A101594 -id:A101594 - cp's OEIS frontend

A083983 If k is a number with exactly two distinct decimal digits, say a and b, neither of which is 0 (i.e., a member of A101594), define the self-complement of k, SC(k), to be the number obtained by replacing a with b and vice versa. E.g. SC(232233) = 323322. Sequence contains primes p such that SC(p) is also a prime.

13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 191, 199, 313, 331, 337, 773, 911, 919, 1171, 7717, 11113, 11119, 11177, 11717, 33331, 77171, 77711, 79999, 97777, 99991, 113111, 131111, 131113, 131311, 191999, 199999, 313133, 313331, 313333, 331333, 333737, 377737

Offset: 1

Amarnath Murthy, May 22 2003

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

More terms from David Wasserman, Dec 07 2004

Offset corrected by Arkadiusz Wesolowski, Oct 17 2011

A083985 If k is a number with exactly two distinct decimal digits, say a and b, neither of which is 0 (i.e., a member of A101594), define the self-complement of k, SC(k), to be the number obtained by replacing a with b and vice versa. Then a(n) = gcd(A101594(n), SC(A101594(n))).

Original entry on oeis.org

3, 1, 1, 3, 1, 1, 9, 1, 3, 1, 6, 1, 2, 9, 2, 1, 1, 1, 1, 1, 9, 1, 1, 3, 1, 6, 1, 9, 2, 1, 12, 1, 3, 1, 1, 9, 1, 3, 1, 1, 1, 2, 9, 2, 1, 1, 2, 3, 1, 9, 1, 1, 3, 1, 3, 1, 9, 2, 1, 12, 1, 2, 3, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 7, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 6

Offset: 1

Amarnath Murthy, May 22 2003

			12 is the term corresponding to k = 48 as (48,84) = 1
a(7) = gcd(18, 81) = 9.

Cf. A083983, A083984, A083986.

Cf. A101594.

Corrected and extended by David Wasserman, Dec 07 2004

Offset corrected by Andrew Howroyd, Sep 29 2024

A083984 If k is a number with exactly two distinct decimal digits, say a and b, neither of which is 0 (i.e., a member of A101594), define the self-complement of k, SC(k), to be the number obtained by replacing a with b and vice versa. E.g. SC(232233) = 323322. Sequence contains numbers n such that n and SC(n) are relatively prime.

Original entry on oeis.org

13, 14, 16, 17, 19, 23, 25, 29, 31, 32, 34, 35, 37, 38, 41, 43, 47, 49, 52, 53, 56, 58, 59, 61, 65, 67, 71, 73, 74, 76, 79, 83, 85, 89, 91, 92, 94, 95, 97, 98, 112, 113, 115, 116, 118, 119, 121, 122, 131, 133, 151, 155, 166, 181, 188, 191, 199, 211, 212, 221, 223, 227

Offset: 1

Amarnath Murthy, May 22 2003

			25 is a member as 25 and 52 are relatively prime, but 24 is not a member as 24 and 42 are not coprime.

Cf. A083983, A083985, A083986.

More terms from David Wasserman, Dec 07 2004

Offset corrected by Andrew Howroyd, Sep 29 2024

A083986 If k is a number with exactly two distinct decimal digits, say a and b, neither of which is 0 (i.e., a member of A101594), define the self-complement of k, SC(k), to be the number obtained by replacing a with b and vice versa. Then a(n) = lcm(A101594(n), SC(A101594(n))).

Original entry on oeis.org

84, 403, 574, 255, 976, 1207, 162, 1729, 84, 736, 168, 1300, 806, 216, 1148, 2668, 403, 736, 1462, 1855, 252, 2701, 3154, 1209, 574, 168, 1462, 270, 1472, 3478, 336, 4606, 255, 1300, 1855, 270, 3640, 1425, 4930, 5605, 976, 806, 252, 1472, 3640, 5092, 2924

Offset: 1

Amarnath Murthy, May 22 2003

			a(7) = lcm(18, 81) = 162.

Cf. A083983, A083984, A083985.

Cf. A101594.

Corrected and extended by David Wasserman, Dec 07 2004

Offset corrected by Andrew Howroyd, Sep 29 2024

A031955 Numbers with exactly two distinct base-10 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, 100, 101, 110, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 133, 141, 144, 151, 155, 161, 166

Offset: 1

Clark Kimberling

The three-digit terms are given by A210666(1,...,244). For numbers with exactly two distinct (but unspecified) digits in other bases, see A031948-A031954. For numbers made of two *given* digits, see A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9), and A032804-A032816 (in other bases). - M. F. Hasler, Apr 04 2015
A235154 is a subsequence. - Altug Alkan, Dec 03 2015
A235717 is a subsequence. - Robert Israel, Dec 03 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Different from A029742.

Cf. A043638, A101594, A031948, A031949, A031950, A031951, A031952, A031953, A031954, A235154, A235717.

a031955 n = a031955_list !! (n-1)
a031955_list = filter ((== 2) . a043537) [0..]
-- Reinhard Zumkeller, Feb 05 2012

M:= 5: # to get all terms < 10^M
sort([seq(seq(seq(seq(add(10^(m-j)*`if`(member(j,S2),d2,d1),j=1..m)  ,
  S2 = combinat:-powerset({$2..m}) minus {{}}),
  d2 = {$0..9} minus {d1}), d1 = 1..9), m=2..M)]); # Robert Israel, Dec 03 2015

Select[Range@ 166, Length@ Union@ IntegerDigits@ # == 2 &] (* Michael De Vlieger, Dec 03 2015 *)

is_A031955(n)=#Set(digits(n))==2 \\ M. F. Hasler, Apr 04 2015

def ok(n): return len(set(str(n))) == 2
print(list(filter(ok, range(167)))) # Michael S. Branicky, Oct 12 2021

A043537(a(n)) = 2. - Reinhard Zumkeller, Dec 03 2009

Name edited by Charles R Greathouse IV, Feb 13 2017

A125290 Numbers with at least two distinct digits in decimal representation, none of which is 0.

Original entry on oeis.org

12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123

Offset: 1

Reinhard Zumkeller, Nov 26 2006

Also numbers having at least two partitions into digit values of their decimal representations: A061827(a(n)) > 1.

First differs from A101594 at a(83) = 123 != 131 = A101594(83). - Michael S. Branicky, Dec 13 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Subsequence of A052382. Supersequence of A101594.

Cf. A125293, A004719, A043537, A061827, A168046.

a125290 n = a125290_list !! (n-1)
a125290_list = filter ((> 1) . a043537) a052382_list
-- Reinhard Zumkeller, Jun 18 2013

Select[Range[200], FreeQ[#, 0] && Length[Union[#]] > 1 & [IntegerDigits[#]] &] (* Paolo Xausa, May 06 2024 *)

def ok(n): s = set(str(n)); return len(s) >= 2 and "0" not in s
print([k for k in range(124) if ok(k)]) # Michael S. Branicky, Dec 13 2021

A043537(A004719(a(n))) > 1.
A168046(a(n)) * A043537(A004719(a(n))) > 1. - Reinhard Zumkeller, Jun 18 2013
a(n) ~ n. - Charles R Greathouse IV, Feb 13 2017

Name clarified by Michael S. Branicky, Dec 13 2021

A370849 Least of the smoothest two-nonzero-digit numbers of length n.

Original entry on oeis.org

16, 144, 3888, 55566, 255552, 1111222, 76776777, 799779977, 4334433444, 61161166611, 292229292292, 1122121111111, 55115551555155, 799777779779979, 1161111111166611, 11112112121222112, 111111222221111112, 3334334333334333333, 55333333335335355355, 222229999999292992929, 3383383883833883388888, 11112221111212222222221, 112122222222122122122112, 2777227772777277722272272, 61666616611611166166161116, 858885585585555585558558858, 3331333133331111313111133133, 98888999899889989898999889999, 111661111111666616661166166616

Offset: 2

Ed Pegg Jr and M. F. Hasler, Mar 02 2024

"Least" means that we list the smallest one if there is more than one solution of length n having the same smoothness. "Smoothest" means having the least greatest prime factor, A006530. Length means the number of digits in base 10. We consider only nonzero digits since otherwise the somewhat uninteresting solution would most often be 10^(n-1) = (2*5)^(n-1). [Alternatively, one might exclude those solutions by only forbidding multiples of 10: see below.]
The two digits are coprime. - David A. Corneth, Mar 05 2024
In an alternate sequence forbidding multiples of 10, 101010110010001010011 replaces 222229999999292992929. - Ed Pegg Jr, Mar 05 2024

			a(2) = 16 = 2^4 is certainly the smallest number made of 2 distinct nonzero digits that has the least largest prime factor. 32 and 64 would have the same smoothness, but we list the smallest solution
a(3) = 144 = 2^4*3^2 is the least 3-digit number made of 2 distinct nonzero digits that has the least largest prime factor, here 3. (288 would have the same smoothness.)
a(4) = 3888 = 2^4*3^5 and 7776 = 2^5*3^5 are the smoothest 4-digit numbers made of 2 distinct nonzero digits.
For n = 7 digits, all of {1111222, 2222444, 3333666, 4444888, 5665556, 7777887} have the same minimum smoothness of 29.
Similarly, for n = 10, all of {4334433444, 4444994444, 8668866888, 8889988888} have the same minimum smoothness of 23 (and all of them also have prime factors 2, 11 and 19; the first and third are also divisible by 3^4, the two others have a second factor 19 and four factors 23).

Ed Pegg Jr, A smooth, 2-digit sequence, Mar 01 2024

Cf. A006530 (greatest prime factor), A101594 (zeroless numbers with exactly 2 distinct digits).

Cf. A370361 (greatest prime factor of the terms).

a(n)={my(s=oo,L); forvec(d=vector(2,i,[1,9]), gcd(d)>1&&next; my(g, f(v) = fromdigits(vecextract(d,v))); forvec(v=vector(n,i,[1,2]), if(s < g=A006530(f(v)), next, s == g, L=concat(L,f(v)), s=g, L=[f(v)])),2); vecmin(L)}

from sympy import factorint
from itertools import combinations
from sympy.utilities.iterables import multiset_permutations
def a(n):
    m = (int('9'*n),)*2
    for c in combinations("123456789", 2):
        for r in multiset_permutations(c[0]*n+c[1]*n, n):
            t = int("".join(r))
            s = max(factorint(t, limit=m[0]))
            m = min(m, (s, t))
    return m[1]
print([a(n) for n in range(2, 12)]) # Michael S. Branicky, Mar 03 2024

a(21)-a(23) from Michael S. Branicky, Mar 05 2024
a(24)-a(25) from David A. Corneth, Mar 05 2024
a(26)-a(30) from Don Reble, Mar 06 2024

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A083985 If k is a number with exactly two distinct decimal digits, say a and b, neither of which is 0 (i.e., a member of A101594), define the self-complement of k, SC(k), to be the number obtained by replacing a with b and vice versa. Then a(n) = gcd(A101594(n), SC(A101594(n))).

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A083986 If k is a number with exactly two distinct decimal digits, say a and b, neither of which is 0 (i.e., a member of A101594), define the self-complement of k, SC(k), to be the number obtained by replacing a with b and vice versa. Then a(n) = lcm(A101594(n), SC(A101594(n))).

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A031955 Numbers with exactly two distinct base-10 digits.

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A125290 Numbers with at least two distinct digits in decimal representation, none of which is 0.

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A370849 Least of the smoothest two-nonzero-digit numbers of length n.

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