768
4 * 4768 = 19072
5 * 19072 = 95360
6 * 95360 = 572160
7 * 572160 = [4005120] => 4512
8 * 4512 = [36096] => 309
...
76 * 27 = [2052] => 5
77 * 5 = 385
78 * 385 = [30030] => 0

Hans Havermann, What's so special about 102735?

Cf. A010784, A321149, A320486.

A320487 a(0) = 1; thereafter a(n) is obtained by applying the "delete multiple digits" map m -> A320485(m) to 2*a(n-1).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 61, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 61, 1

Offset: 0

N. J. A. Sloane, Oct 24 2018, following a suggestion from Eric Angelini

In short, double the previous term and delete any digits appearing more than once.
Periodic with period 28.
Using the variant A320486 yields the same sequence, since the empty string never occurs. - M. F. Hasler, Oct 24 2018
Conjecture: If we start with any nonnegative integer and repeatedly double and apply the "delete multiple digits" map m -> A320485(m), we eventually reach 0 or 1 (see A323835). - N. J. A. Sloane, Feb 03 2019

			2*32768 = 65536 -> 3 since we delete the multiple digits 6 and 5.
2*61 = 122 -> 1 since we delete the multiple 2's.

Eric Angelini, Posting to Sequence Fans Mailing List, Oct 24 2018

N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A000079, A065243, A320485, A323830.

See A035615 for a classic related base-2 sequence.

a[0] = 1;a[n_] := a[n] = FromDigits[First /@ Select[ Tally[IntegerDigits[2 a[n - 1]]], #[[2]] == 1 &]];Table[a[n], {n, 0, 56}] (* Stan Wagon, Nov 17 2018 *)

A=[2];for(i=1,99,A=concat(A,A320486(A[#A]*2)));A \\ M. F. Hasler, Oct 24 2018

A320485 Keep just the digits of n that appear exactly once; write -1 if all digits disappear.

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, -1, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, -1, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, -1, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, -1, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, -1, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, -1, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, -1, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, -1, 1, 0, 102, 103, 104, 105, 106, 107, 108, 109, 0, -1, 2, 3, 4, 5, 6, 7, 8, 9, 120, 2, 1, 123, 124, 125, 126, 127, 128, 129, 130, 3

Offset: 0

N. J. A. Sloane, Oct 24 2018

Digits that appear more than once in n are erased. Leading zeros are erased unless the result is 0. If all digits are erased, we write -1 for the result.
The map n -> a(n) was invented by Eric Angelini and described in a posting to the Sequence Fans Mailing List on Oct 24 2018.
More than the usual number of terms are shown in order to reach some interesting examples.
a(n) = -1 mostly. - David A. Corneth, Oct 24 2018

			1231 becomes 23, 1123 becomes 23, 11231 becomes 23, and 11023 becomes 23 (as we don't accept leading zeros). Note that 112323 disappears immediately and we get -1.
101 and 110 become 0 while 11000 and 10001 become -1.

Eric Angelini, Posting to Sequence Fans Mailing List, Oct 24 2018

Robert Israel, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

See A320486 for another version.

Cf. A010784, A115853, A321008-A321012.

f:= proc(n) local F,S;
  F:= convert(n,base,10);
  S:= select(t -> numboccur(t,F)>1, [$0..9]);
  if S = {} then return n fi;
  F:= subs(seq(s=NULL,s=S),F);
  if F = [] then -1
  else add(F[i]*10^(i-1),i=1..nops(F))
  fi
end proc:
map(f, [$0..200]); # Robert Israel, Oct 24 2018

Array[If[# == {}, -1, FromDigits@ #] &@ Map[If[#[[-1]] > 1, -1, #[[1]] ] /. -1 -> Nothing &, Tally@ IntegerDigits[#]] &, 131] (* Michael De Vlieger, Oct 24 2018 *)

a(n) = {my(d=digits(n), v = vector(10), res = 0, t = 0); for(i=1, #d, v[d[i]+1]++); for(i=1, #d, if(v[d[i]+1]==1, t = 1; res=10 * res + d[i])); res - !t + !n} \\ David A. Corneth, Oct 24 2018

def A320485(n):
    return (lambda x: int(x) if x != '' else -1)(''.join(d if str(n).count(d) == 1 else '' for d in str(n))) # Chai Wah Wu, Nov 19 2018

From Rémy Sigrist, Oct 24 2018: (Start)
a(n) = n iff n belong to A010784.
a(n) <= 9876543210 with equality iff n = 9876543210.
(End)
If n > 9876543210, then a(n) < n. If a(n) < n, then a(n) <= 99n/1000. - Chai Wah Wu, Oct 24 2018

A321801 Delete all consecutive identical decimal digits of n; write 0 if all digits disappear.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 0, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 0, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 0, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 0, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 0, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 0, 1, 101, 102, 103, 104, 105, 106, 107, 108, 109, 0, 0, 2, 3, 4, 5, 6, 7, 8, 9, 120, 121, 1, 123, 124, 125, 126, 127, 128, 129, 130, 131

Offset: 0

Chai Wah Wu, Nov 19 2018

Consecutive identical digits of n are erased. Leading zeros are erased unless the result is 0. If all digits are erased, we write 0 for the result (A321802 is another version, which uses -1 for the empty string).

More than the usual number of terms are shown in order to reach some interesting examples. Agrees with A320486 for n < 101.

			12311 becomes 123, 1123 becomes 23, 11231 becomes 231, and 110232 becomes 232 (as we don't accept leading zeros). Note that 112233 disappears immediately and we get 0.
1110, 11000, 1100011 all become 0.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A320486, A321802.

A321801[n_]:=FromDigits[Flatten[Select[Split[IntegerDigits[n]],Length[#]==1&]]];Array[A321801,100,0] (* Paolo Xausa, Nov 14 2023 *)

A321801(n)={forstep(i=#n=digits(n),2,-1,n[i]!=n[i-1]&&next;if(i<3||n[i-2]!=n[i],n=n[^i];i--);n=n[^i]);fromdigits(n)} \\ M. F. Hasler, Nov 20 2018

from re import split
def A321801(n):
    return int('0'+''.join(d if len(d) == 1 else '' for d in split('(0+)|(1+)|(2+)|(3+)|(4+)|(5+)|(6+)|(7+)|(8+)|(9+)',str(n)) if d != '' and d != None))

cp's OEIS Frontend

A321012 Trajectory of 596 under repeated application of the map k -> A320486(k^2).

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A321008 a(1)=1; thereafter a(n) is obtained by applying Eric Angelini's remove-repeated-digits map, x->A320486(x), to n*a(n-1), stopping when 0 is reached.

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A321011 Trajectory of 86 under repeated application of the map k -> A320486(k^2).

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A321010 Numbers k such that f(k^2) = k, where f is Eric Angelini's remove-repeated-digits map x->A320486(x).

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A321149 a(1) = 102735, a(n) = prime(n-1)*a(n-1) but products that are not in A010784 are first reduced as in A320486 (see comments); continue until zero is reached.

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A321009 a(1)=2; thereafter a(n) is obtained by applying Eric Angelini's remove-repeated-digits map, x->A320486(x), to n*a(n-1), stopping when 0 is reached.

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A321148 a(1) = 24603, a(n) = n*a(n-1) but products that are not in A010784 are first reduced as in A320486 (see comments); continue until zero is reached.

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A320487 a(0) = 1; thereafter a(n) is obtained by applying the "delete multiple digits" map m -> A320485(m) to 2*a(n-1).

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Author

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Mathematica

PARI

A320485 Keep just the digits of n that appear exactly once; write -1 if all digits disappear.

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