A019546 -id:A019546 - cp's OEIS frontend

A213304 Smallest number with n nonprime substrings (Version 3: substrings with leading zeros are counted as nonprime if the corresponding number is not a prime).

2, 1, 10, 14, 101, 104, 144, 1001, 1014, 1044, 1444, 10010, 10014, 10144, 10444, 14444, 100120, 100104, 100144, 101444, 104444, 144444, 1000144, 1001040, 1001044, 1001444, 1014444, 1044444, 1444444, 10001044, 10001444, 10010404, 10010444, 10014444, 10144444, 10444444, 14444444, 100010404, 100010444, 100014444, 100104044, 100104444, 100144444, 101444444, 104444444, 144444444

Offset: 0

Hieronymus Fischer, Aug 26 2012

The sequence is well defined since for each n >= 0 there is a number with n nonprime substrings.

Different from A213303, first difference is at a(16).

			a(0)=2, since 2 is the least number with zero nonprime substrings.
a(1)=1, since 1 has 1 nonprime substrings.
a(2)=10, since 10 is the least number with 2 nonprime substrings, these are 1 and 10 ('0' will not be counted).
a(3)=14, since 14 is the least number with 3 nonprime substrings, these are 1 and 4 and 14. 10, 11 and 12 only have 2 such substrings.

Hieronymus Fischer, Table of n, a(n) for n = 0..100

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079307, A213300 - A213321.

a(m(m+1)/2) = (13*10^(m-1)-4)/9, m>0.
With b(n):=floor((sqrt(8*n-7)-1)/2):
a(n) > 10^b(n), for n>2, a(n) = 10^b(n) for n=1,2.
a(n) >= 10^b(n)+4*10^(n-1-b(n)(b(n)+1)/2)-1)/9, equality holds if n or n+1 is a triangular number > 0 (cf. A000217).
a(n) >= A213303(n).
a(n) <= A213307(n).

A378081 Primes that remain prime if any two of their digits are deleted.

Original entry on oeis.org

223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 1117, 1171, 4111

Offset: 1

Enrique Navarrete, Nov 15 2024

Any term >= 1000 must have its last three digits be from {1, 3, 7, 9}. - Michael S. Branicky, Nov 15 2024
From David A. Corneth, Nov 16 2024: (Start)
Any term < 1000 has exactly three digits and all digits prime (cf. A019546).
Any term >= 1000 is of the form 100*p + r where p is prime and r has only digits coprime to 10 and 11 <= r <= 99.
Also digits in any term >= 1000 are from {1, 4, 7} and at most one digit from {2, 5, 8}. Else at least one of the numbers resulting from removing any two digits is a multiple of 3 and not 3 itself so not prime.
a(19) >= 10^9 if it exists. (End)
a(19) >= 10^10 if it exists. - Michael S. Branicky, Nov 16 2024
a(19) >= 10^100 if it exists. - David A. Corneth, Nov 18 2024

			From _David A. Corneth_, Nov 18 2024: (Start)
4111 is a term since 4111 is prime and removing any to digits from it gives 11 or 11 or 11 or 41 or 41 or 41 and all of those are prime.
No term can end in 12587 as by removing we can (among other numbers) obtain 287, 127, 128, 157, 158, 257 and 587 which are respectively 0 through 6 (mod 7) (all possible residue classes mod 7). So by prepending more digits to 12587 we can always get a multiple of 7 (that is larger than 7) after removing some two digits. (End)

David A. Corneth, PARI program

Cf. A019546, A051362, A226108.

q[n_] := Module[{d = IntegerDigits[n], nd}, nd = Length[d]; nd > 2 && AllTrue[FromDigits /@ Map[d[[#]] &, Subsets[Range[nd], {nd - 2}]], PrimeQ]]; Select[Prime[Range[600]], q] (* Amiram Eldar, Nov 16 2024 *)

PARI
```
\\ See Corneth link
```

from sympy import isprime
from itertools import combinations as C
def ok(n):
    if n < 100 or not isprime(n): return False
    s = str(n)
    return all(isprime(int(t)) for i, j in C(range(len(s)), 2) if (t:=s[:i]+s[i+1:j]+s[j+1:])!="")
print([k for k in range(1, 10**6) if ok(k)]) # Michael S. Branicky, Nov 15 2024

A082755 Smaller of a pair of consecutive primes having only prime digits.

Original entry on oeis.org

2, 3, 5, 223, 727, 3253, 3727, 5233, 5323, 7573, 7723, 7753, 22273, 23327, 25523, 27733, 32233, 32323, 32533, 35323, 35533, 37253, 37273, 52223, 52727, 53323, 53327, 53773, 55333, 72223, 72727, 75223, 75527, 75553, 222527, 222533, 222553, 223273, 223753, 225223

Offset: 1

Amarnath Murthy, Apr 18 2003

			223 is a term as the next prime 227 also has only prime digits.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (Terms for n = 1..1000 from Harvey P. Dale)

Cf. A019546, A082756.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 0; q = 1; pd = {1}; Do[p = q; pd = qd; q = NextPrim[p]; qd = Union[ Join[{2, 3, 5, 7}, IntegerDigits[q]]]; If[pd == qd == {2, 3, 5, 7}, Print[p]], {n, 1, 20000}]
Prime[#]&/@SequencePosition[Table[If[AllTrue[IntegerDigits[n],PrimeQ],1,0],{n,Prime[Range[20000]]}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 31 2017 *)

from sympy import nextprime, isprime
from itertools import count, islice, product
def onlypd(n): return set(str(n)) <= set("2357")
def agen():
    yield from [2, 3, 5]
    for digits in count(2):
        for p in product("2357", repeat=digits-1):
            for end in "37":
                t = int("".join(p) + end)
                if isprime(t) and onlypd(nextprime(t)):
                    yield t
print(list(islice(agen(), 40))) # Michael S. Branicky, Mar 11 2022

Edited and extended by Robert G. Wilson v, Apr 22 2003

A092626 Primes with one nonprime digit.

Original entry on oeis.org

13, 17, 29, 31, 43, 47, 59, 67, 71, 79, 83, 97, 127, 137, 157, 173, 229, 239, 251, 263, 271, 283, 293, 307, 313, 317, 331, 347, 359, 367, 379, 383, 397, 433, 457, 503, 521, 547, 563, 571, 587, 593, 653, 673, 677, 739, 743, 751, 787, 797, 823, 827, 853, 857

Offset: 1

Jani Melik, Apr 11 2004

Heuristically, there are 15/(8 log 10) * n^(log_10 4) members up to n, or about 0.814 * n^0.602.

			13 is prime and it has one nonprime digit, 1;
3259 is prime and it has one nonprime digit, 9.

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

Cf. A019546.

stev_sez:=proc(n) local i, tren, st, ans, anstren; ans:=[ ]: anstren:=[ ]: tren:=n: for i while (tren>0) do st:=round( 10*frac(tren/10) ): ans:=[ op(ans), st ]: tren:=trunc(tren/10): end do; for i from nops(ans) to 1 by -1 do anstren:=[ op(anstren), op(i,ans) ]; od; RETURN(anstren); end:
ts_stnepf:=proc(n) local i, stpf, ans; ans:=stev_sez(n): stpf:=0: for i from 1 to nops(ans) do if (isprime(op(i,ans))='false') then stpf:=stpf+1; # number of nonprime digits fi od; RETURN(stpf) end:
ts_pr_neprn:=proc(n) local i, stpf, ans, ans1, tren; ans:=[ ]: stpf:=0: tren:=1: for i from 1 to n do if ( isprime(i)='true' and ts_stnepf(i) = 1) then ans:=[ op(ans), i ]: tren:=tren+1; fi od; RETURN(ans) end: ts_pr_neprn(4000);

Select[Prime[Range[200]],Count[IntegerDigits[#],?(!PrimeQ[#]&)]==1&] (* _Harvey P. Dale, Feb 18 2018 *)

Edited by R. J. Mathar, Nov 02 2009

Comment from Charles R Greathouse IV, Mar 19 2010

A152312 Primes without odd prime digits.

Original entry on oeis.org

2, 11, 19, 29, 41, 61, 89, 101, 109, 149, 181, 191, 199, 211, 229, 241, 269, 281, 401, 409, 419, 421, 449, 461, 491, 499, 601, 619, 641, 661, 691, 809, 811, 821, 829, 881, 911, 919, 929, 941, 991, 1009, 1019, 1021, 1049, 1061

Offset: 1

Omar E. Pol, Dec 02 2008

Cf. A019546, A034844, A087363.

A179335 a(n) is the smallest prime which appears as a substring of the decimal representation of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 3, 7, 19, 2, 2, 3, 3, 41, 3, 7, 3, 5, 61, 7, 7, 3, 7, 3, 89, 7, 101, 3, 7, 109, 3, 2, 3, 3, 3, 149, 5, 5, 3, 7, 3, 7, 181, 19, 3, 7, 19, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 401, 409, 19, 2, 3, 3, 3, 3, 449, 5, 61, 3, 7, 7, 7

Offset: 1

Reinhard Zumkeller, Jul 11 2010

a(n) < 10 iff prime(n) is in A179336;

a(n) = prime(n) iff prime(n) is in A033274. [Corrected by M. F. Hasler, Aug 27 2012]

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

Cf. A000040, A070024, A019546, A092629, A104250, A019546, A034844, A179336, A193238.

A179335(n)={my(p=prime(n),m=0,M); for(d=1,n, M=10^d; n=p; until(n<=M || !n\=10, isprime(n%M) & (!m || m>n%M) & m=n%M); m & return(m))} \\ M. F. Hasler, Aug 27 2012

from sympy import isprime, prime
def a(n):
    s = str(prime(n))
    ss = set(int(s[i:i+1+l]) for i in range(len(s)) for l in range(len(s)))
    return min(t for t in ss if isprime(t))
print([a(n) for n in range(1, 94)]) # Michael S. Branicky, Jun 29 2022

A124673 Numbers with distinct prime digits.

Original entry on oeis.org

2, 3, 5, 7, 23, 25, 27, 32, 35, 37, 52, 53, 57, 72, 73, 75, 235, 237, 253, 257, 273, 275, 325, 327, 352, 357, 372, 375, 523, 527, 532, 537, 572, 573, 723, 725, 732, 735, 752, 753, 2357, 2375, 2537, 2573, 2735, 2753, 3257, 3275, 3527, 3572, 3725, 3752, 5237

Offset: 1

Tanya Khovanova, Dec 24 2006

Jinyuan Wang, Table of n, a(n) for n = 1..64

Cf. A019546, A124674.

Select[Range[10000], Length[IntegerDigits[ # ]] == Length[Union[IntegerDigits[ # ]]] && Complement[IntegerDigits[ # ], {2, 3, 5, 7}] == {} &]
Rest[FromDigits/@Flatten[Permutations/@Subsets[{2,3,5,7}],1]]//Sort (* Harvey P. Dale, Jun 02 2024 *)

A124888 Primes with prime number of only prime digits (i.e., 2, 3, 5, 7).

Original entry on oeis.org

23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 22273, 22277, 22573, 22727, 22777, 23227, 23327, 23333, 23357, 23537, 23557, 23753, 23773, 25237, 25253, 25357, 25373, 25523, 25537, 25577, 25733, 27253, 27277

Offset: 1

Lekraj Beedassy, Nov 12 2006

Michael S. Branicky, Table of n, a(n) for n = 1..10000
József Bölcsföldi, György Birkás, and Miklós Ferenczi, Bölcsföldi-Birkás-Ferenczi prime numbers (Full prime numbers), International Journal of Mathematics and Statistics Invention (IJMSI), Volume 5, Issue 2, February 2017, pp. 4-7.

Cf. A019546, A120533.

Select[Prime[Range[3000]],ContainsOnly[IntegerDigits[#],{2,3,5,7}]&&PrimeQ[Length[IntegerDigits[#]]]&] (* James C. McMahon, Dec 14 2024 *)

isok(p) = isprime(p) && (d=digits(p)) && isprime(#d) && vecmin(vector(#d, k, isprime(d[k]))); \\ Michel Marcus, Sep 21 2017

from sympy import isprime, prime
from itertools import count, islice, product
def agen(): yield from filter(isprime, (int("".join(s)+e) for i in count(1) for s in product("2357", repeat=prime(i)-1) for e in "37"))
print(list(islice(agen(), 42))) # Michael S. Branicky, Jun 23 2022

Terms 773, 23753 inserted by Georg Fischer, Jun 23 2022

A213303 Smallest number with n nonprime substrings (Version 2: substrings with leading zeros are counted as nonprime if the corresponding number is > 0).

Original entry on oeis.org

2, 1, 10, 14, 101, 104, 144, 1001, 1014, 1044, 1444, 10010, 10014, 10144, 10444, 14444, 100101, 100104, 100144, 101444, 104444, 144444, 1000144, 1001014, 1001044, 1001444, 1014444, 1044444, 1444444, 10001044, 10001444, 10010144, 10010444, 10014444, 10144444, 10444444, 14444444, 100010144

Offset: 0

Hieronymus Fischer, Aug 26 2012

The sequence is well defined since for each n >= 0 there is a number with n nonprime substrings.

Different from A213304, first different term is a(16).

			a(0)=2, since 2 is the least number with zero nonprime substrings.
a(1)=1, since 1 has 1 nonprime substrings.
a(2)=10, since 10 is the least number with 2 nonprime substrings, these are 1 and 10 ('0' will not be counted).
a(3)=14, since 14 is the least number with 3 nonprime substrings, these are 1 and 4 and 14. 10, 11 and 12 only have 2 such substrings.

Hieronymus Fischer, Table of n, a(n) for n = 0..100

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079307, A213300 - A213321.

a(m(m+1)/2) = (13*10^(m-1)-4)/9, m>0.
With b(n):=floor((sqrt(8*n-7)-1)/2):
a(n) > 10^b(n), for n>2, a(n) = 10^b(n) for n=1,2.
a(n) >= 10^b(n)+4*10^(n-1-b(n)(b(n)+1)/2)-1)/9, equality holds if n or n+1 is a triangular number > 0 (cf. A000217).
a(n) <= A213304(n).
a(n) <= A213306(n).

A213306 Minimal prime with n nonprime substrings (Version 2: substrings with leading zeros are counted as nonprime if the corresponding number is > 0).

Original entry on oeis.org

2, 13, 11, 103, 101, 149, 1009, 1021, 1049, 1481, 10039, 10069, 10169, 11681, 14669, 100109, 100189, 100169, 101681, 104681, 146669, 1000669, 1001041, 1001081, 1004669, 1014469, 1046849, 1468469, 10001081, 10004669, 10010851

Offset: 0

Hieronymus Fischer, Aug 26 2012

			a(0) = 2, since 2 is the least number with zero nonprime substrings.
a(1) = 13, since 13 has 1 nonprime substring (=’1’).
a(2) = 11, since 11 is the least number with 2 nonprime substrings (= 2 times ‘1’).
a(3) = 103, since 103 is the least number with 3 nonprime substrings, these are ‘1’ and ‘10’ and ‘03’ (‘0’ is not a valid substring in version 2).

Hieronymus Fischer, Table of n, a(n) for n = 0..100

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079307, A213300 - A213321.

a(n) > 10^floor((sqrt(8*n+1)-1)/2), for n>2.
a(n) >= A213303(n).
a(n) <= A213307(n).

cp's OEIS Frontend

A213304 Smallest number with n nonprime substrings (Version 3: substrings with leading zeros are counted as nonprime if the corresponding number is not a prime).

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A378081 Primes that remain prime if any two of their digits are deleted.

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Mathematica

PARI

Python

A082755 Smaller of a pair of consecutive primes having only prime digits.

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Mathematica

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Extensions

A092626 Primes with one nonprime digit.

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Maple

Mathematica

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A152312 Primes without odd prime digits.

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A179335 a(n) is the smallest prime which appears as a substring of the decimal representation of prime(n).

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PARI

Python

A124673 Numbers with distinct prime digits.

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Mathematica

A124888 Primes with prime number of only prime digits (i.e., 2, 3, 5, 7).

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Mathematica

PARI

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Extensions

A213303 Smallest number with n nonprime substrings (Version 2: substrings with leading zeros are counted as nonprime if the corresponding number is > 0).