A071925 -id:A071925 - cp's OEIS frontend

A087510 Primes consisting only of digits 0 and 1 occurring with equal frequency.

10010101, 10100011, 1000011011, 1000110101, 1001000111, 1001001011, 1001010011, 1010000111, 1010001101, 1010010011, 1010100011, 1010110001, 1011000101, 1100001101, 1101001001, 10000101011101, 10000111100011, 10000111110001, 10001000011111, 10001001011011

Offset: 1

Paul D. Hanna and Amarnath Murthy, Sep 11 2003

There are 18 digit pairs which can produce such primes: (1,0),(1,3),(1,4),(1,6),(1,7),(1,9),(2,3),(2,9),(3,4),(3,5),(3,7),(3,8),(4,7),(4,9),(5,9),(6,7),(7,9),(8,9).

Alois P. Heinz, Table of n, a(n) for n = 1..18167 (first 1000 terms from T. D. Noe)

Primes in A071925.

The 18 sequences in this family are: this sequence (1,0), A087511 (1,3), A087512 (1,4), A087513 (1,6), A087514 (1,7), A087515 (1,9), A087527 (2,3), A087528 (2,9), A087529 (3,4), A087530 (3,5), A087531 (3,7), A087532 (3,8), A087533 (4,7), A087534 (4,9), A087535 (5,9), A087536 (6,7), A087537 (7,9), A087538 (8,9).

Select[FromDigits/@Tuples[{0,1},14],PrimeQ[#] && Length[x=IntegerDigits[#]]==2*Count[x,0] &] (* Jayanta Basu, May 23 2013 *)

\\ B(k,d1,d2,pred) k-digits of (d1,d2) each, satisfying pred.
B(k,d1,d2,pred)={my(L=List(),m=10^(2*k-1)); forsubset([2*k,k], s, my(t=(10^(2*k)-1)/9*d1 + (d2-d1)*sum(i=1, #s, 10^(s[i]-1))); if(t>=m && pred(t), listput(L,t))); vecsort(Vec(L))}
{ concat(vector(7,k,B(k,0,1,isprime)))[1..20] } \\ Andrew Howroyd, Sep 20 2024

A182040 Integers whose decimal representation consists of three distinct digits, one appearing once, one appearing twice, and one appearing three times.

Original entry on oeis.org

100012, 100013, 100014, 100015, 100016, 100017, 100018, 100019, 100021, 100022, 100031, 100033, 100041, 100044, 100051, 100055, 100061, 100066, 100071, 100077, 100081, 100088, 100091, 100099, 100102, 100103, 100104, 100105, 100106, 100107, 100108, 100109, 100112

Offset: 1

Jonathan Vos Post, Apr 09 2012

There are 38880 terms, including 41 squares (A182098) and 3640 primes (A182092). - Zak Seidov, Apr 12 2012

This is the subsequence of A218556 consisting of terms with indices n = 254, ..., 39133. The number of terms is 38880 = A218566(10,3), the starting index is 254 = 1 + A218566(10,1) + A218566(10,2) + 1. - M. F. Hasler, Nov 02 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..38880 (complete sequence)

Cf. A071925, A181986 (digitally balanced numbers: ternary numbers which have the same number of 0's as 1's as 2's), A182051 (primes with a majority of one digit).

t = Select[Range[100000, 999999], Sort[Transpose[Tally[IntegerDigits[#]]][[2]]] == {1, 2, 3} &]; Take[t, 32] (* T. D. Noe, Apr 11 2012 *)

is(n)=n=vecsort(eval(Vec(Str(n))));vecsort(apply(k->sum(i=1, #n,n[i]==k),vecsort(n,,8)))==[1,2,3] \\ Charles R Greathouse IV, Apr 11 2012

A218556 Numbers with d distinct decimal digits (d=1,...,10) such that for each k=1,...,d, some digit occurs exactly k times.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 101, 110, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 133, 141, 144, 151, 155, 161, 166, 171, 177, 181, 188, 191, 199, 200, 202, 211, 212, 220, 221, 223, 224, 225, 226, 227, 228, 229, 232, 233, 242, 244, 252, 255, 262, 266, 272, 277, 282, 288, 292, 299, 300, 303, 311, 313, 322, 323, 330

Offset: 1

M. F. Hasler, Nov 02 2012

For each of the terms, the number of digits is a triangular number A000217.
The number of terms with d = 1,2,3,... different digits is 10, 243, 38880, ... = A218566(10,d) (+ 1 for d=1, accounting for the initial 0).
The sequence is finite, it has N = 1 + sum_{i=1..10} A218566(10,i) = 9083370609101493843078695864582213215764827510991133 terms. The last term is a(N) = 9999999999888888888777777776666666555555444443333222110 (ten "9"s, nine "8"s, ..., one "0").

			The terms a(1)=0 through a(10)=9 have exactly 1 digit occurring exactly once.
The terms a(11)=100 through a(253)=998, listed in A210666, have one digit occurring once and a second, different digit occurring exactly twice.
The terms a(254)=100012 through a(39133)=999887 are listed in A182040.
For d=4, we have the (1+2+3+4 =) 10-digit terms a(39134)=1000011223 through 9999888776 with 4 different digits which occur with frequencies 1,2,3 and 4.

Cf. A071925, A181986, A182040, A182092.

{my(T(n)=n*(n+1)\2); print1(0); for(i=1,2, s=vector(i+1,j,j-1); for(n=10^(T(i)-1),10^T(i)-1,i !=#Set(digits(n)) & next; c=vector(10); for(j=1,#d=digits(n),c[d[j]+1]++); vecsort(c,,8)==s & print1(","n)))}

is_A218556(n)={ my(c=vector(10)); for(i=1,#n=digits(n),c[n[i]+1]++); #(c=vecsort(c,,8))==1+c[#c] && 2*#n==c[#c]*#c }

A181986 Digitally balanced numbers: ternary numbers which have the same number of 0's as 1's as 2's.

Original entry on oeis.org

102, 120, 201, 210, 100122, 100212, 100221, 101022, 101202, 101220, 102012, 102021, 102102, 102120, 102201, 102210, 110022, 110202, 110220, 112002, 112020, 112200, 120012, 120021, 120102, 120120, 120201, 120210, 121002, 121020, 121200, 122001, 122010, 122100

Offset: 1

Jonathan Vos Post, Apr 04 2012

This is to A071925 as base 3 A007089 is to base 2 A007088. A049354 digitally balanced numbers in base 3: equal numbers of 0's, 1's, 2's, beginning 11, 15, 19, 21, 260, ... is the same sequence, but expressed in base 10. The terms of this sequence are represented directly in base 3.

			100122 is an element because it contains two each of "0" and "1" and "2".

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A007089, A031443, A049354, A071925.

Select[FromDigits/@Flatten[Permutations/@Table[PadRight[{},3n,{0,1,2}],{n,2}],1],Mod[IntegerLength[#],3]==0&]//Sort (* Harvey P. Dale, Mar 15 2020 *)

More terms from Alois P. Heinz, Apr 05 2012

A143906 a(n) is A143905(n) written in binary.

Original entry on oeis.org

1001, 10011001, 10100101, 11000011, 100011110001, 100101101001, 100110011001, 101001100101, 101010010101, 101100001101, 110001100011, 110010010011, 110100001011, 111000000111, 1000011111100001, 1000101111010001

Offset: 1

Leroy Quet, Sep 04 2008

This sequence lists all finite binary strings that are palindromes with an equal number of 0's and 1's and that each start and end with 1, listed in order of the strings' numerical values if the strings are considered to be base 2 numbers.

Cf. A006995, A031443, A143905.

Cf. A057148, A071925.

Intersection of A057148 and A071925. - R. J. Mathar, Sep 05 2008

Extended by R. J. Mathar, Sep 05 2008

Corrected comment. - Leroy Quet, Oct 13 2008

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A087510 Primes consisting only of digits 0 and 1 occurring with equal frequency.

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A182040 Integers whose decimal representation consists of three distinct digits, one appearing once, one appearing twice, and one appearing three times.

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A218556 Numbers with d distinct decimal digits (d=1,...,10) such that for each k=1,...,d, some digit occurs exactly k times.

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A181986 Digitally balanced numbers: ternary numbers which have the same number of 0's as 1's as 2's.

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A143906 a(n) is A143905(n) written in binary.

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