A034844 -id:A034844 - cp's OEIS frontend

A092628 Primes with exactly three nonprime digits.

101, 109, 149, 181, 191, 199, 401, 409, 419, 449, 461, 491, 499, 601, 619, 641, 661, 691, 809, 811, 881, 911, 919, 941, 991, 1013, 1021, 1031, 1039, 1051, 1063, 1087, 1093, 1097, 1103, 1117, 1129, 1151, 1163, 1171, 1187, 1193, 1201, 1249, 1289, 1291

Offset: 1

Jani Melik, Apr 11 2004

			101 is prime and it has three nonprime digits, 0 and twice 1;
4261 is prime and it has three nonprime digits, 1, 4 and 6.

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

Cf. A019546, A034844.

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_neprnt:=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) = 3) then ans:=[ op(ans), i ]: tren:=tren+1; fi od; RETURN(ans) end: ts_pr_neprnt(5000);

dgQ[n_]:=Count[IntegerDigits[n],?(!PrimeQ[#]&)]==3; Select[Prime[ Range[300]], dgQ] (* _Harvey P. Dale, Oct 11 2011 *)

A108614 Semiprimes with non-semiprimes digits (no digits 4,6,9 in semiprimes).

Original entry on oeis.org

10, 15, 21, 22, 25, 33, 35, 38, 51, 55, 57, 58, 77, 82, 85, 87, 111, 115, 118, 121, 122, 123, 133, 155, 158, 177, 178, 183, 185, 187, 201, 202, 203, 205, 213, 215, 217, 218, 221, 235, 237, 253, 278, 287, 301, 302, 303, 305, 321, 323, 327, 335, 355, 358, 371

Offset: 1

Zak Seidov, Jun 13 2005

Complement of A107342 in the class of semiprimes.

This is to semiprimes A001358 as A034844 (Primes with nonprime digits) is to primes A000040. [Jonathan Vos Post, Jul 15 2010]

Cf. A107342, A137167.

Cf. A107665, A107666, A111494, A111496, A111697, A179336. [Jonathan Vos Post, Jul 15 2010]

cnd[n_]:=Plus@@Last/@FactorInteger[n]==2&&Union[FreeQ[IntegerDigits[n], # ]&/@{4, 6, 9}]=={True};Select[Range[600], cnd[ # ]&]

{j in A001358 and j not in A179463}. [Jonathan Vos Post, Jul 15 2010]

A157527 Primes using only the composite digits (4, 6, 8, 9) and all of them.

Original entry on oeis.org

46489, 46889, 48649, 48869, 64489, 64849, 68449, 68489, 84649, 84869, 88469, 444869, 448969, 449689, 468499, 468869, 468889, 468899, 469849, 486449, 486869, 486949, 488689, 489689, 489869, 496849, 496889, 498469, 498689, 644489, 644869

Offset: 1

Lekraj Beedassy, Mar 02 2009, Mar 03 2009

Subsequence of A051416.

There are no 4-digit terms so each term must have at least one repeating digit. - Harvey P. Dale, Oct 05 2023

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

Cf. A051416, A138165, A034844, A108419.

a := proc (n) if convert(convert(ithprime(n), base, 10), set) = {4, 6, 8, 9} then ithprime(n) else end if end proc: seq(a(n), n = 1 .. 53000); # Emeric Deutsch, Mar 03 2009
isA157527 := proc(n) local dgs ; if not isprime(n) then false; else dgs := convert(convert(n,base,10),set) ; if dgs intersect {4,6,8,9} <> {4,6,8,9} then false; elif dgs intersect {0,1,2,3,5,7} <> {} then false; else true; fi; fi; end: for n from 1 to 100000 do p := ithprime(n) ; if isA157527(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, Mar 03 2009

With[{c={4,6,8,9}},Select[Flatten[Table[10 FromDigits/@Tuples[c,n]+9,{n,5}]],PrimeQ[#] && Intersection[c,IntegerDigits[#]]==c&]] (* Harvey P. Dale, Oct 05 2023 *)

Corrected and extended by numerous correspondents, Mar 04 2009

A177850 Smallest n-digit emirp with only nonprime digits.

Original entry on oeis.org

149, 1009, 10009, 100049, 1000849, 10000169, 100000891, 1000000009, 10000001041, 100000000669, 1000000000091, 10000000001011, 100000000000469, 1000000000004449, 10000000000001101, 100000000000000049

Offset: 3

Jonathan Vos Post, May 14 2010

Least value of emirps with only nonprime digits (i.e., 0,1,4,6,8,9) A128390, with n digits. This is to primes with nonprime digits (A034844) as smallest n-digit emirp with only prime digits (A177513) is to primes with only prime digits.

			a(6) = 100049 because all digits {0,1,4,9} are nonprime, and 100049 is prime, and R(100049) = A004086(100049) = 940001 is prime, and there is no smaller 6-digit number meeting these criteria.

Cf. A000040, A004086, A006567, A034844, A128390, A177513.

isA084984 := proc(n) convert(convert(n,base,10),set) ; if % intersect {2,3,5,7} = {} then true; else false; end if; end proc:
A177850 := proc(n) local a; a := 10^(n-1) ; while not (isA006567(a) and isA084984(a)) do a := nextprime(a) ; end do; if a < 10^n then return a ; else return -1 ; end if; end proc:
seq(A177850(n),n=3..40) ; # R. J. Mathar, May 24 2010

More terms from R. J. Mathar, May 24 2010

A179924 Primes with embedded primes.

Original entry on oeis.org

13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 103, 107, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 179, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337

Offset: 1

Robert G. Wilson v, Aug 01 2010

A079066(a(n)) > 0. - Reinhard Zumkeller, Jul 19 2011

Is there a prime such that the previous prime is embedded in it? - Ivan N. Ianakiev, Nov 09 2023. Answer from Amiram Eldar: No. If prime(n) is embedded in prime(n+1) then prime(n+1) has at least one digit more than prime(n), so prime(n+1) > 2 * prime(n). But according to Bertrand's postulate, prime(n+1) < 2*prime(n).

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

Cf. A039996, A079397, A033274, A034844, A092621, A179909 - A179919, A033274, A179922.

import Data.List (findIndices)
a179924 n = a179924_list !! (n-1)
a179924_list = map (a000040 . (+ 1)) $ findIndices (> 0) a079066_list
-- Reinhard Zumkeller, Jul 19 2011

f[n_] := Block[{id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[Partition[id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 68, f@# > 1 &]

A215927 Primes having at least one digit that is not prime.

Original entry on oeis.org

11, 13, 17, 19, 29, 31, 41, 43, 47, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 229, 239, 241, 251, 263, 269, 271, 281, 283, 293, 307, 311, 313, 317, 331, 347

Offset: 1

Luca Brigada Villa, Aug 27 2012

Complement of A019546 within the primes A000040.

			19 is in the sequence because neither of its two digits is prime, 1 being a unit and 9 being the square of 3.
23 is not in the sequence because both 2 and 3 are prime.
29 is in the sequence because 9 is not prime (though 2 is).

Cf. A000040, A019546, A034844, A033274, A152313, A141409.

[p: p in PrimesUpTo(500) | not Set(Intseq(p)) subset [2,3,5,7]]; // Vincenzo Librandi Oct 25 2016

Select[Prime[Range[100]], Complement[IntegerDigits[#], {2, 3, 5, 7}] != {} &] (* Alonso del Arte, Aug 27 2012 *)

is_A215927(n)=isprime(n)&apply(x->!isprime(x),eval(Vec(Str(n)))) \\ - M. F. Hasler, Aug 27 2012

a(55) corrected by Vincenzo Librandi, Oct 25 2016

A323391 Primes containing nonprime digits (from 1 to 9) in their decimal expansion and whose digits are distinct, i.e., consisting of only digits 1, 4, 6, 8, 9.

Original entry on oeis.org

19, 41, 61, 89, 149, 419, 461, 491, 619, 641, 691, 941, 1489, 4691, 4861, 6481, 6491, 6841, 8419, 8461, 8641, 8941, 9461, 14869, 46819, 48619, 49681, 64189, 64891, 68491, 69481, 81649, 84691, 84961, 86491, 98641

Offset: 1

Bernard Schott, Jan 13 2019

There are only 36 terms in this sequence, which is a finite subsequence of A152313.
Two particular examples:
6481 is also the smallest prime formed from the concatenation of two consecutive squares.
81649 is the only prime containing all the nonprime positive digits such that every string of two consecutive digits is a square.

			14869 is the smallest prime that contains all the nonprime positive digits; 98641 is the largest one.

Chris K. Caldwell and G. L. Honaker, Jr., 81649, Prime Curios!

Subsequence of A152313. Subsequence of A029743. Subsequence of A155024 (with distinct nonprime digits but with 0) and of A034844.

Cf. A029743 (with distinct digits), A124674 (with distinct prime digits), A155045 (with distinct odd digits), A323387 (with distinct square digits), A323578 (with distinct digits for which parity of digits alternates).

Select[Union@ Flatten@ Map[FromDigits /@ Permutations@ # &, Rest@ Subsets@ {1, 4, 6, 8, 9}], PrimeQ] (* Michael De Vlieger, Jan 19 2019 *)

isok(p) = isprime(p) && (d=digits(p)) && vecmin(d) && (#Set(d) == #d) && (#select(x->isprime(x), d) == 0); \\ Michel Marcus, Jan 14 2019

A355856 Primes, with at least one prime digit, that remain primes when all of their prime digits are removed.

Original entry on oeis.org

113, 131, 139, 151, 179, 193, 197, 211, 241, 311, 389, 421, 431, 541, 613, 617, 631, 719, 761, 829, 839, 859, 1013, 1021, 1031, 1039, 1051, 1093, 1097, 1123, 1153, 1201, 1213, 1217, 1229, 1231, 1249, 1259, 1279, 1291, 1297, 1301, 1321, 1381, 1399, 1429, 1439, 1459, 1493, 1531, 1549

Offset: 1

Tamas Sandor Nagy, Jul 19 2022

Terms of A034844 that only have nonprime digits are not terms here. - Michel Marcus, Jul 19 2022

			The prime 179 is a term because when its prime digit 7 is removed, it remains 19, which is still a prime.
The prime 136457911 is a term because when all of its prime digits, 3, 5, and 7 are removed, it remains 164911, which is still a prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A000040, A034844, A019546.

function a = A355856( max_prime )
    a = []; p = primes( max_prime );
    for n = 1:length(p)
        s = num2str(p(n));
        s = strrep(s,'2',''); s = strrep(s,'3','');
        s = strrep(s,'5',''); s = strrep(s,'7','');
        m = str2double(s);
        if m > 1
            if isprime(m) && m ~= p(n)
                a = [a p(n)];
            end
        end
    end
end % Thomas Scheuerle, Jul 19 2022

q[n_] := (r = FromDigits[Select[IntegerDigits[n], ! PrimeQ[#] &]]) != n && PrimeQ[r]; Select[Prime[Range[250]], q] (* Amiram Eldar, Jul 19 2022 *)

isok(p) = if (isprime(p), my(d=digits(p), v=select(x->(!isprime(x)), d)); (#v != #d) && isprime(fromdigits(v));) \\ Michel Marcus, Jul 19 2022

from sympy import isprime
def ok(n):
    s = str(n)
    if n < 10 or set(s) & set("2357") == set(): return False
    sd = s.translate({ord(c): None for c in "2357"})
    return len(sd) and isprime(int(sd)) and isprime(n)
print([k for k in range(2000) if ok(k)]) # Michael S. Branicky, Jul 23 2022

A092622 Primes with exactly two prime digits.

Original entry on oeis.org

23, 37, 53, 73, 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, 877, 937, 953, 977, 1033, 1123

Offset: 1

Jani Melik, Apr 11 2004

			23 is prime and it has two prime digits, 2 and 3;
127 is prime and it has two prime digits 2 and 7.

Cf. A034844.

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_stpf:=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))='true') then stpf:=stpf+1; # number of prime digits fi od; RETURN(stpf) end: ts_pr_prnd:=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_stpf(i) = 2) then ans:=[ op(ans), i ]: tren:=tren+1; fi od; RETURN(ans) end: ts_pr_prnd(2500);

a(n) >> n^1.285

A092623 Primes with exactly three prime digits.

Original entry on oeis.org

223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 1223, 1237, 1277, 1327, 1373, 1523, 1553, 1723, 1733, 1753, 1777, 2027, 2053, 2137, 2153, 2203, 2207, 2213, 2221, 2239, 2243, 2251, 2267, 2287, 2293, 2297, 2339, 2347, 2351, 2371

Offset: 1

Jani Melik, Apr 11 2004

			223 is prime and it has three prime digits 2,2,3;
1237 is prime and it has three prime digits 2,3,7;

Cf. A034844, 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_stpf:=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))='true') then stpf:=stpf+1; # number of prime digits fi od; RETURN(stpf) end: ts_pr_prnt:=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_stpf(i) = 3) then ans:=[ op(ans), i ]: tren:=tren+1; fi od; RETURN(ans) end: ts_pr_prnt(5000);

Select[Prime[Range[400]],Count[IntegerDigits[#],?PrimeQ]==3&] (* _Harvey P. Dale, Dec 27 2011 *)

a(n) >> x^1.285

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A092628 Primes with exactly three nonprime digits.

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A108614 Semiprimes with non-semiprimes digits (no digits 4,6,9 in semiprimes).

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A157527 Primes using only the composite digits (4, 6, 8, 9) and all of them.

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A177850 Smallest n-digit emirp with only nonprime digits.

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A179924 Primes with embedded primes.

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A215927 Primes having at least one digit that is not prime.

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A323391 Primes containing nonprime digits (from 1 to 9) in their decimal expansion and whose digits are distinct, i.e., consisting of only digits 1, 4, 6, 8, 9.

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A355856 Primes, with at least one prime digit, that remain primes when all of their prime digits are removed.

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