A087515 -id:A087515 - 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

A087527 Primes consisting only of digits 2 and 3 occurring with equal frequency.

Original entry on oeis.org

23, 23223323, 32323223, 2222323333, 2223223333, 2232223333, 2232322333, 2232332233, 2323222333, 2332322233, 2333222323, 2333223223, 3223232323, 3232222333, 3232232233, 3232233223, 3232322323, 3323232223, 22222232333333, 22222322333333, 22222323233333

Offset: 1

Paul D. Hanna and Amarnath Murthy, Sep 12 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).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A087510, A087511, A087515.

\\ Needs B() from A087510.
concat(vector(6,k,B(k,2,3,isprime))) \\ Andrew Howroyd, Sep 21 2024

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def auptodigs(maxdigits):
    alst = []
    for d in range(2, maxdigits + 1, 2):
        ms = "2"*(d//2) + "3"*(d//2 - 1)
        for p in multiset_permutations(ms, d-1):
            t = int("".join(p) + "3")
            if isprime(t):
                alst.append(t)
    return alst
print(auptodigs(10)) # Michael S. Branicky, Jan 11 2022

Offset changed to 1 and a(19) corrected by Georg Fischer, Jan 11 2022

A199306 Palindromic primes in the sense of A007500 with digits '0', '1' and '6' only.

Original entry on oeis.org

11, 101, 1061, 1601, 10061, 10601, 11161, 16001, 16061, 16111, 16661, 101611, 106661, 116101, 166601, 1011601, 1016011, 1016611, 1061101, 1066111, 1106101, 1110611, 1111661, 1116601, 1160111, 1160611, 1166101, 1600061, 1611161, 1616161, 1660661, 1661111, 10011101, 10100161, 10106111

Offset: 1

M. F. Hasler, Nov 06 2011

All terms start and end with the digit '1'. This fact is used in the given PARI program.

Cf. A020449 - A020472, A062350, A087510 - A087515, A199325 - A199329.

[p: p in PrimesUpTo(10^8) | Set(Intseq(p)) subset [0,1,6] and IsPrime(Seqint(Reverse(Intseq(p))))]; // Bruno Berselli, Nov 07 2011

a(n=50,list=0,L=[0,1,6])={ for(d=1,1e9, my(t,u=vector(d-1,i,10^(d-i))~,o=10^d+1);forvec(v=vector(#u,i,[1,#L]),isprime(t=o+vector(#u,i,L[v[i]])*u) || next; isprime(A004086(t)) || next; list & print1(t", "); n-- || return(t)))}  \\ M. F. Hasler, Nov 07 2011

A173002 Primes consisting of two digits only, each digit with frequency f = 4.

Original entry on oeis.org

10010101, 11171777, 11177717, 11313331, 11333131, 11919199, 11919991, 13111333, 13131133, 13131331, 13133311, 13311313, 14441411, 16166611, 16616161, 17111777, 17171177, 17171771, 17177117, 17711717, 17717171

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 07 2010

2 digits, f = 1: 20 primes p 11 < p < =97: 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
2 digits, f = 2: no primes as abab has divisor 101, abba and aabb divisor 11
2 digits, f = 3: no primes as sum of digits 3 * (a+b)
2 digits, f = 4: there are 18 possibilities for (a,b):
(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)
Each possibility occurs, 2+9+3+5+13+11+2+6+3+3+10+2+2+5+2+2+6+4 = 90 = 2 * 3^2 * 5 primes

			Complete list classified according to the 18 possible "pairs":
10010101, 10011101
11313331, 11333131, 13111333, 13131133, 13131331, 13133311, 13311313, 31133131, 33113131
14441411, 41414411, 44114141
16166611, 16616161, 61116661, 61661161, 66161611
11171777, 11177717, 17111777, 17171177, 17171771, 17177117, 17711717, 17717171, 71117177, 71171717, 71717117, 77111717, 77711171
11919199, 11919991, 19111999, 19199119, 19911919, 19991911, 91919911, 91999111, 99111919, 99119191, 99919111
23223323, 32323223
22929299, 29229929, 29299229, 29992229, 92922299, 99292229
34434343, 44334343, 44343433
35553533, 53355353, 53533553
33373777, 33773737, 37373773, 37377337, 73337377, 73337773, 73373737, 73773373, 77337373, 77733373
38383883, 88838333
47447747, 77474447
44994949, 49444999, 49494499, 49499449, 94449499
55599959, 99555959
67766767, 76767667
77997979, 79779979, 79797997, 79997977, 99977797, 99979777
88989899, 98988899, 98989889, 99898889

Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005
Wladyslaw Narkiewicz: The development of prime number theory: from Euclid to Hardy and Littlewood, Springer Monographs in Mathematics, Berlin, New York, 2000
Paulo Ribenboim: The little book of bigger primes, Springer Berlin, New York, 2004

A087511, A087512, A087513, A087514, A087515, A087527, A087528, A087529, A087530, A087531, A087532, A087533, A087534, A087535, A087536, A087537, A087538

Second entry 10011101 deleted (does not comply with definition) and a new term added at the end. Lekraj Beedassy, Jul 17 2010

A199305 Palindromic primes in the sense of A007500 with digits '0', '1' and '5' only.

Original entry on oeis.org

5, 11, 101, 151, 1151, 1511, 10151, 10501, 11551, 15101, 15511, 15551, 100511, 110051, 115001, 150011, 150151, 151051, 1001551, 1051051, 1055501, 1115551, 1150151, 1150511, 1501501, 1510511, 1550551, 1551001, 1551551, 1555111, 10000511, 10011101, 10011511, 10055011, 10101551

Offset: 1

M. F. Hasler, Nov 06 2011

All terms, except for the initial 5, start and end with the digit '1'. This fact could be used to significantly speed up the given program.

Cf. A020449-A020472, A199325-A199329, A087510-A087515.

[p: p in PrimesUpTo(10^8) | Set(Intseq(p)) subset [0,1,5] and IsPrime(Seqint(Reverse(Intseq(p))))]; // Bruno Berselli, Nov 07 2011

a(n=50, list=0, L=[0, 1, 5], needpal=1)={ for(d=1, 1e9, u=vector(d, i, 10^(d-i))~; forvec(v=vector(d, i, [1+(i==1&!L[1]), #L]), isprime(t=vector(d, i, L[v[i]])*u) || next; needpal & !isprime(A004086(t)) & next; list & print1(t", "); n-- || return(t)))}  \\ M. F. Hasler, Nov 06 2011

A214532 List of words over {1,9} with equal numbers of 1's and 9's.

Original entry on oeis.org

19, 91, 1199, 1919, 1991, 9119, 9191, 9911, 111999, 119199, 119919, 119991, 191199, 191919, 191991, 199119, 199191, 199911, 911199, 911919, 911991, 919119, 919191, 919911, 991119, 991191, 991911, 999111, 11119999, 11191999

Offset: 1

Jonathan Vos Post, Jul 20 2012

This is to 9 as A214531 is to 8, as A214530 is to 7, as A214530 is to 6, as A214522 is to 5, as A214521 is to 4, as A214488 is to 3, and as A214218 is to 2.

The subsequence of primes (when interpreted as decimal integers) is A087515.

Cf. A214218, A214488, A214521, A214522, A214524, A214530, A214531.

FromDigits/@(Flatten[Permutations/@Table[Join[Table[1,{n}],Table[9,{n}]],{n,4}],1]) (* Harvey P. Dale, Aug 20 2014 *)

cp's OEIS Frontend

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

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A087527 Primes consisting only of digits 2 and 3 occurring with equal frequency.

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A199306 Palindromic primes in the sense of A007500 with digits '0', '1' and '6' only.

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A173002 Primes consisting of two digits only, each digit with frequency f = 4.

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A199305 Palindromic primes in the sense of A007500 with digits '0', '1' and '5' only.

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Magma

PARI

A214532 List of words over {1,9} with equal numbers of 1's and 9's.

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Mathematica