A011532 -id:A011532 - cp's OEIS frontend

A257219 Numbers that have at least one divisor containing the digit 2 in base 10.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108

Offset: 1

Jaroslav Krizek, Apr 20 2015

Numbers k whose concatenation of divisors A037278(k), A176558(k), A243360(k) or A256824(k) contains a digit 2.
Sequences of numbers k whose concatenation of divisors contains a digit j in base 10 for 0 <= j <= 9: A209932 for j = 0, A000027 for j = 1, A257219 for j = 2, A257220 for j = 3, A257221 for j = 4, A257222 for j = 5, A257223 for j = 6, A257224 for j = 7, A257225 for j = 8, A257226 for j = 9.
All even numbers and all numbers which have a digit "2" themselves are trivially in this sequence. The first terms not of this form are the odd multiples of odd numbers between 21 and 29: { 63 = 3*21, 69 = 3*23, 75 = 3*25, 81 = 3*27, 87 = 3*29, 105 = 5*21, 115 = 5*23, 135 = 5*27, 145 = 5*29, ...}. - M. F. Hasler, Apr 22 2015
A011532 (numbers that contain a 2) is a subsequence. - Michel Marcus, May 19 2015

			18 is in sequence because the list of divisors of 18: (1, 2, 3, 6, 9, 18) contains digit 2.
In the same way all even numbers have the divisor 2 and thus are in this sequence; numbers N in { 20,...,29, 120,...,129, 200,...,299 } have the digit 2 in N which is divisor of itself. - _M. F. Hasler_, Apr 22 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A037278, A176558, A243360, A256824.

[n: n in [1..1000] | [2] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))]

Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 2] > 0 &] (* Michael De Vlieger, Apr 20 2015 *)

is(n)=!bittest(n,0)||setsearch(Set(digits(n)),2)||fordiv(n,d,setsearch(Set(digits(d)),2)&&return(1)) \\ M. F. Hasler, Apr 22 2015

a(n) ~ n. - Charles R Greathouse IV, Apr 22 2015

A175688 Numbers k with property that arithmetic mean of its digits is both an integer and one of the digits of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 102, 111, 120, 123, 132, 135, 147, 153, 159, 174, 195, 201, 204, 210, 213, 222, 231, 234, 240, 243, 246, 258, 264, 285, 306, 312, 315, 321, 324, 333, 342, 345, 351, 354, 357, 360, 369, 375, 396, 402

Offset: 1

Claudio Meller, Aug 09 2010

Subsequence of A061383.

A180160(a(n)) = 0. - Reinhard Zumkeller, Aug 15 2010

			135 is in the list because (1+3+5)/3 = 3 and 3 is a digit of 135.

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

Cf. A007953, A055642, A004426, A004427; A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539.

Cf. A052018. - Reinhard Zumkeller, Aug 15 2010

a175688 n = a175688_list !! (n-1)
a175688_list = filter f [0..] where
   f x = m == 0 && ("0123456789" !! avg) `elem` show x
         where (avg, m) = divMod (a007953 x) (a055642 x)
-- Reinhard Zumkeller, Jun 18 2013

idQ[n_]:=Module[{idn=IntegerDigits[n],m},m=Mean[idn];IntegerQ[m] && MemberQ[idn,m]]; Select[Range[0,500],idQ] (* Harvey P. Dale, Jun 10 2011 *)

Edited by Reinhard Zumkeller, Aug 13 2010

A043498 Numbers having two 2's in base 10.

Original entry on oeis.org

22, 122, 202, 212, 220, 221, 223, 224, 225, 226, 227, 228, 229, 232, 242, 252, 262, 272, 282, 292, 322, 422, 522, 622, 722, 822, 922, 1022, 1122, 1202, 1212, 1220, 1221, 1223, 1224, 1225, 1226, 1227, 1228, 1229, 1232, 1242, 1252

Offset: 1

Clark Kimberling

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Subsequence of A011532.

q:= n-> numboccur(2, convert(n, base, 10))=2:
select(q, [$2..2222])[];  # Alois P. Heinz, Mar 15 2020

Select[Range[5000], DigitCount[#, 10, 2] == 2 &] (* Vincenzo Librandi, Nov 20 2015 *)

c(k, d, b) = {my(c=0, f); while (k>b-1, f=k-b*(k\b); if (f==d, c++); k\=b); if (k==d, c++); return(c)}
for(n=0, 2000, if(c(n, 2, 10)==2, print1(n, ", "))) \\ Altug Alkan, Nov 20 2015

A381087 The smallest positive integer that produces a product that contains the digit 2 when multiplied by 2 a total of n times.

Original entry on oeis.org

2, 1, 6, 31, 64, 64, 331, 331, 814, 1607, 4107, 5129, 5129, 5129, 10283, 12819, 16163, 16163, 16163, 40108, 40108, 40108, 40108, 40108, 40108, 80313, 80313, 80313, 80313, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153, 100153

Offset: 0

Scott R. Shannon, Feb 13 2025

The last known distinct term is a(148) = 3130008; all subsequent terms studied also equal 3130008, and it is plausible, although unproven, that this is the last distinct value as n -> infinity.

			a(2) = 6 as 6*2 = 12 and 12*2 = 24, and the two products contain the digit 2.
a(8) = 814 as 814*2 = 1628, 1628*2 = 3256, 3256*2 = 6512, 6512*2 = 13024, 13024*2 = 26048, 26048*2 = 52096, 52096*2 = 104192, 104192*2 = 208384, and the eight products contain the digit 2.

Scott R. Shannon, Table of n, a(n) for n = 0..500

Cf. A378138 (distinct values), A381183, A011532.

A381183 a(n) = the smallest positive integer that produces a product that contains the digit 2 when multiplied by 2 at most n times, and where a further multiplication by 2 produces a number that does not contain the digit 2. Set a(n) = -1 if no such number exists.

Original entry on oeis.org

2, 1, 6, 31, 128, 64, 516, 331, 814, 1607, 4107, 10158, 10258, 5129, 10283, 12819, 25633, 28141, 16163, 51404, 80134, 80864, 40633, 80216, 40108, 128129, 250627, 160626, 80313, 125641, 208141, 383814, 391628, 195814, 156766, 196314, 391563, 490641, 806166, 785313, 628222, 314111, 625322, 312661, 1563305, 2630104, 1315052, 657526, 328763, 1643815

Offset: 0

Michael De Vlieger and Scott R. Shannon, Feb 16 2025

It is plausible that there are many terms, likely almost all terms, such that a(n) = -1, since the products as n increases become so large it is almost certain that subsequent products also contain the digit 2. It is therefore extremely unlikely that the series of products will terminate for very large values of n. See A381087.

For all starting values up to 10^9 the lowest undetermined term is a(263), while the largest determined term is a(370) = 357131067. The largest term value in this range is a(301) = 957107659.

			a(2) = 6 as 6*2 = 12, 12*2 = 24, 24*2 = 48, and the first two products contain the digit 2 while the third does not.
a(6) = 516 as 516*2 = 1032, 1032*2 = 2064, 2064*2 = 4128, 4128*2 = 8256, 8256*2 = 16512, 16512*2 = 33024, 33024*2 = 66048, and the first six products contain the digit 2 while the seventh does not.

Scott R. Shannon, Table of n, a(n) for n = 0..262

Cf. A381087, A378138, A011532.

A043499 Numbers having three 2's in base 10.

Original entry on oeis.org

222, 1222, 2022, 2122, 2202, 2212, 2220, 2221, 2223, 2224, 2225, 2226, 2227, 2228, 2229, 2232, 2242, 2252, 2262, 2272, 2282, 2292, 2322, 2422, 2522, 2622, 2722, 2822, 2922, 3222, 4222, 5222, 6222, 7222, 8222, 9222, 10222, 11222, 12022, 12122, 12202, 12212

Offset: 1

Clark Kimberling

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Subsequence of A011532.

Select[Range[13000],DigitCount[#,10,2]==3&] (* Harvey P. Dale, Aug 23 2021 *)

A208273 Composite numbers containing a digit 2.

Original entry on oeis.org

12, 20, 21, 22, 24, 25, 26, 27, 28, 32, 42, 52, 62, 72, 82, 92, 102, 112, 120, 121, 122, 123, 124, 125, 126, 128, 129, 132, 142, 152, 162, 172, 182, 192, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221

Offset: 1

Jaroslav Krizek, Mar 04 2012

Subsequence of A011532. Complement of A208272 with respect to A011532.

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

Cf. A208272 (primes containing a digit 2), A011532 (numbers containing a digit 2).

filter:= proc(n) not isprime(n) and member(2,convert(n,base,10)) end proc:
select(filter, [$4..300]); # Robert Israel, Oct 09 2024

Select[Range[300], ! PrimeQ[#] && MemberQ[IntegerDigits[#], 2] &] (* T. D. Noe, Mar 06 2012 *)
Select[Range[300],CompositeQ[#]&&DigitCount[#,10,2]>0&] (* Harvey P. Dale, Sep 08 2024 *)

A378138 The distinct values, in order of appearance, of A381087.

Original entry on oeis.org

2, 1, 6, 31, 64, 331, 814, 1607, 4107, 5129, 10283, 12819, 16163, 40108, 80313, 100153, 256379, 1281895, 2571143, 3130008

Offset: 0

Scott R. Shannon, Feb 16 2025

See A381087 for further details. It is plausible, although unproven, that 3130008 is the final term.

Cf. A381087, A381183, A011532.

A285470 Numbers k where "2" appears as the second digit of the decimal representation.

Original entry on oeis.org

12, 22, 32, 42, 52, 62, 72, 82, 92, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 620, 621, 622, 623, 624, 625, 626, 627

Offset: 1

Jamie Robert Creasey, Apr 19 2017

To find a(n), concatenate the first digit of n with 2 and then the other digits (if any) from n. See example. - David A. Corneth, Jun 12 2017

			a(21) = 221, a(36) = 326.
As the first digit of 983 is 9, and the others are 83, a(983) = 9283. - _David A. Corneth_, Jun 12 2017

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A011532 (containing 2), A052404 (without 2), A217394 (starting with 2).

seq(seq(seq(a*10^d + 2*10^(d-1)+c, c=0..10^(d-1)-1),a=1..9),d=1..2); # Robert Israel, Jun 12 2017

Table[FromDigits@ Apply[Join, {{First@ #}, {2}, Rest@ #}] &@ IntegerDigits@ n, {n, 67}] (* Michael De Vlieger, Jun 12 2017 *)
Select[Range[700],NumberDigit[#,IntegerLength[#]-2]==2&] (* Harvey P. Dale, Aug 15 2025 *)

isok(n) = (n>9) && digits(n)[2] == 2; \\ Michel Marcus, Jun 12 2017

a(n) = my(d = digits(n)); fromdigits(concat([d[1], [2], vector(#d-1, i, d[i+1])])) \\ David A. Corneth, Jun 12 2017

nxt(n) = if(isok(n+1), n+1, d = digits(n); t = 9*10^(#d-2); if(d[1]==9,t*=3); n+=t++) \\ David A. Corneth, Jun 12 2017

def a(n): s = str(n); return int(s[0] + "2" + s[1:])
print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Dec 22 2021

From Robert Israel, Jun 12 2017: (Start)
a(10*n+j) = 10*a(n)+j for 0<=j<=9 and n >= 1.
G.f. g(x) satisfies g(x) = 10*(1-x^10)*g(x^10)/(1-x) + (x + 2*x + ... + 9*x^9)*x^10/(1-x^10) + 12*x + 22*x^2 + ... + 92*x^9. (End)

A095790 Numbers whose name in English contains an "r".

Original entry on oeis.org

3, 4, 13, 14, 23, 24, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 53, 54, 63, 64, 73, 74, 83, 84, 93, 94, 103, 104, 113, 114, 123, 124, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148

Offset: 1

Michael Joseph Halm, Jul 10 2004

A008520 are numbers which contain an "e", A008540 an "f", A011538 a "g", A008536 an "n", A008519 an "o", A008538 an "s", A008522 a "t", A011534 a "u", A011532 a "w", A011536 an "x" and A008553 a "y"

			a(1) = 3 because "three" contains an "r", 0, 1 and 2 do not

Cf. A008520, A008540, A011538, A008536, A008519, A008538, A008522, A011534, A011532, A011536, A008553.

cp's OEIS Frontend

A257219 Numbers that have at least one divisor containing the digit 2 in base 10.

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A175688 Numbers k with property that arithmetic mean of its digits is both an integer and one of the digits of k.

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A043498 Numbers having two 2's in base 10.

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A381087 The smallest positive integer that produces a product that contains the digit 2 when multiplied by 2 a total of n times.

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A381183 a(n) = the smallest positive integer that produces a product that contains the digit 2 when multiplied by 2 at most n times, and where a further multiplication by 2 produces a number that does not contain the digit 2. Set a(n) = -1 if no such number exists.

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A043499 Numbers having three 2's in base 10.

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A208273 Composite numbers containing a digit 2.

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Maple

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A378138 The distinct values, in order of appearance, of A381087.

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A285470 Numbers k where "2" appears as the second digit of the decimal representation.

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