A050288 -id:A050288 - cp's OEIS frontend

A368034 Pandigital primes that are the concatenation of 7 primes.

23401589677, 23401675789, 23401675897, 23401756789, 23401767589, 23401896757, 23415780961, 23415809761, 23415896017, 23415897601, 23416017589, 23416075789, 23417809561, 23418095761, 23475601789, 23475809761, 23476018957, 23540176789, 23540178967, 23541601789

Offset: 1

Eric Angelini and Giorgos Kalogeropoulos, Dec 08 2023

To be prime, a pandigital must use at least 11 digits. There are 4397 pandigital primes with 11 digits that can be cut into 7 prime-chunks.

			a(1) = 23401589677 can be cut into the 7 prime-chunks 2,3,401,5,89,67,7;
a(2) = 23401675789 can be cut into the 7 prime-chunks 2,3,401,67,5,7,89; etc.

Éric Angelini, Six prime chunks, Personal blog, December 2023.

Cf. A368035, A368044, A050288.

A368035 Pandigital primes that are the concatenation of 8 primes.

Original entry on oeis.org

1032341589677, 1032341758967, 1032343678957, 1032343758967, 1032343767589, 1032347617589, 1032347675897, 1032347789567, 1032347896157, 1032354161897, 1032354178967, 1032354189761, 1032354361789, 1032354378967, 1032354389767, 1032354761897, 1032354767897, 1032356174189, 1032356747897, 1032356778941

Offset: 1

Eric Angelini and Giorgos Kalogeropoulos, Dec 08 2023

The least prime number that can be cut into 8 prime-chunks has at least 13 digits. There are in total 875653 such 13-digit primes.

			a(1) = 1032341589677 can be cut into the 8 prime-chunks 103,2,3,41,5,89,67,7;
a(2) = 1032341758967 can be cut into the 8 prime-chunks 103,2,3,41,7,5,89,67; etc.

Éric Angelini, Six prime chunks, Personal blog, December 2023.

Cf. A368034, A368044, A050288.

A368044 Pandigital primes that are the concatenation of 9 primes.

Original entry on oeis.org

103132343589761, 103132343675897, 103132343761589, 103132343789561, 103132343895677, 103132343895767, 103132357614389, 103132357674389, 103132357894361, 103132361743589, 103132367743589, 103132367754389, 103132367894357, 103132367895743, 103132374358961, 103132375674389

Offset: 1

Eric Angelini and Giorgos Kalogeropoulos, Dec 09 2023

The least prime number that can be cut into 9 prime-chunks has at least 15 digits. There are 3764 sets of 9 prime-chunks that produce in total 69061759 primes with 15 digits.

			a(1) = 103132343589761 can be cut into the 9 prime-chunks 103, 13, 2, 3, 43, 5, 89, 7, 61;
a(2) = 103132343675897 can be cut into the 9 prime-chunks 103, 13, 2, 3, 43, 67, 5, 89, 7; etc.

Éric Angelini, Six prime chunks, Personal blog, December 2023.

Cf. A368034, A368035, A050288.

A159292 Pandigital emirps.

Original entry on oeis.org

10124389567, 10124563789, 10124597683, 10124635897, 10124673859, 10124687359, 10124695783, 10124735689, 10124795683, 10124867359, 10124958673, 10124965387, 10124965783, 10125364897, 10125693847, 10125749863, 10125784639, 10125938467, 10126387549, 10126457893, 10126498573

Offset: 1

Lekraj Beedassy, Apr 08 2009

There are 413842 11-digit terms. - Jud McCranie, Jul 03 2013 [in light of the comment below, this was independently computed and confirmed to be correct by Michael S. Branicky, Apr 06 2024]

The above statement [by Jud McCranie] is uncertain, as the contributed b-file was wrong (missing terms) from a(436) on. At this point, one has to consider permutations of 10223456789, before coming back, for n > 495, to permutations of 10123456789 starting with 10231.... - M. F. Hasler, Apr 06 2024

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..435 from Jud McCranie, terms 436..500 corrected and added by M. F. Hasler)

Cf. A006567 (emirps), A050288 (pandigital primes).

L=List(); append(N=10123456789, M=Vecsmall([2,3,3]))=forperm(digits(N),p, cmp(p[3..5],M)>0 && break; isprime(P=fromdigits(Vec(p)))&& isprime(fromdigits(Vecrev(p)))&& listput(L, P))
append(); append(10223456789); #A159292=Set(L) \\ M. F. Hasler, Apr 05 2024

from sympy import isprime
from itertools import count, islice, product
def emirp(s):
    r = s[::-1]
    return r != s and isprime(int(s)) and isprime(int(r))
def agen(): # generator of terms
    for d in count(11):
        for f in "1379":
            for m in product("0123456789", repeat=d-2):
                for e in "1379":
                    t = f + "".join(m) + e
                    if len(set(t)) == 10 and emirp(t):
                        yield int(t)
print(list(islice(agen(), 100))) # Michael S. Branicky, Apr 09 2024

Intersection of A006567 and A050288. - M. F. Hasler, Apr 05 2024

Erroneous terms corrected and more terms from M. F. Hasler, Apr 05 2024

A159569 Pandigital primes that become zeroless pandigital primes when the digit 0 is deleted.

Original entry on oeis.org

10123465789, 10123685749, 10123746859, 10123854679, 10123945687, 10123956487, 10124356789, 10124378569, 10124563987, 10124568793, 10124683759, 10124695783, 10124736859, 10124763589, 10124785639, 10124867539, 10124867593, 10124935687, 10125367849, 10125368749

Offset: 1

Lekraj Beedassy, Apr 15 2009

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..13117

Cf. A050288, A050290.

remz(d) = {nd = []; for (i=1, #d, if (d[i] != 0, nd = concat(nd, d[i]))); subst(Pol(nd), x, 10);}
isok(n) = isprime(n) && (d=digits(n)) && (#vecsort(d,,8)==10) && isprime(remz(d));
lista() = forprime(n=10123465789,, if (isok(n), print1(n, ", "))); \\ Michel Marcus, Oct 06 2014

Missing terms a(2)-a(6) and a(15)-a(20) added by Hiroaki Yamanouchi, Oct 06 2014

A168334 Pandigital palindromic primes.

Original entry on oeis.org

1023456987896543201, 1023458697968543201, 1023459768679543201, 1023469587859643201, 1023489657569843201, 1023497568657943201, 1023549678769453201, 1023568794978653201, 1023568947498653201

Offset: 1

Lekraj Beedassy, Nov 23 2009

Intersection of A050288 and A002385.

M. F. Hasler, Table of n, a(n), for n = 1..2309

Cf. A050288, A002385.

{n=1011111111111111101; p=vector(8,i,if(i<8,10^i*10)+10^(17-i))~; a=[]; for(k=0,8!, ispseudoprime(n+numtoperm(8,k)*p) & a=concat(a,n+numtoperm(8,k)*p)); a=vecsort(a)} \\ M. F. Hasler, Dec 07 2009

More terms from M. F. Hasler, Dec 07 2009

A225295 Pandigital numbers with exactly 4 prime factors (with multiplicity).

Original entry on oeis.org

1023456987, 1023458679, 1023458967, 1023465798, 1023465897, 1023465978, 1023465987, 1023467589, 1023467859, 1023468579, 1023468597, 1023468795, 1023469758, 1023478569, 1023479586, 1023479865, 1023485967, 1023486579, 1023486957, 1023487659, 1023487965, 1023489657, 1023495678

Offset: 1

Jonathan Vos Post, May 04 2013

			a(1) = 1023456987 = 3^2 * 7 * 16245349.
a(2) = 1023458679 = 3^3 * 37905877.
a(3) = 1023458967 = 3^2 * 113 * 1006351.
a(4) = 1023465897 = 3^2 * 6379 * 17827.
a(5) = 1023465987 = 3^2 * 53 * 2145631.
a(511032) = 10123456987 = 7^2 * 9833 * 21011.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A014613, A221646.
Cf. A050288 (pandigital primes), A175845 (pandigital numbers).
Cf. A171102 (3-almost primes).

is(n)=#vecsort(digits(n),,8)>9 && bigomega(n)==4 \\ Charles R Greathouse IV, May 04 2013

A014613 INTERSECTION A171102.

a(n) ~ 6n log n / (log log n)^3. - Charles R Greathouse IV, May 04 2013

a(6)-a(23) from Charles R Greathouse IV, May 04 2013

A358020 Least prime number > prime(n) (n >= 5) whose set of decimal digits coincides with the set of decimal digits of prime(n), or -1 if no such prime exists.

Original entry on oeis.org

1111111111111111111, 31, 71, 191, 223, 229, 113, 73, 4111, 433, 4447, 353, 599, 661, 677, 1117, 337, 97, 383, 8999, 797, 10111, 1013, 701, 1009, 131, 271, 311, 173, 193, 419, 1151, 571, 613, 617, 317, 197, 811, 199, 1193, 719, 911, 2111, 233, 277, 929, 2333, 293, 421, 521, 2557

Offset: 5

Jean-Marc Rebert, Oct 24 2022

			prime(6) = 13 and the prime number 31 have the same set of digits {1,3}, and 31 is the smallest such number, hence a(6) = 31.
prime(13) = 41 and the prime number 4111 have the same set of digits {1,4}, and 4111 is the smallest such number, hence a(13) = 4111.
prime(20) = 71 and the prime number 1117 have the same set of digits {1,7}, and 1117 is the smallest such number, hence a(20) = 1117.

Jean-Marc Rebert, Table of n, a(n) for n = 5..10000

Cf. A000040, A357096, A004022, A004023, A020451, A020455, A050288, A166681.

N:= 60: # for a(5)..a(N)
A:= Array(5..N):
R:= 1111111111111111111:
A[5]:= R: count:= 1:
for k from 6 while count < N-4 do
  p:= ithprime(k);
  S:= convert(convert(p,base,10),set);
  if assigned(V[S]) and V[S]<=N then A[V[S]]:= p; count:=count+1;  fi;
  V[S]:= k;
od:
convert(A,list); # Robert Israel, Oct 25 2022

a(n)=my(m=Set(digits(prime(n)))); if(n<5, return()); if(n==5,return(1111111111111111111));forprime(p=prime(n+1), , if(Set(digits(p))==m, return(p)))

from sympy import isprime, prime
from itertools import count, product
def a(n):
    pn = prime(n)
    s = str(pn)
    for d in count(len(s)):
        for p in product(set(s), repeat=d):
            if p[0] == "0": continue
            t = int("".join(p))
            if t > pn and isprime(t):
                return t
print([a(n) for n in range(5, 56)]) # Michael S. Branicky, Oct 25 2022

A371361 The first of two consecutive primes p, q such that p, q and p + q are all pandigital.

Original entry on oeis.org

10234568791, 10234685971, 10234756849, 10234786589, 10234865779, 10235678449, 10237845649, 10243756981, 10245836789, 10245936781, 10245968371, 10247658389, 10247658923, 10247685893, 10248357659, 10248756893, 10256734879, 10256839447, 10256839477, 10257384679, 10257486913, 10258367429, 10258367489

Offset: 1

Robert Israel, Mar 19 2024

The first case where a(n) and a(n+1) are consecutive primes is n = 18. Thus p = a(18) = 10256839447, q = a(19) = 10256839477 and r = 10256839487 are three consecutive primes with p, q, r, p + q and q + r all pandigital. In addition, p + r = 20513678934 ia pandigital.

			a(3) =  10234756849 is a term because it is prime and pandigital, the next prime 10234756859 is also pandigital, and  10234756849 + 10234756859 = 20469513708 is pandigital.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..100 from Robert Israel)

Cf. A171102, A050288

ispd:= proc(n) convert(convert(n,base,10),set) = {$0..9} end proc:
q:=nextprime(10^10): qgood:= false: Res:= NULL: count:= 0:
while count < 25 do
  p:= q; pgood:= qgood;
  q:= nextprime(p); qgood:= ispd(q);
  if pgood and qgood and ispd(p+q) then
    Res:= Res, p; count:= count+1;
  fi;
od:
Res;

A332265 a(n) is the number of prime numbers created when concatenating all the arrangements of the decimal integers from 0 to 3*n+4.

Original entry on oeis.org

20, 3202, 2056675, 3500185228

Offset: 0

Scott R. Shannon, May 04 2020

Only 4 and every third integer after 4 can create primes when concatenating the integer arrangements of 0,...,3*n+4 as the other integer values will create numbers with digit sums divisible by 3, and hence are divisible by 3. The digit 0 is allowed to be the first digit in the number but is then ignored when determining if the remaining digits form a prime.

			a(0) = 20 as there are twenty primes created when concatenating the integer arrangements of 0,1,2,3,4. They are 1423, 2143, 2341, 4231, 10243, 12043, 20143, 20341, 20431, 23041, 24103, 30241, 32401, 40123, 40213, 40231, 41023, 41203, 42013, 43201.
a(1) = 3202. The smallest prime created using integers 0..7 is 1234657 while the largest is 76540231.
a(2) = 2056675. The smallest prime created using integers 0..10 is 10123457689 while the largest is 987654310021.

Cf. A000040, A295206, A006879, A050288, A088628, A185122, A176009, A099182.

Table[Count[FromDigits /@  Flatten /@ IntegerDigits /@ Permutations[Range[0, 3 n + 4]], ?PrimeQ], {n, 0, 2}] (* _Robert Price, Sep 16 2020 *)
(* OR, if the above runs low on memory to store all the Permutations at once... *)
Table[p0 = Range[0, 3n+4]; p = NextPermutation[p0]; c = 0;
 While[p != p0,
  If[PrimeQ[FromDigits[Flatten[IntegerDigits /@ p]]], c++];
p = NextPermutation[p]]; c, {n, 0, 2}] (* Robert Price, Sep 16 2020 *)

a(3) from Giovanni Resta, May 04 2020

cp's OEIS Frontend

A368034 Pandigital primes that are the concatenation of 7 primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A368035 Pandigital primes that are the concatenation of 8 primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A368044 Pandigital primes that are the concatenation of 9 primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A159292 Pandigital emirps.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Formula

Extensions

A159569 Pandigital primes that become zeroless pandigital primes when the digit 0 is deleted.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A168334 Pandigital palindromic primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A225295 Pandigital numbers with exactly 4 prime factors (with multiplicity).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A358020 Least prime number > prime(n) (n >= 5) whose set of decimal digits coincides with the set of decimal digits of prime(n), or -1 if no such prime exists.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Python

A371361 The first of two consecutive primes p, q such that p, q and p + q are all pandigital.

Views