A063850 -id:A063850 - cp's OEIS frontend

A005151 Summarize the previous term (digits in increasing order), starting with a(1) = 1.

1, 11, 21, 1112, 3112, 211213, 312213, 212223, 114213, 31121314, 41122314, 31221324, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314

Offset: 1

N. J. A. Sloane

a(n) = 21322314 for n > 12. - Reinhard Zumkeller, Jan 25 2014

The digits of each term a(n) are a permutation of those of the corresponding term A063850(n). - Chayim Lowen, Jul 16 2015

			The term after 312213 is obtained by saying "Two 1's, two 2's, two 3's", which gives 21-22-23, i.e., 212223.

C. Fleenor, "A litteral sequence", Solution to Problem 2562, Journal of Recreational Mathematics, vol. 31 No. 4 pp. 307 2002-3 Baywood NY.
Problem in J. Recreational Math., 30 (4) (1999-2000), p. 309.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Colin Barker, Table of n, a(n) for n = 1..1000
V. Bronstein and A. S. Fraenkel, On a curious property of counting sequences, Amer. Math. Monthly, 101 (1994), 560-563.
Onno M. Cain and Sela T. Enin, Inventory Loops (i.e. Counting Sequences) have Pre-period 2 max S_1 + 60, arXiv:2004.00209 [math.NT], 2020.
X. Gourdon and B. Salvy, Effective asymptotics of linear recurrences with rational coefficients, Discrete Mathematics, vol. 153, no. 1-3, 1996, pages 145-163.
James Henle, Is (some) mathematics poetry?, Journal of Humanistic Mathematics 1:1 (2011), pp. 94-100.
Madras Math's Amazing Number Facts, Fact No. 13
Madras Math, Descriptive Number
Trevor Scheopner, The Cyclic Nature (and Other Intriguing Properties) of Descriptive Numbers, Princeton Undergraduate Mathematics Journal, Issue 1, Article 4.
L. J. Upton, Letter to N. J. A. Sloane, Jan 8 1991.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A005150, A047842. See A083671 for another version.

Cf. A023989, A118628, A060857.

import Data.List (group, sort, transpose)
a005151 n = a005151_list !! (n-1)
a005151_list = 1 : f [1] :: [Integer] where
   f xs = (read $ concatMap show ys) : f ys where
          ys = concat $ transpose [map length zss, map head zss]
          zss = group $ sort xs
-- Reinhard Zumkeller, Jan 25 2014

RunLengthEncode[x_List] := (Through[{Length, First}[ #1]] &) /@ Split[ Sort[x]]; LookAndSay[n_, d_:1] := NestList[ Flatten[ RunLengthEncode[ # ]] &, {d}, n - 1]; F[n_] := LookAndSay[n, 1][[n]]; Table[ FromDigits[ F[n]], {n, 25}] (* Robert G. Wilson v, Jan 22 2004 *)
a[1] = 1; a[n_] := a[n] = FromDigits[Reverse /@ Sort[Tally[a[n-1] // IntegerDigits], #1[[1]] < #2[[1]]&] // Flatten]; Array[a, 26] (* Jean-François Alcover, Jan 25 2016 *)

say(n) = {digs = digits(n); d = vecsort(digs,,8); s = ""; for (k=1, #d, nbk = #select(x->x==d[k], digs); s = concat(s, Str(nbk)); s = concat(s, d[k]);); eval(s);}
lista(nn) = {print1(n = 1, ", "); for (k=1, nn, m = say(n); print1(m, ", "); n = m;);} \\ Michel Marcus, Feb 12 2016

a(n,show_all=1,a=1)={for(i=2,n,show_all&&print1(a",");a=A047842(a));a} \\ M. F. Hasler, Feb 25 2018

Vec(x*(1 + 10*x + 10*x^2 + 1091*x^3 + 2000*x^4 + 208101*x^5 + 101000*x^6 - 99990*x^7 - 98010*x^8 + 31007101*x^9 + 10001000*x^10 - 9900990*x^11 - 9899010*x^12) / (1 - x) + O(x^40)) \\ Colin Barker, Aug 23 2018

from itertools import accumulate, groupby, repeat
def summarize(n, _):
  return int("".join(str(len(list(g)))+k for k, g in groupby(sorted(str(n)))))
def aupton(nn): return list(accumulate(repeat(1, nn+1), summarize))
print(aupton(25)) # Michael S. Branicky, Jan 11 2021

a(n+1) = A047842(a(n)). - M. F. Hasler, Feb 25 2018

G.f.: x*(1 + 10*x + 10*x^2 + 1091*x^3 + 2000*x^4 + 208101*x^5 + 101000*x^6 - 99990*x^7 - 98010*x^8 + 31007101*x^9 + 10001000*x^10 - 9900990*x^11 - 9899010*x^12) / (1 - x). - Colin Barker, Aug 23 2018

A138484 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 0.

Original entry on oeis.org

0, 10, 1011, 3110, 102113, 13311210, 10411223, 1322311410, 1041142322, 3213243110, 1031331422, 2214313310, 1031331422, 2214313310, 1031331422, 2214313310, 1031331422, 2214313310, 1031331422, 2214313310, 1031331422

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

After a while sequence has period 2 -> {1031331422,2214313310}

			To get the term after 102113, we say: one 3's, three 1's, one 2's, one 0's, so 13311210.

Cf. A063850, A022482, A005150, A005151, A006751, A006715, A006711, A022506-A022513, A138485-A138493.

A138493 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 9.

Original entry on oeis.org

9, 19, 1911, 3119, 192113, 13311219, 19411223, 1322311419, 1941142322, 3213243119, 1931331422, 2214313319, 1931331422, 2214313319, 1931331422, 2214313319, 1931331422, 2214313319, 1931331422, 2214313319, 1931331422

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

After a while sequence has period 2 -> {1931331422,2214313319}

			To get the term after 192113, we say: one 3's, three 1's, one 2's, one 9's, so 13311219

Cf. A063850, A022482, A005150, A005151, A006751, A006715, A006711, A022506-A022513, A138484-A138492.

A123132 Describe prime factorization of n (primes in ascending order and with repetition) (method A - initial term is 2).

Original entry on oeis.org

12, 13, 22, 15, 1213, 17, 32, 23, 1215, 111, 2213, 113, 1217, 1315, 42, 117, 1223, 119, 2215, 1317, 12111, 123, 3213, 25, 12113, 33, 2217, 129, 121315, 131, 52, 13111, 12117, 1517, 2223, 137, 12119, 13113, 3215, 141, 121317, 143, 22111, 2315, 12123

Offset: 2

Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 30 2006

Method A = 'frequency' followed by 'digit'-indication. Say 'what you see' in prime factors of n, n>1.

			2 has "one 2" in its prime decomposition, so a(2)=12.
3 has "one 3" in its prime decomposition, so a(3)=13.
4=2*2 has "two 2" in its prime decomposition, so a(4)=22.
5 has "one 5" in its prime decomposition, so a(5)=15.
6=2*3 has "one 2 and one 3" in its prime decomposition, so a(6)=1213.
.....

Paolo P. Lava, Table of n, a(n) for n = 2..10000
A. Frank & P. Jacqueroux, International Contest, 2001.

Cf. A006751, A027746, A063850.

a[n_] := FromDigits@ Flatten@ IntegerDigits[ Reverse /@ FactorInteger@ n]; a/@ Range[2,30] (* Giovanni Resta, Jun 16 2013 *)

for(n=2,25,factn=factor(n); for(i=1,omega(n),print1(factn[i,2],factn[i,1])); print1(","))

a(n) = my(factn=factor(n), sout = ""); for(i=1, omega(n), sout = concat(sout, Str(factn[i, 2])); sout = concat(sout, Str(factn[i, 1]))); eval(sout); \\ Michel Marcus, Jun 29 2017

A138485 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 1.

Original entry on oeis.org

1, 11, 21, 1112, 1231, 211312, 223113, 232122, 421113, 13311214, 14411223, 13223124, 14322123, 23322114, 14213223, 23322114, 14213223, 23322114, 14213223, 23322114, 14213223, 23322114, 14213223, 23322114, 14213223

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

After a while sequence has period 2 -> {23322114,14213223}

			To get the term after 211312, we say: two 2's, three 1's, one 3's, so 223113.

Cf. A063850, A022482, A005150, A005151, A006751, A006715, A006711, A022506-A022513, A138484, A138486-A138493.

A138486 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 2.

Original entry on oeis.org

2, 12, 1211, 3112, 122113, 133122, 222123, 134211, 31121413, 23411412, 22312413, 23211432, 32231421, 21321423, 23321421, 21321423, 23321421, 21321423, 23321421, 21321423, 23321421, 21321423, 23321421, 21321423

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

After a while sequence has period 2 -> {21321423,23321421}

			To get the term after 122113, we say: one 3's, three 1's, two 2's, so 133122

Cf. A063850, A022482, A005150, A005151, A006751, A006715, A006711, A022506-A022513, A138484, A138485, A138487-A138493.

Duplicate term 21321423 removed by Georg Fischer, Sep 18 2023

A138487 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 3.

Original entry on oeis.org

3, 13, 1311, 3113, 2321, 112213, 133122, 222123, 134211, 31121413, 23411412, 22312413, 23211432, 32231421, 21321423, 23321421, 21321423, 23321421, 21321423, 23321421, 21321423, 23321421, 21321423, 23321421, 21321423

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

After a while sequence has period 2 -> {21321423,23321421}

			To get the term after 112213, we say: one 3's, three 1's, two 2's, so 133122

Cf. A063850, A022482, A005150, A005151, A006751, A006715, A006711, A022506-A022513, A138484-A138486, A138488-A138493.

A138488 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 4.

Original entry on oeis.org

4, 14, 1411, 3114, 142113, 13311214, 14411223, 13223124, 14322123, 23322114, 14213223, 23322114

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

After a while sequence has period 2 -> {23322114,14213223}

			To get the term after 142113, we say: one 3's, three 1's, one 2's, one 4's, so 13311214

Cf. A063850, A022482, A005150, A005151, A006751, A006715, A006711, A022506-A022513, A138484-A138487, A138489-A138493.

A138489 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 5.

Original entry on oeis.org

5, 15, 1511, 3115, 152113, 13311215, 15411223, 1322311415, 1541142322, 3213243115, 1531331422, 2214313315, 1531331422, 2214313315, 1531331422, 2214313315, 1531331422, 2214313315, 1531331422, 2214313315, 1531331422

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

After a while sequence has period 2 -> {1531331422,2214313315}

			To get the term after 152113, we say: one 3's, three 1's, one 2's, one 5's, so 13311215

Cf. A063850, A022482, A005150, A005151, A006751, A006715, A006711, A022506-A022513, A138484-A138488, A138490-A138493.

A138490 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 6.

Original entry on oeis.org

6, 16, 1611, 3116, 162113, 13311216, 16411223, 1322311416, 1641142322, 3213243116, 1631331422, 2214313316, 1631331422, 2214313316, 1631331422, 2214313316, 1631331422, 2214313316, 1631331422, 2214313316, 1631331422

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

After a while sequence has period 2 -> {1631331422,2214313316}

			To get the term after 162113, we say: one 3's, three 1's, one 2's, one 6's, so 13311216

Cf. A063850, A022482, A005150, A005151, A006751, A006715, A006711, A022506-A022513, A138484-A138489, A138491-A138493.

cp's OEIS Frontend

A005151 Summarize the previous term (digits in increasing order), starting with a(1) = 1.

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A138484 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 0.

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A138493 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 9.

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A123132 Describe prime factorization of n (primes in ascending order and with repetition) (method A - initial term is 2).

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A138485 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 1.

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A138486 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 2.

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A138487 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 3.

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A138488 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 4.

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A138489 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 5.

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