A019546 -id:A019546 - cp's OEIS frontend

A217112 Greatest number (in decimal representation) with n nonprime substrings in binary representation (substrings with leading zeros are considered to be nonprime).

1, 3, 7, 6, 15, 14, 31, 29, 30, 63, 61, 62, 127, 54, 125, 126, 255, 117, 251, 254, 189, 511, 479, 509, 510, 379, 502, 1023, 1021, 1007, 1022, 958, 1018, 1014, 2047, 2045, 1791, 2046, 2042, 2027, 2037, 4091, 4095, 4063, 3069, 4094, 4090, 4085, 8159, 8187, 8191, 8189, 8127

Offset: 1

Hieronymus Fischer, Dec 20 2012

There are no numbers with zero nonprime substrings in binary representation. For all bases > 2 there is always a number (=2) with zero nonprime substrings.

The set of numbers with n nonprime substrings is finite. Proof: Evidently, each 1-digit binary number represents 1 nonprime substring. Hence, each (n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 2^n, such that all numbers > b have more than n nonprime substrings.

			(1) = 1, since 1 = 1_2 (binary) is the greatest number with 1 nonprime substring.
a(2) = 3 = 11_2 has 3 substrings in binary representation (1, 1 and 11), two of them are nonprime substrings (1 and 1), and 11_2 = 3 is the only prime substrings. 3 is the greatest number with 2 nonprime substrings.
a(8) = 29 = 11101_2 has 15 substrings in binary representation (0, 1, 1, 1, 1, 11, 11, 10, 01, 111, 110, 101, 1110, 1101, 11101), exactly 8 of them are nonprime substrings (0, 1, 1, 1, 1, 01, 110, 1110). There is no greater number with 8 nonprime substrings in binary representation.
a(14) = 54 = 110110_2 has 21 substrings in binary representation, only 7 of them are prime substrings (10, 10, 11, 11, 101, 1011, 1101), which implies that exactly 14 substrings must be nonprime. There is no greater number with 14 nonprime substrings in binary representation.

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300 - A213321.
Cf. A217102 - A217109, A217113 - A217119.
Cf. A217302 - A217309.

a(n) >= A217102(n).
a(n) >= A217302(A000217(A070939(a(n)))-n).
Example: a(9)=30=11110_2, A000217(A070939(31))=15, hence, a(9)>=A217302(15-9)=27.
a(n) <= 2^n.
a(n) <= 2^min(6 + n/6, 20*floor((n+125)/126)).
a(n) <= 64*2^(n/6).
With m := floor(log_2(a(n))) + 1:
a(n+m+1) >= 2*a(n), if a(n) is even.
a(n+m) >= 2*a(n), if a(n) is odd.

A217119 Greatest number (in decimal representation) with n nonprime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

47, 428, 1721, 6473, 14033, 35201, 58961, 58967, 465743, 530701, 530710, 1733741, 4250788, 4723108, 4776398, 25051529, 37327196, 42450640, 42986860, 42987589, 42996409, 225463817, 382055767, 382571822, 386888308, 386888419, 387356789

Offset: 0

Hieronymus Fischer, Dec 20 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 9^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n-(k(k+1)/2). For n=0,1,2,3,... the m(n) in base-9 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s. Thus, the number of nonprime substrings of m(n) is ((k+1)(k+2)/2)-k-1+i=(k(k+1)/2)+i=n. This proves the statement of existence. Proof of finiteness: Each 3-digit base-9 number has at least 1 nonprime substring. Hence, each 3(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 9^(3n+2) such that all numbers > b have more than n nonprime substrings. It follows, that the set of numbers with n nonprime substrings is finite.

			a(0) = 47, since 47 = 52_9 (base-9) is the greatest number with zero nonprime substrings in base-9 representation.
a(1) = 428 = 525_9 has 1 nonprime substring in base-9 representation (= 525_9). All the other base-9 substrings (2, 5, 5, 25, 52) are prime substrings. 525_9 is the greatest number with 1 nonprime substring.
a(2) = 1721 = 2322_9 has 10 substrings in base-9 representation, exactly 2 of them are nonprime substrings (22_9 and 23_3=8), and there is no greater number with 2 nonprime substrings in base-9 representation.
a(7) = 58967= 88788_9 has 15 substrings in base-9 representation, exactly 7 of them are nonprime substrings (4-times 8, 2-times 88, and 8788), and there is no greater number with 7 nonprime substrings in base-9 representation.

Hieronymus Fischer, Table of n, a(n) for n = 0..40

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300 - A213321.
Cf. A217102 - A217109, A217112 - A217118.
Cf. A217302 - A217309.

a(n) >= A217109(n).
a(n) >= A217309(A000217(num_digits_9(a(n)))-n), where num_digits_9(x)=floor(log_9(x))+1 is the number of digits of the base-9 representation of x.
a(n) <= 9^(n+2).
a(n) <= 9^min(n+2, 6*floor((n+7)/8)).
a(n) <= 9^((3/4)*(n + 3)).
a(n+m+1) >= 9*a(n), where m := floor(log_9(a(n))) + 1.

A152313 Primes without 0's or primes in their decimal expansion.

Original entry on oeis.org

11, 19, 41, 61, 89, 149, 181, 191, 199, 419, 449, 461, 491, 499, 619, 641, 661, 691, 811, 881, 911, 919, 941, 991, 1181, 1481, 1489, 1499, 1619, 1669, 1699, 1811, 1861, 1889, 1949, 1999, 4111, 4441, 4481, 4649, 4691, 4861, 4889, 4919, 4969, 4999

Offset: 1

Omar E. Pol, Dec 02 2008

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A019546, A034844, A087363, A152312.

[p: p in PrimesUpTo(5000) | Set(Intseq(p))  subset [1,4,6,8,9]]; // Vincenzo Librandi, Oct 25 2016

F:= proc(d) local T, R, L, r;
   R:= NULL;
   T:= combinat:-cartprod([[1,4,6,8,9]$d]);
   while not T[finished] do
     L:= T[nextvalue]();
     r:= add(L[i]*10^(d-i),i=1..d);
     if isprime(r) then R:= R,r fi
   od;
R
end proc:
seq(F(d),d=2..5); # Robert Israel, Dec 07 2017

Select[Prime[Range[800]], Complement[IntegerDigits[#], {1, 4, 6, 8, 9}] == {} &] (* Vincenzo Librandi, Oct 25 2016 *)

A214705 Primes that contain only the digits (2, 5, 7).

Original entry on oeis.org

2, 5, 7, 227, 257, 277, 557, 577, 727, 757, 2557, 2777, 5227, 5527, 5557, 7577, 7727, 7757, 22277, 22727, 22777, 25577, 27277, 27527, 52727, 52757, 57527, 57557, 57727, 72227, 72277, 72577, 72727, 75227, 75277, 75527, 75557, 75577, 77527, 77557, 222527, 222557

Offset: 1

Vincenzo Librandi, Jul 28 2012

The digits are prime numbers.

Jason Bard, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Subsequence of A019546.

Cf. A385768 - A385800.

[p: p in PrimesUpTo(100000) | Set(Intseq(p)) subset [2,5,7]];

Flatten[Table[Select[FromDigits/@Tuples[{2,5,7},n],PrimeQ],{n,6}]]

from sympy import primerange
def ok(p): return set(str(p)) <= set("257")
def aupto(limit): return [p for p in primerange(2, limit+1) if ok(p)]
print(aupto(222557)) # Michael S. Branicky, Feb 05 2021

A083185 Palindromic primes using only nonprime digits (0,1,4,6,8,9).

Original entry on oeis.org

11, 101, 181, 191, 919, 10601, 11411, 16061, 16661, 18181, 18481, 19891, 19991, 91019, 94049, 94649, 94849, 94949, 96469, 98689, 1008001, 1114111, 1160611, 1180811, 1186811, 1190911, 1196911, 1409041, 1411141, 1444441, 1461641

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 26 2003

Harvey P. Dale, Table of n, a(n) for n = 1..100
P. De Geest, World!Of Palindromic Primes

Cf. A045336, A019546, A002385.

Palindromes in A034844.

Select[ Prime[ Range[111500]], IntegerDigits[ # ] == Reverse[ IntegerDigits[ # ]] && Union[ Join[ IntegerDigits[ # ], {0, 1, 4, 6, 8, 9}]] == {0, 1, 4, 6, 8, 9} & ]
Select[Prime[Range[120000]],PalindromeQ[#]&&NoneTrue[IntegerDigits[#], PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 08 2018 *)

Edited and extended by Patrick De Geest, Jun 11 2003

A085557 Numbers that have more prime digits than nonprime digits.

Original entry on oeis.org

2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 122, 123, 125, 127, 132, 133, 135, 137, 152, 153, 155, 157, 172, 173, 175, 177, 202, 203, 205, 207, 212, 213, 215, 217, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232

Offset: 1

Jason Earls, Jul 04 2003

Begins to differ from A046034 at the 21st term (which is the first 3-digit term).

			133 is in the sequence as the prime digits are 3 and 3 (those are two digits; counted with multiplicity) and one nonprime digit 1 and so there are more prime digits than nonprime digits. - _David A. Corneth_, Sep 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A193238, A046034, A046035, A118950, A019546, A203263, A035232, A039996, A085823, A052382, A084544, A084984, A017042, A001743, A001744, A014261, A014263, A202267, A202268, A211681.

is(n) = my(d = digits(n), c = 0); for(i = 1, #d, if(isprime(d[i]), c++)); c<<1 > #d \\ David A. Corneth, Sep 06 2020

from itertools import count, islice
def A085557_gen(startvalue=1): # generator of terms
    return filter(lambda n:len(s:=str(n))<(sum(1 for d in s if d in {'2','3','5','7'})<<1),count(max(startvalue,1)))
A085557_list = list(islice(A085557_gen(),20)) # Chai Wah Wu, Feb 08 2023

A092624 Numbers with exactly two prime digits.

Original entry on oeis.org

22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 122, 123, 125, 127, 132, 133, 135, 137, 152, 153, 155, 157, 172, 173, 175, 177, 202, 203, 205, 207, 212, 213, 215, 217, 220, 221, 224, 226, 228, 229, 230, 231, 234, 236, 238, 239, 242, 243, 245

Offset: 1

Jani Melik, Apr 11 2004

A193238(a(n))=2; subsequence of A118950. [Reinhard Zumkeller, Jul 19 2011]

			25 has two prime digits, 2 and 5;
207 has two prime digits, 2 and 7.

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

Cf. A034895, A046034, A085557, A019546, A034844.

Cf. A084984, A092620, A092625.

import Data.List (elemIndices)
a092624 n = a092624_list !! (n-1)
a092624_list = elemIndices 2 a193238_list
-- Reinhard Zumkeller, Jul 19 2011

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_nd:=proc(n) local i, stpf, ans, ans1, tren; ans:=[ ]: stpf:=0: tren:=1: for i from 1 to n do if ( ts_stpf(i) = 2) then ans:=[ op(ans), i ]: tren:=tren+1; fi od; RETURN(ans) end: ts_pr_nd(500);

Select[Range[300],Count[IntegerDigits[#],?PrimeQ]==2&] (* _Harvey P. Dale, Apr 20 2025 *)

A092625 Numbers with exactly three prime digits.

Original entry on oeis.org

222, 223, 225, 227, 232, 233, 235, 237, 252, 253, 255, 257, 272, 273, 275, 277, 322, 323, 325, 327, 332, 333, 335, 337, 352, 353, 355, 357, 372, 373, 375, 377, 522, 523, 525, 527, 532, 533, 535, 537, 552, 553, 555, 557, 572, 573, 575, 577, 722, 723, 725

Offset: 1

Jani Melik, Apr 11 2004

It is the same as A046034 from two digit numbers from 22 up to four digit numbers from 1222.

A193238(a(n))=3; subsequence of A118950. [Reinhard Zumkeller, Jul 19 2011]

			222 has three prime digits, three times 2;
1235 has three prime digits, 2, 3 and 5.

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

Cf. A034895, A046034, A085557, A019546, A034844.

Cf. A084984, A092620, A092624.

import Data.List (elemIndices)
a092625 n = a092625_list !! (n-1)
a092625_list = elemIndices 3 a193238_list
-- Reinhard Zumkeller, Jul 19 2011

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_nt:=proc(n) local i, stpf, ans, ans1, tren; ans:=[ ]: stpf:=0: tren:=1: for i from 1 to n do if ( ts_stpf(i) = 3) then ans:=[ op(ans), i ]: tren:=tren+1; fi od; RETURN(ans) end: ts_pr_nt(2000);

Select[Range[800],Total[Boole[PrimeQ[IntegerDigits[#]]]]==3&] (* Harvey P. Dale, Dec 31 2023 *)

A108419 Primes with at least one of each prime digit.

Original entry on oeis.org

2357, 2753, 3257, 3527, 5237, 5273, 7253, 7523, 22573, 23357, 23537, 23557, 23753, 25237, 25357, 25373, 25537, 25733, 27253, 32257, 32537, 32573, 35227, 35257, 35327, 35527, 37253, 52237, 52733, 53327, 53527, 57223, 72253, 72353, 72533, 73523, 75223, 75253

Offset: 1

Rick L. Shepherd, Jun 02 2005

This is a subsequence of A019546.

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

Cf. A019546 (primes whose digits are primes).

Select[Table[Prime[n],{n,7200}],ContainsExactly[IntegerDigits[#],{2,3,5,7}]&] (* James C. McMahon, Mar 05 2024 *)

from sympy import isprime
from itertools import count, islice, product
def agen(): yield from (t for d in count(4) for b in product("2357", repeat=d-1) for e in "37" if len(set(b+(e,)))==4 and isprime(t:=int("".join(b)+e)))
print(list(islice(agen(), 40))) # Michael S. Branicky, Mar 05 2024

Offset corrected by Arkadiusz Wesolowski, Oct 18 2011

A124674 Primes with distinct prime digits.

Original entry on oeis.org

2, 3, 5, 7, 23, 37, 53, 73, 257, 523, 2357, 2753, 3257, 3527, 5237, 5273, 7253, 7523

Offset: 1

Tanya Khovanova, Dec 24 2006

Caldwell and Honaker, 2357, Prime Curios!

Cf. A019546 (primes whose digits are primes), A124673.

Select[Range[10000], PrimeQ[ # ] && Length[IntegerDigits[ # ]] == Length[Union[IntegerDigits[ # ]]] && Complement[IntegerDigits[ # ], {2, 3, 5, 7}] == {} &]

is(k) = isprime(k) && setintersect([2, 3, 5, 7], v=vecsort(digits(k))) == v; \\ Jinyuan Wang, Mar 27 2020

cp's OEIS Frontend

A217112 Greatest number (in decimal representation) with n nonprime substrings in binary representation (substrings with leading zeros are considered to be nonprime).

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A217119 Greatest number (in decimal representation) with n nonprime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime).

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A152313 Primes without 0's or primes in their decimal expansion.

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Magma

Maple

Mathematica

A214705 Primes that contain only the digits (2, 5, 7).

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Magma

Mathematica

Python

A083185 Palindromic primes using only nonprime digits (0,1,4,6,8,9).

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Mathematica

Extensions

A085557 Numbers that have more prime digits than nonprime digits.

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PARI

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A092624 Numbers with exactly two prime digits.

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Haskell

Maple

Mathematica

A092625 Numbers with exactly three prime digits.

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