A087513 -id:A087513 - 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

A087511 Primes consisting only of digits 1 and 3 occurring with equal frequency.

Original entry on oeis.org

13, 31, 11313331, 11333131, 13111333, 13131133, 13131331, 13133311, 13311313, 31133131, 33113131, 1113131333, 1131131333, 1131311333, 1131331133, 1133111333, 1133113133, 1133133311, 1133311313, 1133313113, 1133313131, 1133331113, 1311113333, 1311311333, 1311313313

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). - corrected by Robert Israel, Jul 10 2018

The number of digits is even and not divisible by 3. - Robert Israel, Jul 09 2018

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

Cf. A087510, A087512, A087513.

sort(select(isprime,[seq(seq((10^(2*d)-1)/9+2*add(10^i, i=s), s=combinat:-choose([$0..(2*d-1)], d)), d=[1,2,4,5,7,8,10])])); # Robert Israel, Jul 09 2018

Union[FromDigits/@Select[Flatten[Table[Tuples[{1,3},k],{k,10}],1], PrimeQ[FromDigits[#]] && Count[#,1]==Count[#,3] &]] (* Jayanta Basu, May 19 2013 *)

\\ Needs B() from A087510.
concat(vector(6,k,B(k,1,3,isprime))) \\ Andrew Howroyd, Sep 21 2024

A087512 Primes consisting only of digits 1 and 4 occurring with equal frequency.

Original entry on oeis.org

41, 14441411, 41414411, 44114141, 1144141441, 1144144411, 1144441141, 1411414441, 1441411441, 1441444111, 1444114141, 4111144441, 4111444411, 4114414141, 4141141441, 4141144141, 4144111441, 4411114441, 4411144141, 4411414411, 4414141141, 4414441111

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).

Andrew Howroyd, Table of n, a(n) for n = 1..5319 (first 5001 terms from Harvey P. Dale)

Cf. A087510, A087511, A087513.

Union[Flatten[Table[Select[FromDigits/@Permutations[Flatten[Table[{1,4},{n}]]],PrimeQ],{n,5}]]] (* Harvey P. Dale, Nov 21 2014 *)

\\ Needs B() from A087510.
concat(vector(6,k,B(k,1,4,isprime))) \\ Andrew Howroyd, Sep 21 2024

More terms from Harvey P. Dale, Nov 21 2014

Offset changed by Andrew Howroyd, Sep 21 2024

A087514 Primes consisting only of digits 1 and 7 occurring with equal frequency.

Original entry on oeis.org

17, 71, 11171777, 11177717, 17111777, 17171177, 17171771, 17177117, 17711717, 17717171, 71117177, 71171717, 71717117, 77111717, 77711171, 1111717777, 1117117777, 1117177177, 1117771177, 1117771717, 1177117177, 1177177711, 1177711177, 1177717171, 1177717711

Offset: 1

Amarnath Murthy and Paul D. Hanna, 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).

Andrew Howroyd, Table of n, a(n) for n = 1..10444 (first 1001 terms from Harvey P. Dale)

Cf. A087510, A087511, A087513.

Sort[Select[FromDigits/@Flatten[Table[Permutations[PadRight[{},2n,{1,7}]],{n,5}],1],PrimeQ]] (* Harvey P. Dale, Jul 27 2019 *)

\\ Needs B() from A087510.
concat(vector(6,k,B(k,1,7,isprime))) \\ Andrew Howroyd, Sep 21 2024

Offset changed by Andrew Howroyd, Sep 21 2024

A214524 List of words over {1,6} with equal numbers of 1's and 6's.

Original entry on oeis.org

16, 61, 1166, 1616, 1661, 6116, 6161, 6611, 111666, 116166, 116616, 116661, 161166, 161616, 161661, 166116, 166161, 166611, 611166, 611616, 611661, 616116, 616161, 616611, 661116, 661161, 661611, 666111

Offset: 1

Jonathan Vos Post, Jul 19 2012

This 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 A087513.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A214218, A214488, A214521, A214522.

Sort[FromDigits/@Flatten[Table[Permutations[PadRight[{},2n,{1,6}]],{n,3}],1]] (* Harvey P. Dale, May 11 2015 *)

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

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

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A087511 Primes consisting only of digits 1 and 3 occurring with equal frequency.

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A087512 Primes consisting only of digits 1 and 4 occurring with equal frequency.

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A087514 Primes consisting only of digits 1 and 7 occurring with equal frequency.

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A214524 List of words over {1,6} with equal numbers of 1's and 6's.

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

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