A091633 -id:A091633 - cp's OEIS frontend

A091871 A091633 indexed by A000040.

2, 4, 5, 6, 7, 8, 11, 12, 20, 21, 22, 25, 30, 32, 33, 34, 40, 41, 43, 44, 45, 46, 64, 65, 66, 67, 68, 74, 75, 78, 128, 130, 131, 137, 139, 156, 157, 159, 164, 165, 167, 168, 187, 193, 196, 215, 220, 222, 270, 275, 293, 294, 295, 298, 299, 301, 302, 303, 444, 446

Offset: 1

Ray Chandler, Feb 07 2004

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

Cf. A049084, A091633, A136333.

a091871 n = a091871_list !! (n-1)
a091871_list = f [1..] a000040_list where
   f (i:is) (p:ps) = if (null $ show p `intersect` "024568")
                        then i : f is ps else f is ps
-- Reinhard Zumkeller, Jul 18 2014

a(n)=k such that A000040(k) = A091633(n).

a(n) = A049084(A091633(n)). - Reinhard Zumkeller, Jul 18 2014

A030096 Primes whose digits are all odd.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 53, 59, 71, 73, 79, 97, 113, 131, 137, 139, 151, 157, 173, 179, 191, 193, 197, 199, 311, 313, 317, 331, 337, 353, 359, 373, 379, 397, 557, 571, 577, 593, 599, 719, 733, 739, 751, 757, 773, 797, 911, 919, 937, 953, 971, 977, 991

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Intersection of A000040 and A014261. Subsequence of A066640 and hence A014261. Subsequence of A038604. A091633 is a subsequence.

Cf. A076704 = odd-digit prime powers of prime numbers; A091296 = odd-digit semiprimes; A000040 = prime numbers; A001358 = semiprimes.

a030096 n = a030096_list !! (n-1)
a030096_list = filter f a000040_list where
   f x = odd d && (x < 10 || f x') where (x', d) = divMod x 10
-- Reinhard Zumkeller, Apr 07 2014, Jan 29 2013

[p: p in PrimesUpTo(1000) | forall{d: d in [0,2,4,6,8] | d notin Set(Intseq(p))}]; // Vincenzo Librandi, Apr 29 2019

Select[Prime[Range[500]],And@@OddQ[IntegerDigits[#]]&] (* Harvey P. Dale, Jan 28 2013 *)

is(n)=isprime(n) && #setintersect([0,2,4,6,8],Set(digits(n)))==0 \\ Charles R Greathouse IV, Feb 07 2017

Edited by N. J. A. Sloane at the suggestion of T. D. Noe and Jonathan Vos Post, Sep 15 2007

A068652 Numbers such that every cyclic permutation is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993

Offset: 1

Amarnath Murthy, Feb 28 2002

See the closely related sequence A016114 for further information. - N. J. A. Sloane, May 04 2017

These numbers are sometimes called circular primes. - Tanya Khovanova, Jul 29 2024

			197 is a member as all the three cyclic permutations 197,971,719 are primes.

Ray Chandler, Table of n, a(n) for n = 1..57
K. S. Brown, On General Palindromic Numbers
C. K. Caldwell, Circular Primes
Patrick De Geest, Circular Primes
H. Heinz, Prime Patterns (Illustration using 19937)
Gianni A. Sarcone, Tourbillonnants nombres premiers, Tangente Web Site, No date.
Wikipedia, Circular prime

Cf. A003459, A016114, A091633.

fQ[p_] := Module[{b = IntegerDigits[p]}, And @@ Table[PrimeQ[FromDigits[b = RotateLeft[b]]], {Length[b] - 1}]]; Select[Prime[Range[100000]], fQ] (* T. D. Noe, Mar 22 2012 *)
ecppQ[n_]:=AllTrue[FromDigits/@Table[RotateLeft[IntegerDigits[n],i],{i, IntegerLength[n]}],PrimeQ]; Select[Range[400000],ecppQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 25 2015 *)

More terms from Martin Renner, Apr 10 2002

A270083 Near-miss circular primes: Primes p where all but one of the numbers obtained by cyclically permuting the digits of p are prime.

Original entry on oeis.org

19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 101, 103, 107, 127, 149, 157, 163, 173, 181, 191, 271, 277, 307, 313, 317, 331, 359, 367, 379, 397, 419, 479, 491, 571, 577, 593, 617, 631, 673, 701, 709, 727, 739, 757, 761, 787, 797, 811, 839, 877, 907, 911

Offset: 1

Felix Fröhlich, Mar 10 2016

Prime p is a term of the sequence iff A262988(p) = A055642(p) - 1.
If a(512) exists, it is larger than 10^16. - Giovanni Resta, Apr 27 2017
If one of the digits is even or 5, the miss occurs when that digit is permuted to the ones place. Avoiding that simple obstruction, this sequence intersected with A091633 is 19, 173, 191, 313, 317, 331, 379, 397, 739, 797, 911, 937, 977, 1319, 1777, 1913, 1979, 1993, 3191, 3373, 3719, 3733, 3793, ... . Is this an infinite subsequence? - Danny Rorabaugh, May 15 2017

Felix Fröhlich and Giovanni Resta, Table of n, a(n) for n = 1..511 (first 487 terms from Felix Fröhlich)

Cf. A045978, A068652, A262988.

NearCircPrmsUpTo10powerK[k_]:= Union @ Flatten[ Table[ParallelMap[If[(Count[FromDigits /@ NestList[RotateLeft, IntegerDigits[#], IntegerLength[#]-1], ?PrimeQ] == IntegerLength[#]-1), #, Nothing] &, Select[FromDigits /@ Tuples[Range[0, 9], n], PrimeQ]], {n, k}], 1]; NearCircPrmsUpTo10powerK[7] (* _Mikk Heidemaa, Apr 26 2017 *)

rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
eva(n) = subst(Pol(n), x, 10)
is(n) = my(r=rot(digits(n)), i=0); while(r!=digits(n), if(ispseudoprime(eva(r)), i++); r=rot(r)); if(ispseudoprime(eva(r)), i++); if(n==1 || n==11, return(0)); if(i==#Str(n)-1, 1, 0)
forprime(p=1, 1e3, if(is(p), print1(p, ", ")))

A108386 Primes p such that p's set of distinct digits is {1,3,7,9}.

Original entry on oeis.org

1973, 3719, 3917, 7193, 9137, 9173, 9371, 13397, 13799, 13997, 17393, 17939, 19373, 19379, 19739, 19793, 19937, 19973, 31379, 31397, 31793, 31799, 31973, 33179, 33791, 37139, 37199, 37991, 39317, 39371, 39719, 39791, 39971, 71339, 71399

Offset: 1

Rick L. Shepherd, Jun 01 2005

The digits in {1,3,7,9} are the possible ending digits of multidigit primes. [Corrected by Lekraj Beedassy, Apr 04 2009]

Subsequence of A091633. - Michel Marcus, Jun 08 2014

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A108382 ({1, 3, 7}), A108383 ({1, 3, 9}), A108384 ({1, 7, 9}), A108385 ({3, 7, 9}), A030096 (Primes whose digits are all odd).

A136333 Numbers containing only digits coprime to 10 in their decimal representation.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 17, 19, 31, 33, 37, 39, 71, 73, 77, 79, 91, 93, 97, 99, 111, 113, 117, 119, 131, 133, 137, 139, 171, 173, 177, 179, 191, 193, 197, 199, 311, 313, 317, 319, 331, 333, 337, 339, 371, 373, 377, 379, 391, 393, 397, 399, 711, 713, 717, 719, 731

Offset: 1

Reinhard Zumkeller, Mar 26 2008

Numbers containing digits 1,3,7,9 only, or, numbers written in base 4 (cf. A007090) with digits mapped by: 0->1, 1->3, 2->7 and 3->9. - Reinhard Zumkeller, Jul 17 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 10-automatic sequences.

Cf. A007090, A091633 (primes), A245193.

import Data.List (intersect)
a136333 n = a136333_list !! (n-1)
a136333_list = filter (null . intersect "024568" . show) [1..]
-- Reinhard Zumkeller, Jul 17 2014

fQ[n_] := Block[{s = {1, 3, 7, 9}}, Union[Join[s, IntegerDigits@ n]] == s]; Select[ Range@ 1000, fQ] (* or *)
depth = 3; FromDigits@# & /@ FlattenAt[ Table[ Tuples[{1, 3, 7, 9}, n], {n, depth}], {#} & /@ Range[depth]] (* Robert G. Wilson v, Jul 02 2014 *)

isok(m) = my(d=digits(m)); apply(x->gcd(x, 10), d) == vector(#d, k, 1); \\ Michel Marcus, Feb 25 2022

Sum_{n>=1} 1/a(n) = 2.395867871130444522329053889312125689319669370758630349552737883715872077555... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

A245193 Smallest prime having in decimal representation A136333(n) as suffix.

Original entry on oeis.org

11, 3, 7, 19, 11, 13, 17, 19, 31, 233, 37, 139, 71, 73, 277, 79, 191, 193, 97, 199, 2111, 113, 1117, 3119, 131, 4133, 137, 139, 1171, 173, 4177, 179, 191, 193, 197, 199, 311, 313, 317, 1319, 331, 2333, 337, 2339, 2371, 373, 2377, 379, 3391, 2393, 397, 1399

Offset: 1

Reinhard Zumkeller, Jul 18 2014

a(n) = A136333(n) iff A136333(n) itself is a prime number, cf. A091633.

			.    n  |   a(n)  | A136333(n)
. ------+---------+-----------
.   10  |    233  |      33
.   11  |     37  |      37
.   12  |    139  |      39
.   13  |     71  |      71
.   14  |     73  |      73
.   15  |    277  |      77
.   16  |     79  |      79
.   17  |    191  |      91
.   18  |    193  |      93
.   19  |     97  |      97
.   20  |    199  |      99
.   21  |   2111  |     111
.   22  |    113  |     113
.   23  |   1117  |     117
.   24  |   3119  |     119
.   25  |    131  |     131
.   26  |   4133  |     133
.   27  |    137  |     137
.   28  |    139  |     139
.   29  |   1171  |     171
.   30  |    173  |     173 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000040, A091633, A136333.

import Data.List (isSuffixOf); import Data.Function (on)
a245193 n = head [p | p <- a000040_list,
                      (isSuffixOf `on` show) (a136333 n) p]

isok(m) = my(d=digits(m)); apply(x->gcd(x, 10), d) == vector(#d, k, 1); \\ A136333
f(m) = my(p=nextprime(m), s=10^#Str(m)); while ((p-m) % s, p = nextprime(p+1)); p;
lista(nn) = apply(x->f(x), select(isok, [1..nn]));
lista(1000) \\ Michel Marcus, Feb 25 2022

A281093 Primes having only {3, 4, 7, 9} as digits.

Original entry on oeis.org

3, 7, 37, 43, 47, 73, 79, 97, 337, 347, 349, 373, 379, 397, 433, 439, 443, 449, 479, 499, 733, 739, 743, 773, 797, 937, 947, 977, 997, 3343, 3347, 3373, 3433, 3449, 3499, 3733, 3739, 3779, 3793, 3797, 3943, 3947, 4337, 4339, 4349, 4373, 4397, 4447, 4493, 4733

Offset: 1

Bobby Jacobs, Jan 24 2017

The Fibonacci prime 433494437 is in this sequence.

These are primes that only use digits that survived in the movie 9: {3, 4, 7, 9}.

Jason Bard, Table of n, a(n) for n = 1..10000
Wikipedia, 9 (2009 animated film)
Index to entries for primes with digits in a given set

Cf. A091633.

is(n)=#setunion(Set(digits(n)), [3,4,7,9])==4 && isprime(n) \\ Charles R Greathouse IV, Jan 24 2017

a(n) >> n^k, where k = log(10)/log(4) = 1.660964.... - Charles R Greathouse IV, Jan 24 2017

A334050 Lexicographically earliest sequence of distinct positive integers such that the concatenation of any four successive digits forms a prime.

Original entry on oeis.org

1, 2, 3, 7, 19, 31, 9, 13, 73, 33, 137, 39, 311, 93, 11, 931, 91, 37, 331, 373, 313, 733, 191, 3733, 319, 1373, 3137, 333, 1913, 739, 3119, 337, 193, 119, 3191, 3739, 31193, 371, 933, 71, 9311, 9319, 13733, 13739, 31913, 7331, 913, 7333, 19137, 393, 1193, 3719, 3371, 9337, 1931, 1933, 719, 3373

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Sep 06 2020

			The first terms 1, 2, 3, 7, 19, 31, 9,... form (when 4 successive digits are concatenated) the prime numbers 1237, 2371, 3719, 7193, 1931, 9319, ...

Jean-Marc Falcoz, Table of n, a(n) for n = 1..500

Cf. A091633, A337613 (same idea, 2 successive digits), A337614 (3 successive digits).

A368051 Successive primes building the lexicographically earliest k X k 'Futility Squares' (see the Comment and Example sections for more explanations).

Original entry on oeis.org

13, 31, 113, 101, 313, 1117, 1009, 1019, 7993, 11113, 10007, 10009, 10039, 37997, 111119, 100003, 100019, 100129, 101207, 939973, 1111151, 1000003, 1000033, 1000037, 1000099, 5033981, 1337911, 11111117, 10000019, 10000079, 10000103, 10000121, 10011163, 11702641, 79931311

Offset: 1

Eric Angelini and Giorgos Kalogeropoulos, Dec 09 2023

A  k X k 'Futility Square' is a stack of k primes, each one being of length k. The 1st horizontal prime is also the 1st vertical one; the 2nd horizontal prime is also the 2nd vertical one, and so on. The first horizontal prime must be zeroless. The number on the main diagonal (running from top left to bottom right) is also a prime. The k + 1 primes involved in a k X k square must be distinct and the smallest possible not leading to a contradiction.
There might be more than one k X k 'Futility Square' for some k >= 2. For example another such square for k = 2 is
.
  6 7
  7 1
. - David A. Corneth, Dec 23 2023

			Here is the lexicographically earliest 3 X 3 'Futility Square':
.
  1 1 3
  1 0 1
  3 1 3
.
We see that 113, 101, 313 and the diagonal 103 are distinct primes.
Hereunder is the lexicographically earliest 6 X 6 'Futility Square':
.
  1 1 1 1 1 9
  1 0 0 0 0 3
  1 0 0 0 1 9
  1 0 0 1 2 9
  1 0 1 2 0 7
  9 3 9 9 7 3
.
We see that 111119, 100003, 100019, 100129, 101207, 939973 and the diagonal 100103 are distinct primes.
The sequence is formed by the two horizontal primes of the 2 X 2 square [13, 31], then the three horizontal primes of the 3 X 3 square [113, 101, 313], then the four horizontal primes of the 4 X 4 square [1117, 1009, 1019, 7993], etc.

Michael S. Branicky, Table of n, a(n) for n = 1..5049 (through k=101; terms 1..44 from Michael S. Branicky, terms 45..1539 from David A. Corneth)
Éric Angelini, Squaring Primes, Personal blog, December 2023.
Michael S. Branicky, Python program
David A. Corneth, PARI program
Futility Closet, Squaring Words, Futility Closet, December 2023.

Cf. A038618, A091633.

PARI
```
\\ See PARI link
```
Python
```
# See Python link
```

a(28)-a(35) for k=8 from Michael S. Branicky, Dec 23 2023

cp's OEIS Frontend

A091871 A091633 indexed by A000040.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Formula

A030096 Primes whose digits are all odd.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Extensions

A068652 Numbers such that every cyclic permutation is a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A270083 Near-miss circular primes: Primes p where all but one of the numbers obtained by cyclically permuting the digits of p are prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A108386 Primes p such that p's set of distinct digits is {1,3,7,9}.

Views

Author

Keywords

Comments

Links

Crossrefs

A136333 Numbers containing only digits coprime to 10 in their decimal representation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A245193 Smallest prime having in decimal representation A136333(n) as suffix.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

A281093 Primes having only {3, 4, 7, 9} as digits.

Views

Author

Keywords

Comments

Links

Crossrefs