A001637 -id:A001637 - cp's OEIS frontend

A005150 Look and Say sequence: describe the previous term! (method A - initial term is 1).

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, 11131221133112132113212221, 3113112221232112111312211312113211, 1321132132111213122112311311222113111221131221, 11131221131211131231121113112221121321132132211331222113112211, 311311222113111231131112132112311321322112111312211312111322212311322113212221

Offset: 1

N. J. A. Sloane

Method A = "frequency" followed by "digit"-indication.
Also known as the "Say What You See" sequence.
Only the digits 1, 2 and 3 appear in any term. - Robert G. Wilson v, Jan 22 2004
All terms end with 1 (the seed) and, except the third a(3), begin with 1 or 3. - Jean-Christophe Hervé, May 07 2013
Proof that 333 never appears in any a(n): suppose it appears for the first time in a(n); because of "three 3" in 333, it would imply that 333 is also in a(n-1), which is a contradiction. - Jean-Christophe Hervé, May 09 2013
This sequence is called "suite de Conway" in French (see Wikipédia link). - Bernard Schott, Jan 10 2021
Contrary to many accounts (including an earlier comment on this page), Conway did not invent the sequence. The first mention of the sequence appears to date back to the 1977 International Mathematical Olympiad in Belgrade, Yugoslavia. See the Editor's note on page 4, directly preceding Conway's article in Eureka referenced below. - Harlan J. Brothers, May 03 2024

			The term after 1211 is obtained by saying "one 1, one 2, two 1's", which gives 111221.

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 208.
S. R. Finch, Mathematical Constants, Cambridge, 2003, section 6.12 Conway's Constant, pp. 452-455.
M. Gilpin, On the generalized Gleichniszahlen-Reihe sequence, Manuscript, Jul 05 1994.
A. Lakhtakia and C. Pickover, Observations on the Gleichniszahlen-Reihe: An Unusual Number Theory Sequence, J. Recreational Math., 25 (No. 3, 1993), 192-198.
Clifford A. Pickover, Computers and the Imagination, St Martin's Press, NY, 1991.
Clifford A. Pickover, Fractal horizons: the future use of fractals, New York: St. Martin's Press, 1996. ISBN 0312125992. Chapter 7 has an extensive description of the elements and their properties.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 486.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, 1999, p. 23.
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 4.

T. D. Noe, Table of n, a(n) for n = 1..25
Henry Bottomley, Evolution of Conway's 92 Look and Say audioactive elements.
Éric Brier, Rémi Géraud-Stewart, David Naccache, Alessandro Pacco, and Emanuele Troiani, Stuttering Conway Sequences Are Still Conway Sequences, arXiv:2006.06837 [math.DS], 2020.
Éric Brier, Rémi Géraud-Stewart, David Naccache, Alessandro Pacco, and Emanuele Troiani, The Look-and-Say The Biggest Sequence Eventually Cycles, arXiv:2006.07246 [math.DS], 2020.
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.
Ben Chen, Richard Chen, Joshua Guo, Tanya Khovanova, Shane Lee, Neil Malur, Nastia Polina, Poonam Sahoo, Anuj Sakarda, Nathan Sheffield, and Armaan Tipirneni, On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.
J. H. Conway, The weird and wonderful chemistry of audioactive decay, Eureka 46 (1986) 5-16.
J. H. Conway, The weird and wonderful chemistry of audioactive decay, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Computation, Springer, NY 1987, pp. 173-188.
J. H. Conway and Brady Haran, Look-and-Say Numbers (2014), Numberphile video.
S. B. Ekhad and D. Zeilberger, Proof of Conway's Lost Cosmological Theorem, arXiv:math/9808077 [math.CO], 1998.
S. B. Ekhad and D. Zeilberger, Proof of Conway's lost cosmological theorem, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 78-82.
S. Eliahou and M. J. Erickson, Mutually describing multisets and integer partitions, Discrete Mathematics, Volume 313, Issue 4, Feb 28 2013, Pages 422-433. - From _N. J. A. Sloane_, Jan 03 2013
S. R. Finch, Conway's Constant [From the Wayback Machine]
Steven Finch, The On-Line Encyclopedia of Integer Sequences, founded in 1964 by N. J. A. Sloane, A Tribute to John Horton Conway, The Mathematical Intelligencer (2021) Vol. 43, 146-147.
X. Gourdon and B. Salvy, Effective asymptotics of linear recurrences with rational coefficients, Discrete Mathematics, vol. 153, no. 1-3, 1996, pages 145-163. See p. 161.
M. Hilgemeier, Die Gleichniszahlen-Reihe, in Bild der Wissenschaft, 12 (1986), 194-195, with permission from the Konradin Medien GmbH.
M. Hilgemeier, One metaphor fits all, in Fractal Horizons, ed. C. A Pickover, St. Martins, NY, 1996, pp. 137-161.
R. A. Litherland, Conway's Cosmological Theorem (Overview).
R. A. Litherland, Conway's Cosmological Theorem, 12 pages, Apr 14 2006 (pdf file).
R. A. Litherland, Programs for Conway's Cosmological Theorem, (gzipped tar ball).
R. A. Litherland, The audioactive package.
M. Lothaire, Algebraic Combinatorics on Words, Cambridge, 2002, see p. 37, etc.
MacTutor History of Mathematics, John H. Conway
O. Martin, Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA, Amer. Math. Monthly, 113 (No. 4, 2006), 289-307. - From _N. J. A. Sloane_, Feb 19 2013
Thomas Morrill, Look, Knave, arXiv:2004.06414 [math.CO], 2020.
Paulo Ortolan, Java program for A005150.
Matt Parker, Can you trust an elegant conjecture?, Stand-Up Maths, 2022, video.
Rosetta Code, Look and say sequence programs in over 60 languages.
J. Sauerberg and L. Shu, The long and the short on counting sequences, Amer. Math. Monthly, 104 (1997), 306-317.
T. Sillke, Conway sequence.
L. J. Upton, Letter to N. J. A. Sloane, Jan 08 1991.
Kevin Watkins, Abstract Interpretation Using Laziness: Proving Conway's Lost Cosmological Theorem.
Kevin Watkins, Proving Conway's Lost Cosmological Theorem, POP seminar talk, CMU, Dec 2006.
Eric Weisstein's World of Mathematics, Look and Say Sequence.
Wikipedia, Look-and-say sequence.
Wikipédia, Suite de Conway.
W. W. Zadrozny, Abstraction, Reasoning and Deep Learning: A Study of the "Look and Say" Sequence, arXiv:2109.12755 [cs.AI], 2022.
Julia Witte Zimmerman, Denis Hudon, Kathryn Cramer, Jonathan St. Onge, Mikaela Fudolig, Milo Z. Trujillo, Christopher M. Danforth, and Peter Sheridan Dodds, A blind spot for large language models: Supradiegetic linguistic information, arXiv:2306.06794 [cs.CL], 2023.
Julia Witte Zimmerman, Denis Hudon, Kathryn Cramer, Alejandro J. Ruiz, Calla Beauregard, Ashley Fehr, Mikaela Irene Fudolig, Bradford Demarest, Yoshi Meke Bird, Milo Z. Trujillo, Christopher M. Danforth, and Peter Sheridan Dodds, Tokens, the oft-overlooked appetizer: Large language models, the distributional hypothesis, and meaning, arXiv:2412.10924 [cs.CL], 2024. See pp. 21, 28.

Cf. A001155, A006751, A006715, A001140, A001141, A001143, A001145, A001151, A001154, A007651, A060857.
Cf. A001387, Periodic table: A119566.
Cf. A225224, A221646, A225212 (continuous versions).
Apart from the first term, all terms are in A001637.
About digits: A005341 (number of digits), A022466 (number of 1's), A022467 (number of 2's), A022468 (number of 3's), A004977 (sum of digits), A253677 (product of digits).
About primes: A079562 (number of distinct prime factors), A100108 (terms that are primes), A334132 (smallest prime factor).
Cf. A014715 (Conway's constant), A098097 (terms interpreted as written in base 4).

import List
say :: Integer -> Integer
say = read . concatMap saygroup . group . show
where saygroup s = (show $ length s) ++ [head s]
look_and_say :: [Integer]
look_and_say = 1 : map say look_and_say
-- Josh Triplett (josh(AT)freedesktop.org), Jan 03 2007

a005150 = foldl1 (\v d -> 10 * v + d) . map toInteger . a034002_row
-- Reinhard Zumkeller, Aug 09 2012

Java
```
See Paulo Ortolan link.
```

RunLengthEncode[ x_List ] := (Through[ {First, Length}[ #1 ] ] &) /@ Split[ x ]; LookAndSay[ n_, d_:1 ] := NestList[ Flatten[ Reverse /@ RunLengthEncode[ # ] ] &, {d}, n - 1 ]; F[ n_ ] := LookAndSay[ n, 1 ][ [ n ] ]; Table[ FromDigits[ F[ n ] ], {n, 1, 15} ]
A005150[1] := 1; A005150[n_] := A005150[n] = FromDigits[Flatten[{Length[#], First[#]}&/@Split[IntegerDigits[A005150[n-1]]]]]; Map[A005150, Range[25]] (* Peter J. C. Moses, Mar 21 2013 *)

A005150(n,a=1)={ while(n--, my(c=1); for(j=2,#a=Vec(Str(a)), if( a[j-1]==a[j], a[j-1]=""; c++, a[j-1]=Str(c,a[j-1]); c=1)); a[#a]=Str(c,a[#a]); a=concat(a)); a }  \\ M. F. Hasler, Jun 30 2011

$str="1"; for (1 .. shift(@ARGV)) { print($str, ","); @a = split(//,$str); $str=""; $nd=shift(@a); while (defined($nd)) { $d=$nd; $cnt=0; while (defined($nd) && ($nd eq $d)) { $cnt++; $nd = shift(@a); } $str .= $cnt.$d; } } print($str);
# Jeff Quilici (jeff(AT)quilici.com), Aug 12 2003

# This outputs the first n elements of the sequence, where n is given on the command line.
$s = 1;
for (2..shift @ARGV) {
print "$s, ";
$s =~ s/(.)\1*/(length $&).$1/eg;
}
# Arne 'Timwi' Heizmann (timwi(AT)gmx.net), Mar 12 2008
print "$s\n";

def A005150(n):
    p = "1"
    seq = [1]
    while (n > 1):
        q = ''
        idx = 0 # Index
        l = len(p) # Length
        while idx < l:
            start = idx
            idx = idx + 1
            while idx < l and p[idx] == p[start]:
                idx = idx + 1
            q = q + str(idx-start) + p[start]
        n, p = n - 1, q
        seq.append(int(p))
    return seq
# Olivier Mengue (dolmen(AT)users.sourceforge.net), Jul 01 2005

def A005150(n):
    seq = [1] + [None] * (n - 1) # allocate entire array space
    def say(s):
        acc = '' # initialize accumulator
        while len(s) > 0:
            i = 0
            c = s[0] # char of first run
            while (i < len(s) and s[i] == c): # scan first digit run
                i += 1
            acc += str(i) + c # append description of first run
            if i == len(s):
                break # done
            else:
                s = s[i:] # trim leading run of digits
        return acc
    for i in range(1, n):
        seq[i] = int(say(str(seq[i-1])))
    return seq
# E. Johnson (ejohnso9(AT)earthlink.net), Mar 31 2008

# program without string operations
def sign(n): return int(n > 0)
def say(a):
    r = 0
    p = 0
    while a > 0:
        c = 3 - sign((a % 100) % 11) - sign((a % 1000) % 111)
        r += (10 * c + (a % 10)) * 10**(2*p)
        a //= 10**c
        p += 1
    return r
a = 1
for i in range(1, 26):
    print(i, a)
    a = say(a)
# Volker Diels-Grabsch, Aug 18 2013

import re
def lookandsay(limit, sequence = 1):
    if limit > 1:
        return lookandsay(limit-1, "".join([str(len(match.group()))+match.group()[0] for matchNum, match in enumerate(re.finditer(r"(\w)\1*", str(sequence)))]))
    else:
        return sequence
# lookandsay(3) --> 21
# Nicola Vanoni, Nov 29 2016

import itertools
x = "1"
for i in range(20):
    print(x)
    x = ''.join(str(len(list(g)))+k for k,g in itertools.groupby(x))
# Matthew Cotton, Nov 12 2019

a(n+1) = A045918(a(n)). - Reinhard Zumkeller, Aug 09 2012
a(n) = Sum_{k=1..A005341(n)} A034002(n,k)*10^(A005341(n)-k). - Reinhard Zumkeller, Dec 15 2012
a(n) = A004086(A007651(n)). - Bernard Schott, Jan 08 2021
A055642(a(n+1)) = A005341(n+1) = 2*A043562(a(n)). - Ya-Ping Lu, Jan 28 2025
Conjecture: DC(a(n)) ~ k * (Conway's constant)^n where k is approximately 1.021... and DC denotes the number of digit changes in the decimal representation of n (e.g., DC(13112221)=4 because 1->3, 3-1, 1->2, 2->1). - Bill McEachen, May 09 2025
Conjecture: lim_{n->infinity} (c2+c3-c1)/(c1+c2+c3) = 0.01 approximately, where ci is the number of appearances of 'i' in a(n). - Ya-Ping Lu, Jun 05 2025

A001633 Numbers with an odd number of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145

Offset: 1

N. J. A. Sloane

The lower and upper asymptotic densities of this sequence are 1/11 and 10/11, respectively. - Amiram Eldar, Feb 01 2021

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001637 (complement), A055642.

a001633 n = a001633_list !! (n-1)
a001633_list = filter (odd . a055642) [0..]
-- Reinhard Zumkeller, Jul 14 2014

Select[Range[0, 145], OddQ[Length[IntegerDigits[#]]] &] (* T. D. Noe, Aug 09 2012 *)
Table[Range[10^n,10^(n+1)-1],{n,0,4,2}]//Flatten (* Harvey P. Dale, Nov 29 2022 *)

is(n)=#Str(n)%2 \\ Charles R Greathouse IV, Nov 26 2018

(0 to 199).filter(Integer.toString().length % 2 == 1) // _Alonso del Arte, Feb 03 2020

A055642(a(n)) mod 2 = 1. - Reinhard Zumkeller, Jul 14 2014

Except for k = 0, if k is in this sequence, floor(log_10 k) is even; e.g., if k has three digits, 2 <= log_10 k < 3. - Alonso del Arte, Feb 03 2020

A227870 Numbers with equal number of even and odd digits.

Original entry on oeis.org

10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 1001, 1003, 1005, 1007, 1009, 1010, 1012, 1014, 1016, 1018, 1021, 1023, 1025

Offset: 1

Jon Perry, Nov 02 2013

Numbers with an odd digit length cannot be in this sequence. - Alonso del Arte, Nov 02 2013

			1009 has 2 even digits (00) and 2 odd digits (19) and so is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A001637.

Cf. A030141, A031443.

for (i = 1; i < 5000; i++) {
s = i.toString();
odds = 0; evens = 0;
for (j = 0; j < s.length; j++) if (s.charAt(j)%2 == 0) evens++; else odds++;
if (odds == evens) document.write(i + ", ");
}

Select[Range[1025], (d = Differences[Tally[Mod[IntegerDigits[#], 2]]]) != {} && d[[1, 2]] == 0 &] (* Amiram Eldar, Oct 01 2020 *)
eneodQ[n_]:=With[{id=IntegerDigits[n]},Count[id,?(OddQ[#]&)]==Count[id,?(EvenQ[#]&)]]; Select[Range[1100],eneodQ] (* Harvey P. Dale, Jul 19 2024 *)

isok(m) = my(d=digits(m)); #select(x->(x%2), d) == #select(x->!(x%2), d); \\ Michel Marcus, Oct 01 2020

def ok(i):
  stri = str(i)
  se = sum(1 for d in stri if d in "02468")
  so = sum(1 for d in stri if d in "13579")
  return se == so
def aupto(nn):
  alst, an = [None], 0
  for n in range(1, nn+1):
    while len(alst) < nn+1:
      if ok(an): alst.append(an)
      an += 1
  return alst[1:] # use alst[n] for a(n)
print(aupto(58))  # Michael S. Branicky, Dec 14 2020

A132285 Odd palindromes with an even number of digits.

Original entry on oeis.org

11, 33, 55, 77, 99, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 3003, 3113, 3223, 3333, 3443, 3553, 3663, 3773, 3883, 3993, 5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 7007, 7117, 7227, 7337, 7447, 7557, 7667, 7777, 7887

Offset: 1

Artur Jasinski, Aug 16 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..555

Cf. A005408, A001637, A002113, A132286, A132288.

monQ[n_]:=Module[{len=IntegerLength[n],idn=IntegerDigits[n]},EvenQ[ len] &&Take[idn,len/2]==Reverse[Take[idn,-len/2]]]; Select[Range[11,9999,2], monQ] (* Harvey P. Dale, Apr 18 2014 *)

isok(n) = {if (n % 2 == 0, return (0)); d = digits(n); if (#d % 2 == 1, return (0)); for (i=1, #d/2, if (d[i] != d[#d-i+1], return (0));); return (1);} \\ Michel Marcus, Nov 05 2013

A005408 INTERSECT A001637 INTERSECT A002113. - R. J. Mathar, Aug 22 2007

A280824 Numbers with an even number of digits and with an even number of distinct digits.

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 1000, 1001, 1010

Offset: 1

Ilya Gutkovskiy, Jan 08 2017

Differs from A139819 (the latter contains 100, for example). - R. J. Mathar, Jan 17 2017

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit
Index entries for 10-automatic sequences.

Cf. A000035, A001637, A029742, A043537, A055642, A280823, A280825, A280826.

isA280824 := proc(n)
    if n < 10 then
        return false;
    end if;
    dgs := convert(n,base,10) ;
    if type(nops(dgs),'even') then
        type(nops(convert(dgs,set)),'even') ;
    else
        false;
    end if;
end proc: # R. J. Mathar, Jan 17 2017

Select[Range[1010], Mod[Length[IntegerDigits[#1]], 2] == 0 && Mod[Length[Union[IntegerDigits[#1]]], 2] == 0 & ]

def ok(n): s = str(n); return len(s)%2 == 0 == len(set(s))%2
print(list(filter(ok, range(1011)))) # Michael S. Branicky, Oct 12 2021

A000035(A055642(a(n))) = 0.
A000035(A043537(a(n))) = 0.
a(n) = A029742(n) for n < 82.

A280826 Numbers with an even number of digits and with an odd number of distinct digits.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1030, 1031, 1033, 1040, 1041, 1044, 1050, 1051, 1055, 1060, 1061, 1066, 1070, 1071, 1077, 1080, 1081, 1088, 1090, 1091, 1099, 1102, 1103, 1104, 1105, 1106, 1107, 1108, 1109

Offset: 1

Ilya Gutkovskiy, Jan 08 2017

Eric Weisstein's World of Mathematics, Digit
Index entries for 10-automatic sequences.

Cf. A000035, A001637, A043537, A055642, A280823, A280824, A280825.

Select[Range[1110], Mod[Length[IntegerDigits[#1]], 2] == 0 && Mod[Length[Union[IntegerDigits[#1]]], 2] == 1 & ]
endQ[n_]:=Module[{idn=IntegerDigits[n]},EvenQ[Length[idn]]&&OddQ[ Length[ Union[ idn]]]]; Select[Range[1200],endQ] (* Harvey P. Dale, Mar 31 2019 *)

A000035(A055642(a(n))) = 0.

A000035(A043537(a(n))) = 1.

A358421 Primes that are the concatenation of two primes with the same number of digits.

Original entry on oeis.org

23, 37, 53, 73, 1117, 1123, 1129, 1153, 1171, 1319, 1361, 1367, 1373, 1723, 1741, 1747, 1753, 1759, 1783, 1789, 1913, 1931, 1973, 1979, 1997, 2311, 2341, 2347, 2371, 2383, 2389, 2917, 2953, 2971, 3119, 3137, 3167, 3719, 3761, 3767, 3779, 3797, 4111, 4129, 4153, 4159, 4337, 4373, 4397, 4723, 4729

Offset: 1

J. M. Bergot and Robert Israel, Nov 15 2022

It appears that there are ~ 0.81*100^k/(k^2 log^2 10) 2k-digit numbers in this sequence, making their relative density 0.9/(k^2 log^2 10) among 2k-digit numbers. Of course there are no 2k+1-digit terms in the sequence. - Charles R Greathouse IV, Nov 15 2022

The second Mathematica program below generates all 2-digit and 4-digit terms of the sequence. To generate all 2,753 6-digit terms of the sequence, use this Mathematica program: Select[1000#[[1]]+#[[2]]&/@Tuples[Prime[Range[26,168]],2],PrimeQ]. There are 112,649 8-digit terms of the sequence. - Harvey P. Dale, Feb 28 2023

			a(5) = 1117 is a term because 11 and 17 are both 2-digit primes and 1117 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A000040 and of A001637.

Res:= NULL: count:= 0:
for d from 2 by 2 while count < 100 do
  pq:= 10^(d-1);
  while count < 100 do
    pq:= nextprime(pq);
    if pq > 10^d then break fi;
    q:= pq mod 10^(d/2);
    if q < 10^(d/2-1) then next fi;
    p:= (pq-q)/10^(d/2);
    if isprime(p) and isprime(q) then Res:= Res,pq; count:= count+1 fi
od od:
Res;

Select[Prime[Range[640]], EvenQ[(s = IntegerLength[#])] && IntegerDigits[#][[s/2 + 1]] > 0 && And @@ PrimeQ[QuotientRemainder[#, 10^(s/2)]] &] (* Amiram Eldar, Nov 15 2022 *)
Join[Select[FromDigits/@Tuples[Prime[Range[4]],2],PrimeQ],Select[100#[[1]]+ #[[2]]&/@ Tuples[Prime[Range[5,25]],2],PrimeQ]] (* Harvey P. Dale, Feb 28 2023 *)

from itertools import count, islice
from sympy import isprime, primerange
def agen(): # generator of terms
    for d in count(1):
        pow = 10**d
        for p in primerange(10**(d-1), pow):
            for q in primerange(10**(d-1), pow):
                t = p*pow + q
                if isprime(t): yield t
print(list(islice(agen(), 51))) # Michael S. Branicky, Nov 15 2022

A375213 Even numbers with equal numbers of even and odd digits.

Original entry on oeis.org

10, 12, 14, 16, 18, 30, 32, 34, 36, 38, 50, 52, 54, 56, 58, 70, 72, 74, 76, 78, 90, 92, 94, 96, 98, 1010, 1012, 1014, 1016, 1018, 1030, 1032, 1034, 1036, 1038, 1050, 1052, 1054, 1056, 1058, 1070, 1072, 1074, 1076, 1078, 1090, 1092, 1094, 1096, 1098, 1100

Offset: 1

Jake L Lande, Aug 05 2024

Numbers with an odd digit length cannot be in this sequence.

			1010 is even and has two even digits (0,0) and two odd digits (1,1).

Jake L Lande, Table of n, a(n) for n = 1..1000

Subsequence of A227870 and hence A001637.

Complement of A375214 within A227870.

eeo[n_] := (id = IntegerDigits[n]; Count[EvenQ@id, True] == Count[OddQ@id, True]); Select[Select[Range[1100], eeo], Mod[#, 2] == 0 &]

A289911 Prime vampire numbers: semiprimes xy such that x and y have the same number of digits and the union of the multisets of the digits of x and y is the same as the multiset of digits of xy.

Original entry on oeis.org

117067, 124483, 146137, 371893, 536539, 10349527, 10429753, 10687513, 11722657, 11823997, 12451927, 12484057, 12894547, 13042849, 14145799, 14823463, 17204359, 18517351, 18524749, 18647023, 19262587, 19544341, 19554277, 20540911, 20701957, 21874387, 30189721

Offset: 1

Felix Fröhlich, Jul 15 2017

Subsequence of A014575.

			117067 = 167 * 701. A055642(117067) mod 2 = 0, A055642(167) = A055642(701) and the multiset of digits of 117067 is {0, 1, 1, 6, 7, 7}, which is also the multiset resulting from the union of the multisets of digits of 167 and 701, so 117067 is a term of the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Shyam Sunder Gupta, Cab and Vampire Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 20, 499-512.
Carlos Rivera, Puzzle 199. The Prime-Vampire numbers, The Prime Puzzles & Problems Connection.
G. Villemin's Almanach of Numbers, Nombres Vampires, gives the first 5 terms.
Wikipedia, Vampire number.

Cf. A001358, A001637, A014575, A055642.

is_a001637(n) = #Str(n)%2==0
is_a001358(n) = omega(n)==2
samefactorlength(v) = #Str(v[1])==#Str(v[2])
samedigitmultiset(v) = vecsort(concat(digits(v[1]), digits(v[2])))==vecsort(digits(v[1]*v[2]))
is(n) = if(!is_a001637(n) || !is_a001358(n) || (!issquarefree(n) && bigomega(n) > 2), return(0), my(f=factor(n)[, 1]~); if(samefactorlength(f) && samedigitmultiset(f), return(1), return(0)))

\\ terms with n digits (if n is odd then returns terms with n + 1 digits).
ndigits(n) = {n-=2; n+=(n%2); my(res=List()); forprime(p=ceil(10^(n/2)), 10^(n/2+1)-1, forprime(q = max(p, ceil(10^(n+1)/p)), 10^(n/2+1)-1, if(Set(vecsort(digits(p*q)) -vecsort(concat(digits(p),digits(q))))==[0], listput(res, p*q)))); listsort(res); res} \\ David A. Corneth, Jul 24 2017

A098760 Smallest a(n) not already in the sequence and not having the same length as a(n-1).

Original entry on oeis.org

0, 10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 100, 20, 101, 21, 102, 22, 103, 23, 104, 24, 105, 25, 106, 26, 107, 27, 108, 28, 109, 29, 110, 30, 111, 31, 112, 32, 113, 33, 114, 34, 115, 35, 116, 36, 117, 37, 118, 38, 119, 39, 120, 40, 121, 41, 122, 42

Offset: 0

Eric Angelini, Oct 01 2004

The length of an integer is the number of its digits (10023 has length 5).

Bisections give A001633 and A001637. - David Wasserman, Feb 26 2008

More terms from Sam Alexander, Jan 04 2005

More terms from David Wasserman, Feb 26 2008

cp's OEIS Frontend

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A289911 Prime vampire numbers: semiprimes xy such that x and y have the same number of digits and the union of the multisets of the digits of x and y is the same as the multiset of digits of xy.