A050249 Weakly prime numbers (changing any one decimal digit always produces a composite number). Also called digitally delicate primes.

294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139, 5152507, 5564453, 5575259, 6173731, 6191371, 6236179, 6463267, 6712591, 7204777, 7469789, 7469797

Offset

1,1

Comments

Tao proved that this sequence is infinite. - T. D. Noe, Mar 01 2011

For k = 5, 6, 7, 8, 9, 10, the number of terms < 10^k in this sequence is 0, 5, 35, 334, 3167, 32323. - Jean-Marc Rebert, Nov 10 2015

References

Michael Filaseta and Jeremiah Southwick, Primes that become composite after changing an arbitrary digit, Math. Comp. (2021) Vol. 90, 979-993. doi:10.1090/mcom/3593

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (terms 1..1317 from Klaus Brockhaus, terms 1318..3167 from Jean-Marc Rebert).

Michael Filaseta and Jacob Juillerat, Consecutive primes which are widely digitally delicate, arXiv:2101.08898 [math.NT], 2021.

Jon Grantham, Finding a Widely Digitally Delicate Prime, arXiv:2109.03923 [math.NT], 2021.

Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Jackson Hopper and Paul Pollack, Digitally delicate primes, arXiv:1510.03401 [math.NT], 2015.

Dana Jacobsen, Digitally delicate primes up to 1e11

Jeremiah T. Southwick, Two Inquiries Related to the Digits of Prime Numbers, Ph. D. Dissertation, University of South Carolina (2020).

Terence Tao, A remark on primality testing and decimal expansions, arXiv:0802.3361 [math.NT], 2008-2010; Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.

Eric Weisstein's World of Mathematics, Weakly Prime

Mathematica

fQ[n_] := Block[{d = IntegerDigits@ n, t = {}}, Do[AppendTo[t, FromDigits@ ReplacePart[d, i -> #] & /@ DeleteCases[Range[0, 9], x_ /; x == d[[i]]]], {i, Length@ d}]; ! AnyTrue[Flatten@ t, PrimeQ]] ; Select[Prime@ Range[10^5], fQ] (* Michael De Vlieger, Nov 10 2015, Version 10 *)

Prog

(Magma) IsA118118:=function(n); D:=Intseq(n); return forall{ <k, j>: k in [1..#D], j in [0..9] | j eq D[k] or not IsPrime(Seqint(S)) where S:=Insert(D, k, k, [j]) }; end function; [ p: p in PrimesUpTo(8000000) | IsA118118(p) ]; // Klaus Brockhaus, Feb 28 2011
(PARI) isokp(n) = {v = digits(n); for (k=1, #v, w = v; for (j=0, 9, if (j != v[k], w[k] = j; ntest = subst(Pol(w), x, 10); if (isprime(ntest), return(0)); ); ); ); return (1); }
lista(nn) = {forprime(p=2, nn, if (isokp(p), print1(p, ", ")); ); } \\ Michel Marcus, Dec 15 2015
(Python)
from sympy import isprime
def h1(n): # hamming distance 1 neighbors of n
    s = str(n); d = "0123456789"; L = len(s)
    yield from (int(s[:i]+c+s[i+1:]) for c in d for i in range(L) if c!=s[i])
def ok(n): return isprime(n) and all(not isprime(k) for k in h1(n) if k!=n)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jun 19 2022

Crossrefs

Cf. A118118, A158124 (weakly primes), A158125 (weakly primes).

Cf. A137985 (analogous base-2 sequence), A186995 (weak primes in base n).

Keyword

nonn,base

Author

Eric W. Weisstein

Extensions

Edited by Charles R Greathouse IV, Aug 02 2010

Status

approved

A347424 Digitally delicate truncatable primes: every suffix is prime, changing any one decimal digit always produces a composite number, except the first to zero.

7810223, 19579907, 909001523, 984960937, 78406036607, 90124536947, 99020400307, 190002706337, 393086079907, 500708906197, 509000702017, 600180367883, 780430098443, 3534900290107, 5046024021013, 6006006800743, 6009000432797, 9001924501223, 12090900340283

Offset

1,1

Comments

These prime numbers are both:

- digitally delicate primes (also called weakly prime numbers) A158124: changing any one decimal digit always produces a composite number, with restriction that first digit may not be changed to a 0 (that means no change of the number of significant digits from its original value).

- left-truncatable primes A033664: every suffix is prime, means repeatedly deleting the most significant digit gives a prime at every step until a single-digit prime remains.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..9175 (all terms with <= 29 digits)

Prog

(Python)
from sympy import isprime, primerange
def is_digitally_delicate(p):
    s = str(p)
    for i in range(len(s)):
        for d in "0123456789":
            if d != s[i] and not (i == int(d) == 0):
                if isprime(int(s[:i] + d + s[i+1:])): return False
    return True
def A033664gen(maxdigits):
    yield from [2, 3, 5, 7]
    primestrs, digits, d = ["2", "3", "5", "7"], "0123456789", 1
    while len(primestrs) > 0 and d < maxdigits:
        cands = (d+p for p in primestrs for d in "0123456789")
        primestrs = [c for c in cands if c[0] == "0" or isprime(int(c))]
        yield from sorted(map(int, (p for p in primestrs if p[0] != "0")))
        d += 1
def afind(maxdigits):
    for p in A033664gen(maxdigits):
        if is_digitally_delicate(p): print(p, end=", ")
afind(12) # Michael S. Branicky, Sep 01 2021

Crossrefs

Cf. A158124, A033664.

Keyword

nonn,base

Author

Marc Morgenegg, Sep 01 2021

Extensions

a(3)-a(4) from Amiram Eldar, Sep 01 2021

a(5)-a(19) from Michael S. Branicky, Sep 01 2021

Status

approved

A343075 Digitally delicate square numbers (changing any one decimal digit always produces a nonsquare).

25, 121, 144, 169, 196, 256, 289, 324, 1024, 1089, 1156, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2500, 3136, 3249, 3364, 3481, 3721, 3844, 3969, 4096, 4356, 4489, 4624, 4761, 5041, 5184, 6084, 6241, 6561, 6724, 6889, 7056, 7396

Offset

1,1

Comments

If k is the count of digitally delicate square numbers <= n, then empirically lim_{n->oo} k/n = sqrt(5)/3.

Links

Table of n, a(n) for n=1..41.

Example

n = 25, changing the digit 2 in 25 to d5, d from {0,1,3,4,5,6,7,8,9} gives no square, changing the digit 5 in 25 to 2d, d from {0,1,2,3,4,6,7,8,9} gives no square. Thus n = 25 is a member of the sequence.

Mathematica

changes[n_] := Module[{d = IntegerDigits[n]}, FromDigits @ ReplacePart[d, First[#] -> Last[#]] & /@ Tuples[{Range[Length[d]], Range[0, 9]}]]; q[n_] := AllTrue[changes[n], # == n || ! IntegerQ @ Sqrt[#] &]; Select[Range[100]^2, q] (* Amiram Eldar, Apr 04 2021 *)

Prog

(Python)
from sympy import integer_nthroot
def is_square(n): return integer_nthroot(n, 2)[1]
def change1(n):
  s = str(n)
  for i in range(len(s)):
    for d in set("0123456789") - {s[i]}:
      yield int(s[:i] + d + s[i+1:])
def ok(sqr): return not any(is_square(t) for t in change1(sqr))
print(list(filter(ok, (k*k for k in range(87))))) # Michael S. Branicky, Apr 04 2021

Crossrefs

Cf. A000290, A050249.

Keyword

nonn,base

Author

Ctibor O. Zizka, Apr 04 2021

Status

approved

A354440 Digitally delicate primes where the number of digits appended on the left needed to get a prime increases.

294001, 604171, 971767, 2690201, 10564877, 104097043, 354975121, 1378229029, 1444623667, 1594371379, 3979115747, 15737262803, 22090236251, 28198307351, 35373071549, 49430022721, 67580736437, 142243533671, 659956292591, 1385321944133

Offset

1,1

Comments

Digitally delicate primes (A050249) are primes such that if any single digit is changed the new number is composite. This sequence gives the smallest such prime that needs more digits added to the left to get to another prime. While this list is not complete it has been shown to be finite. A widely digitally delicate prime is known which never becomes prime regardless of the number of extra digits.

294001 can add 1

604171 can add 3

971767 can add 4

2690201 can add 5

10564877 can add 6

104097043 can add 7

354975121 can add 10

1378229029 can add 11

1444623667 can add 12

1594371379 can add 14

3979115747 can add 15

15737262803 can add 16

22090236251 can add 20

28198307351 can add 26

35373071549 can add 27

49430022721 can add 28

67580736437 can add 30

142243533671 can add 47

659956292591 can add 59

1385321944133 can add 76

References

Michael Filaseta and Jeremiah Southwick, Primes that become composite after changing an arbitrary digit, Math. Comp. (2021) Vol. 90, 979-993. doi:10.1090/mcom/3593

Links

Table of n, a(n) for n=1..20.

Example

You can add any 1 extra digit on the left to 294001 without getting a prime but adding two digits would allow for the creation of a prime. For example 10294001 is prime but none of X294001 are.
Starting at 604171 you could add 3 extra digits to the left but not 4 without being able to produce a prime number.
X604171 is not prime
X0604171 is not prime
X00604171 is not prime
however 4000604171 is a prime number
For the largest one found so far
X1385321944133
X01385321944133
X001385321944133
...
X000000000000000000000000000000000000000000000000000000000000000000000000001385321944133 are all composite
but 900000000000000000000000000000000000000000000000000000000000000000000000000001385321944133 is prime

Prog

(Java)
import java.math.BigInteger;
public class delicateprimes {
public static void main(String[] args) {
  BigInteger i, reci=new BigInteger("0");
  i= new BigInteger("1");
  long count=0, v, rec=-1;
  for(long loop=1;; loop++)
  {
    i=i.nextProbablePrime();
    v = delicate(i, true);
    if(v>rec) {count++; rec=v; reci=i; System.out.println("REC=("+reci+", "+rec+") " +loop +"   "+count); }
    if(loop%100000==0)System.out.println("Still running, last prime seen was "+i);
  }
}
static int delicate(BigInteger a, boolean f) // Returns how many digits can be tacked on the delicate prime. f=false just tests the prime with no extra 0s
{
  int e, length, max=200;
  if(!f)max=1;
  String num="", num2="";
  if(!prime(a))return -1;
  for(e=0; e<max; e++)   //While widely digitally delicate primes do exist, this needs to have a stop point
  {
    num=zeros(e)+a.toString();
    if(e>0)length=e; else length = num.length();
    for(int j=0; j<length; j++)
    {
      for(int k=0; k<=9; k++)
      {
        num2=num.substring(0, j)+k+num.substring(j+1);
        if(num2.contentEquals(num))continue;
        if(prime(new BigInteger(num2))) {return e-1; }
      }
    }
  }
  return e-1;
}
static boolean prime(BigInteger a)
{
  return a.isProbablePrime(100);
}
static String zeros(int n)
{
  StringBuffer temp=new StringBuffer("");
  for(int i=0; i<n; i++)temp=temp.append("0");
  return temp.toString();
}
}

Crossrefs

Cf. A050249 (digitally delicate primes).

Keyword

nonn,base,more

Author

Jason Rodgers, May 29 2022

Extensions

Partially edited by N. J. A. Sloane, Sep 03 2022

Status

approved

A076336 (Provable) Sierpiński numbers: odd numbers n such that for all k >= 1 the numbers n*2^k + 1 are composite.

78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, 1259779, 1290677, 1518781, 1624097, 1639459, 1777613, 2131043, 2131099, 2191531, 2510177, 2541601, 2576089, 2931767, 2931991, 3083723, 3098059, 3555593, 3608251

Offset

1,1

Comments

It is only a conjecture that this sequence is complete up to 3000000 - there may be missing terms.

It is conjectured that 78557 is the smallest Sierpiński number. - T. D. Noe, Oct 31 2003

Sierpiński numbers are proved by exhibiting a periodic sequence p of prime divisors with p(k) | n*2^k+1 and disproved by finding prime n*2^k+1. It is conjectured that numbers that cannot be proved Sierpiński in this way are non-Sierpiński. However, some numbers resist both proof and disproof. - David W. Wilson, Jan 17 2005

Sierpiński showed that this sequence is infinite.

There are 4 related sequences that arise in this context:

S1: Numbers n such that n*2^k + 1 is composite for all k (this sequence)

S2: Odd numbers n such that 2^k + n is composite for all k (apparently it is conjectured that S1 and S2 are the same sequence)

S3: Numbers n such that n*2^k + 1 is prime for all k (empty)

S4: Numbers n such that 2^k + n is prime for all k (empty)

The following argument, due to Michael Reid, attempts to show that S3 and S4 are empty: If p is a prime divisor of n + 1, then for k = p - 1, the term (either n*2^k + 1 or 2^k + n) is a multiple of p (and also > p, so not prime). [However, David McAfferty points that for the case S3, this argument fails if p is of the form 2^m-1. So it may only be a conjecture that the set S3 is empty. - N. J. A. Sloane, Jun 27 2021]

a(1) = 78557 is also the smallest odd n for which either n^p*2^k + 1 or n^p + 2^k is composite for every k > 0 and every prime p greater than 3. - Arkadiusz Wesolowski, Oct 12 2015

n = 4008735125781478102999926000625 = (A213353(1))^4 is in this sequence but is thought not to satisfy the conjecture mentioned by David W. Wilson above. For this multiplier, all n*2^(4m + 2) + 1 are composite by an Aurifeuillean factorization. Only the remaining cases, n*2^k + 1 where k is not 2 modulo 4, are covered by a finite set of primes (namely {3, 17, 97, 241, 257, 673}). See Izotov link for details (although with another prime set). - Jeppe Stig Nielsen, Apr 14 2018

Conjecture: if S is a (provable) Sierpiński number, then there exists a prime P such that S^p is also a Sierpiński number for every prime p > P. - Thomas Ordowski, Jul 12 2022

Problem: are there odd numbers K such that K - 2^m is a Sierpiński number for every 1 < 2^m < K? If so, then all positive values of (K - 2^m)*2^n + 1 are composite. Also, by the dual Sierpiński conjecture, K - 2^m + 2^n is composite for every 1 < 2^m < K and for every n > 0. Note that, by the dual Sierpiński conjecture, if p is an odd prime and 1 < 2^m < p, then there exists n such that (p - 2^m)*2^n + 1 is prime. So if such a number K exists, it must be composite. - Thomas Ordowski, Jul 20 2022

From M. F. Hasler, Jul 23 2022: (Start)

1) The above Conjecture is true for Sierpiński numbers provable by a "covering set", with P equal to the largest prime factor of the elements of that set*, according to the explanation from Michael Filaseta posted Jul 12 2022 on the SeqFan mailing list, cf. links. (*More generally: for S^p with any p coprime to all elements of the covering set, but not necessarily prime.)

2) Wilson's comment from 2005 (also the first part, not only the conjecture) is misleading if not wrong because there are provable Sierpiński numbers for which a covering set is not known (maybe even believed not to exist), as explained by Nielsen in his above comment from 2018. It should be clarified that (or: whether) "provable" in the definition of this sequence does not only mean "through a covering set". (End)

References

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 420.

P. Ribenboim, The Book of Prime Number Records, 2nd. ed., 1989, p. 282.

Links

T. D. Noe and Arkadiusz Wesolowski, Table of n, a(n) for n = 1..15000 (T. D. Noe supplied 13394 terms which came from McLean. a(1064), a(7053), and a(13397)-a(15000) from Arkadiusz Wesolowski.)

Chris Caldwell, Riesel number

Chris Caldwell, Sierpinski number

Michael Filaseta, quoted by T. Ordowski, Re: Is it true? SeqFan mailing list, Jul 12 2022

Michael Filaseta, Jacob Juillerat, and Thomas Luckner, Consecutive primes which are widely digitally delicate and Brier numbers, arXiv:2209.10646 [math.NT], 2022.

Yves Gallot, A search for some small Brier numbers, 2000.

Dan Ismailescu and Peter Seho Park, On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences, Journal of Integer Sequences, 16 (2013), #13.9.8.

Anatoly S. Izotov, A Note on Sierpinski Numbers, Fibonacci Quarterly (1995), pp. 206-207.

G. Jaeschke, On the Smallest k Such that All k*2^N + 1 are Composite, Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 381-384.

J. McLean, Searching for large Sierpinski numbers [Cached copy]

J. McLean, Brier Numbers [Cached copy]

Don Reble, Proofs concerning S3 and S4

Carlos Rivera, Problem 29. Brier numbers, The Prime Puzzles and Problems Connection.

Payam Samidoost, Dual Sierpinski problem search page [Broken link?]

Payam Samidoost, Dual Sierpinski problem search page [Cached copy]

Payam Samidoost, 4847 [Broken link?]

Payam Samidoost, 4847 [Cached copy]

W. Sierpiński, Sur un problème concernant les nombres k * 2^n + 1, Elem. Math., 15 (1960), pp. 73-74.

Seventeen or Bust, A Distributed Attack on the Sierpinski Problem

Jeremiah T. Southwick, Two Inquiries Related to the Digits of Prime Numbers, Ph. D. Dissertation, University of South Carolina (2020).

Eric Weisstein's World of Mathematics, Sierpiński Number of the Second Kind

Wikipedia, Sierpiński number.

Crossrefs

Cf. A003261, A052333, A076335, A076337, A101036, A137715, A263169, A305473, A306151.

Keyword

nonn,hard,nice

Author

N. J. A. Sloane, Nov 07 2002

Status

approved

A101036 Riesel numbers (n*2^k-1 is composite for all k>0, n odd) that have a covering set.

509203, 762701, 777149, 790841, 992077, 1106681, 1247173, 1254341, 1330207, 1330319, 1715053, 1730653, 1730681, 1744117, 1830187, 1976473, 2136283, 2251349, 2313487, 2344211, 2554843, 2924861, 3079469, 3177553, 3292241, 3419789, 3423373, 3580901

Offset

1,1

Comments

Conjecture: there are infinitely many Riesel numbers that do not arise from a covering system. See page 16 of the Filaseta et al. reference. - Arkadiusz Wesolowski, Nov 17 2014

a(1) = 509203 is also the smallest odd n for which either n^p*2^k - 1 or abs(n^p - 2^k) is composite for every k > 0 and every prime p > 3. - Arkadiusz Wesolowski, Oct 12 2015

Theorem 11 of Filaseta et al. gives a Riesel number which is thought to violate the assumption of a periodic sequence of prime divisors mentioned in the title of this sequence. - Jeppe Stig Nielsen, Mar 16 2019

If the Riesel number mentioned in the previous comment does in fact not have a covering set, then this sequence is different from A076337, because then that number, 3896845303873881175159314620808887046066972469809^2, is a term of A076337, but not of this sequence. - Felix Fröhlich, Sep 09 2019

Named after the Swedish mathematician Hans Ivar Riesel (1929-2014). - Amiram Eldar, Jun 20 2021

Conjecture: if R is a Riesel number (that has a covering set), then there exists a prime P such that R^p is also a Riesel number for every prime p > P. - Thomas Ordowski, Jul 12 2022

Problem: are there numbers K such that K + 2^m is a Riesel number for every m > 0? If so, then (K + 2^m)*2^n - 1 is composite for every pair of positive integers m,n. Also, by the dual Riesel conjecture, |K + 2^m - 2^n| are always composite. Note that, by the dual Riesel conjecture, if p is an odd prime and n is a positive integer, then there exists n such that (p + 2^m)*2^n - 1 is prime. So if such a number K exists, it must be composite. - Thomas Ordowski, Jul 20 2022

Links

Pierre CAMI and Arkadiusz Wesolowski, Table of n, a(n) for n = 1..15000 (P. CAMI supplied the first 335 terms)

Michael Filaseta, Carrie Finch and Mark Kozek, On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture, Journal of Number Theory, Volume 128, Issue 7 (July 2008), pp. 1916-1940.

Michael Filaseta, Jacob Juillerat, and Thomas Luckner, Consecutive primes which are widely digitally delicate and Brier numbers, arXiv:2209.10646 [math.NT], 2022.

Marcos J. González, Alberto Mendoza, Florian Luca, and V. Janitzio Mejía Huguet, On Composite Odd Numbers k for Which 2^n * k is a Noncototient for All Positive Integers n, J. Int. Seq., Vol. 24 (2021), Article 21.9.6.

Wikipedia, Riesel number.

Crossrefs

See A076337 for references and additional information. Cf. A076336.

Keyword

nonn

Author

David W. Wilson, Jan 17 2005

Extensions

Up to 3292241, checked by Don Reble, Jan 17 2005, who comments that up to this point each n*2^k-1 has a prime factor <= 241.

New name from Felix Fröhlich, Sep 09 2019

Status

approved

A076335 Brier numbers: numbers that are both Riesel and Sierpiński [Sierpinski], or odd n such that for all k >= 1 the numbers n*2^k + 1 and n*2^k - 1 are composite.

3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, 21444598169181578466233, 28960674973436106391349, 32099522445515872473461, 32904995562220857573541

Offset

1,1

Comments

a(1), a(4), and a(6)-a(8) computed by Christophe Clavier, Dec 31 2013 (see link below). 10439679896374780276373 had been found earlier in 2013 by Dan Ismailescu and Peter Seho Park (see reference below). a(3), a(5), and a(9) computed in 2014 by Emmanuel Vantieghem.

These are just the smallest examples known - there may be smaller ones.

There are no Brier numbers below 10^9. - Arkadiusz Wesolowski, Aug 03 2009

Other Brier numbers are 143665583045350793098657, 1547374756499590486317191, 3127894363368981760543181, 3780564951798029783879299, but these may not be the /next/ Brier numbers after those shown. From 2002 to 2013 these four numbers were given here as the smallest known Brier numbers, so the new entry A234594 has been created to preserve that fact. - N. J. A. Sloane, Jan 03 2014

143665583045350793098657 computed in 2007 by Michael Filaseta, Carrie Finch, and Mark Kozek.

It is a conjecture that every such number has more than 10 digits. In 2011 I have calculated that for any n < 10^10 there is a k such that either n*2^k + 1 or n*2^k - 1 has all its prime factors greater than 1321. - Arkadiusz Wesolowski, Feb 03 2016 [Editor's note: The comment below states that the conjecture is now proved. - M. F. Hasler, Oct 06 2021]

There are no Brier numbers below 10^10. For each n < 10^10, there exists at least one prime of the form n*2^k-1 or n*2^k+1 with k <= 356981. The largest necessary prime is 1355477231*2^356981+1. - Kellen Shenton, Oct 25 2020

Links

Table of n, a(n) for n=1..9.

D. Baczkowski, J. Eitner, C. E. Finch, B. Suminski, and M. Kozek, Polygonal, Sierpinski, and Riesel numbers, Journal of Integer Sequences, 2015 Vol 18. #15.8.1.

Chris Caldwell, The Prime Glossary, Riesel number

Chris Caldwell, The Prime Glossary, Sierpinski number

Christophe Clavier, 14 new Brier numbers

Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comput. 29 (1975), pp. 79-81.

P. Erdős, On integers of the form 2^k + p and some related problems, Summa Brasil. Math. 2 (1950), pp. 113-123.

M. Filaseta et al., On Powers Associated with Sierpiński Numbers, Riesel Numbers and Polignac’s Conjecture, Journal of Number Theory, Volume 128, Issue 7, July 2008, Pages 1916-1940. (See pages 9-10)

Michael Filaseta and Jacob Juillerat, Consecutive primes which are widely digitally delicate, arXiv:2101.08898 [math.NT], 2021.

Michael Filaseta, Jacob Juillerat, and Thomas Luckner, Consecutive primes which are widely digitally delicate and Brier numbers, arXiv:2209.10646 [math.NT], 2022. See also Integers (2023) Vol. 23, #A75.

Yves Gallot, A search for some small Brier numbers, 2000.

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 6992565235279559197457863

Dan Ismailescu and Peter Seho Park, On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences, Journal of Integer Sequences, 16 (2013), #13.9.8.

Joe McLean, Brier Numbers [Cached copy]

Carlos Rivera, Problem 29. Brier numbers, The Prime Puzzles and Problems Connection.

Carlos Rivera, Problem 58. Brier numbers revisited, The Prime Puzzles and Problems Connection.

Carlos Rivera, Problem 68. More on Brier numbers, The Prime Puzzles and Problems Connection.

Carlos Rivera, See here for latest information about progress on this sequence

Eric Weisstein's World of Mathematics, Brier Number

Crossrefs

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194638, A194639, A076336, A076337, A040081, A040076, A103963, A103964, A038699, A050921, A064699, A052333, A003261, A364412, A364413.

A180247 gives the primes.

See also A076336, A076337.

A234594 is the old, incorrect version.

Keyword

nonn

Author

Olivier Gérard, Nov 07 2002

Extensions

Many terms reported in Problem 29 from "The Prime Problems & Puzzles Connection" from Carlos Rivera, May 30 2010

Entry revised by Arkadiusz Wesolowski, May 17 2012

Entry revised by Carlos Rivera and N. J. A. Sloane, Jan 03 2014

Entry revised by Arkadiusz Wesolowski, Feb 15 2014

Status

approved

A137985 Complementing any single bit in the binary representation of these primes produces a composite number.

127, 173, 191, 223, 233, 239, 251, 257, 277, 337, 349, 373, 431, 443, 491, 509, 557, 653, 683, 701, 733, 761, 787, 853, 877, 1019, 1193, 1201, 1259, 1381, 1451, 1453, 1553, 1597, 1709, 1753, 1759, 1777, 1973, 2027, 2063, 2333, 2371, 2447, 2633, 2879, 2917

Offset

1,1

Comments

If 2^m is the highest power of 2 in the binary representation of the prime p, there is no requirement that p+2^(m+1) be composite. Sequence A065092 imposes this extra requirement. The prime 223 is the first number in this sequence that is not in A065092.

Mentioned Feb 25 2008 by Terence Tao in his blog http://terrytao.wordpress.com. Tao proves that there are an infinite number of these primes in every fixed base.

Digitally delicate primes in base 2. - Marc Morgenegg, Apr 21 2021

References

Cohen, Fred; Selfridge, J. L., Not every number is the sum or difference of two prime powers. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 79-81. MR0376583 (51 #12758).

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Warren D. Smith et al., Primes such that every bit matters?, Yahoo group "primenumbers", April 2013.

Warren D. Smith and others, Primes such that every bit matters?, digest of 14 messages in primenumbers Yahoo group, Apr 3 - Apr 9, 2013. [Cached copy]

Terence Tao, A remark on primality testing and decimal expansions, arXiv:0802.3361 [math.NT], 2008-2010; Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.

Example

The numbers produced by complementing each of the 8 bits of 223 are 95, 159, 255, 207, 215, 219, 221 and 222, which are all composite.

Mathematica

t={}; k=1; While[Length[t]<100, k++; p=Prime[k]; d=IntegerDigits[p, 2]; n=Length[d]; i=0; While[i<n && (d[[n-i]]==1 && !PrimeQ[p-2^i]) || (d[[n-i]]==0 && !PrimeQ[p+2^i]), i++ ]; If[i==n, AppendTo[t, p]]]; t  (* T. D. Noe *)
isWPbase2[z_] := NestWhile[#*2 &, 2, (# < z && ! PrimeQ@BitXor[z, #] &)] > z; Select[Prime /@ Range[3, PrimePi[10^6]], isWPbase2@# &] (* Terentyev Oleg, Jul 17 2011 *)

Prog

(PARI)f(p)={pow2=1; v=binary(p); L=#v;
forstep(k=L, 1, -1, if(v[k], p-=pow2; if(isprime(p), return(0), p+=pow2), p+=pow2; if(isprime(p), return(0), p-=pow2)); pow2*=2); return(1)}; forprime(p=2, 2879, if(f(p), print1(p, ", "))) \\ Washington Bomfim, Jan 18 2011
(PARI) is_A137985(n)=!for(k=1, n, isprime(bitxor(n, k)) && return; k+=k-1) && isprime(n) \\ Note: A bug in early versions of PARI 2.6 (execute "for(i=0, 1, i>3 && error(buggy); i=9)" to check) makes that this is is_A065092 rather than is_A137985 as expected. For these versions, replace the upper limit n with n\2. \\ M. F. Hasler, Apr 05 2013
(Python)
from sympy import isprime, primerange
def ok(p): # p assumed prime
  return not any(isprime((1<<k)^p) for k in range(p.bit_length()))
def aupto(limit):
  alst = []
  for p in primerange(2, limit+1):
    if ok(p): alst.append(p)
  return alst
print(aupto(2917)) # Michael S. Branicky, Feb 16 2021

Crossrefs

Cf. A050249 (analogous base 10 sequence), A186995 (weak primes in base n).

A065092 is a very similar sequence.

Keyword

nonn,base

Author

T. D. Noe, Feb 26 2008

Extensions

Definition clarified by Chai Wah Wu, Jan 03 2019

Status

approved

A146088 Numbers n with the property that shifting the rightmost digit of n to the left end doubles the number.

0, 105263157894736842, 157894736842105263, 210526315789473684, 263157894736842105, 315789473684210526, 368421052631578947, 421052631578947368, 473684210526315789, 105263157894736842105263157894736842, 157894736842105263157894736842105263

Offset

0,2

Comments

The sequence is infinite, since repeating 105263157894736842 any number of times (e.g. 105263157894736842105263157894736842) gives another number with the same property.

A number N = 10n+m is in the sequence iff 2N = m*10^d+n, where d is the number of digits of n = [N/10]. This is equivalent to 19n = m(10^d-2), i.e. 10^d=2 (mod 19) and n = m(10^d-2)/19, m=2..9 (to ensure that n has d digits). Thus for each d = 18j-1, j=1,2,3... we have exactly 8 solutions which are the j-fold repetition of one among {a(1),...,a(8)}. - M. F. Hasler, May 04 2009

Normally lists have offset 1, but there are good reasons to make an exception in this case. - N. J. A. Sloane, Dec 24 2012

Links

Table of n, a(n) for n=0..10.

G. P. Michon, Deriving A146088 from linear decadic equations.

Formula

a(n) = ((10^d-2)/19*10+1)m, where m=(n-1)%8+2 is the trailing digit and d=(n+7)\8*18-1 is the number of other digits. - M. F. Hasler, May 04 2009

a(8k+i) = A217592(9k+i+1)/2 for i=1..8 with any k.

Example

The sequence starts with a(0)=0 because rotating a lone 0 does double 0. That initial trivial term was not given an index of 1 when it was added, so that the index of other terms of A146088 would not change and invalidate delicate prior cross-references within OEIS (e.g., A217592) or outside of it.
a(4) = 263157894736842105 because 2*a(4) = 526315789473684210.

Mathematica

a[n_] := (m = Mod[n - 1, 8] + 2; d = Floor[(n + 7)/8]*18 - 1; ((10/19)*(10^d - 2) + 1)*m); Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jan 16 2013, after M. F. Hasler *)

Prog

(PARI) A146088(n) = ((10^((n+7)\8*18-1)-2)/19*10+1)*((n-1)%8+2)
/* or a more experimental approach: */ for(d=1, 99, Mod(10, 19)^k-2 & next; for(m=2, 9, print1(", ", m*(10^k-2)/19, m))) \\\\ M. F. Hasler, May 04 2009

Crossrefs

Cf. A092697, A146561, A146569, A146754, A217592.

Keyword

nonn,base

Author

N. J. A. Sloane, based on correspondence from William A. Hoffman III (whoff(AT)robill.com), Apr 10 2009

Extensions

More terms from M. F. Hasler, May 04 2009

a(0)=0 added by G. P. Michon, Oct 29 2012

Status

approved

A269323 Discriminants of real quadratic fields with second 3-class group <729,54>

540365, 945813, 1202680, 1695260, 1958629, 3018569, 3236657, 3687441, 4441560, 5512252, 5571377, 5701693, 6027557, 6049356, 6054060, 6274609, 6366029, 6501608, 6773557, 7573868, 8243464, 8251521, 9054177, 9162577, 9967837

Offset

1,1

Comments

The Artin transfer homomorphisms of the assigned second 3-class group M with SmallGroups identifier <729,54> [Besche, Eick, O'Brien] determine the capitulation type (2,0,3,4) (TKT with two fixed points 3 and 4) of the real quadratic field K in its four unramified cyclic cubic extensions N_i|K (i=1,...,4) and the abelian type invariants of the 3-class groups Cl(3,K)=(3,3) (whence A269323 is a subsequence of A269319) and [Cl(3,N_i)]=[(9,9),(9,3),(9,3),(9,3)] (TTT or IPAD). Conversely, the group M=<729,54> is determined uniquely not only by its Artin pattern AP(G)=(TTT,TKT) but even by the TTT component alone [Mayer, 2015], where TKT, TTT, IPAD are abbreviations for transfer kernel type, transfer target type, index-p abelianization data, respectively [Mayer, 2016]. Consequently, it suffices that the MAGMA program only determines the TTT component of the Artin pattern. This is an instance of the "Principalization algorithm via class group structure" [Mayer, 2014] and saves a considerable amount of CPU time, since the determination of the TKT component is very delicate.

An important Theorem by I.R. Shafarevich [Mayer, 2015, Thm.5.1] disables the metabelian group M=<729,54> as a candidate for the 3-class tower group G, since the relation rank of M is too big. In [Mayer, 2015] it is proved that exactly the two non-metabelian groups <2187,307> and <2187,308> [Besche, Eick, O'Brien] are permitted for G, and the decision is possible with the aid of multi-layered iterated IPADs of second order (which require computing 3-class groups of number fields with absolute degree 54). Since the derived length of both groups is equal to 3, the Hilbert 3-class field tower of all these real quadratic fields has certainly exact length 3.

The MAGMA program requires A269319 as its input list.

Links

Table of n, a(n) for n=1..25.

H. U. Besche, B. Eick, and E. A. O'Brien, The SmallGroups Library - a library of groups of small order, 2005, an accepted and refereed GAP package, available also in MAGMA.

M. R. Bush, private communication, 11 July 2015.

D. C. Mayer, The real quadratic base field K with discriminant d=540365, Targets 2007/2008.

D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2012), no. 2, 471-505.

D. C. Mayer, Principalization algorithm via class group structure, J. Thèor. Nombres Bordeaux 26 (2014), no. 2, 415-464.

D. C. Mayer, New number fields with known p-class tower, Tatra Mt. Math. Publ. 64 (2015), 21-57.

D. C. Mayer, Artin transfer patterns on descendant trees of finite p-groups, Adv. Pure Math. 6 (2016), no. 2, 66-104.

Example

The leading term, 540365, and thus the first real quadratic field K with capitulation type c.21, (2,0,3,4), has been identified on 01 January 2008 [Mayer, 2007/2008]. However, it required seven further years to determine the pro-3 Galois group G=<2187,307|308>, with metabelianization M=G/G''=<729,54>, of the Hilbert 3-class field tower of K in August 2015. (See [Mayer, 2015] for more details.) The first 25 terms of A269323 up to 10^7 have been determined in [Mayer, 2012] and [Mayer, 2014]. The 358, resp. 4377, terms up to 10^8, resp. 10^9, have been computed by [Bush].

Prog

(Magma) SetClassGroupBounds("GRH"); p:=3; dList:=A269319; for d in dList do
ZX<X>:=PolynomialRing(Integers()); K:=NumberField(X^2-d); O:=MaximalOrder(K); C, mC:=ClassGroup(O); sS:=Subgroups(C: Quot:=[p]); sI:=[]; for j in [1..p+1] do Append(~sI, 0); end for; n:=Ngens(C); g:=(Order(C.(n-1)) div p)*C.(n-1); h:=(Order(C.n) div p)*C.n; ct:=0;
for x in sS do ct:=ct+1; if g in x`subgroup then sI[1]:=ct; end if; if h in x`subgroup then sI[2]:=ct; end if; for e in [1..p-1] do if g+e*h in x`subgroup then sI[e+2]:=ct; end if; end for; end for;
sA:=[AbelianExtension(Inverse(mQ)*mC) where Q, mQ:=quo<C|x`subgroup>: x in sS]; sN:=[NumberField(x): x in sA]; sR:=[MaximalOrder(x): x in sA];
sF:=[AbsoluteField(x): x in sN]; sM:=[MaximalOrder(x): x in sF];
sM:=[OptimizedRepresentation(x): x in sF];
sA:=[NumberField(DefiningPolynomial(x)): x in sM]; sO:=[Simplify(LLL(MaximalOrder(x))): x in sA]; TTT:=[]; epsilon:=0; polarization1:=3; polarization2:=3; for j in [1..#sO] do CO:=ClassGroup(sO[j]); Append(~TTT, pPrimaryInvariants(CO, p));
if (3 eq #pPrimaryInvariants(CO, p)) then epsilon:=epsilon+1; end if;
val:=Valuation(Order(CO), p); if (2 eq val) then polarization2:=val; elif (4 le val) then if (3 eq polarization1) then polarization1:=val; else polarization2:=val; end if; end if; end for; if (3 eq polarization2) and (4 eq polarization1) and (0 eq epsilon) then printf "%o, ", d; end if; end for;

Crossrefs

Subsequence of A269319

Keyword

nonn,hard

Author

Daniel Constantin Mayer, Mar 10 2016

Status

approved