Displaying 61-70 of 101 results found.
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Minimal natural number (in decimal representation) with n prime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).
+10
2
1, 2, 7, 11, 23, 43, 93, 151, 239, 373, 479, 727, 1495, 2015, 2775, 5591, 6133, 7919, 12271, 22367, 24303, 30431, 48991, 89527, 95607, 98143, 129887, 357883, 358111, 382431, 744797, 519551, 1431007, 1432447, 1556319, 2457439
COMMENTS
The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=4*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 4^j = 2*(4^n - 1)/3 or m(n)=1, 2, 22, 222, 2222, 22222, …,for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-4 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.
No term is divisible by 4. a(1) = 2 is the only even term.
FORMULA
a(n) > 4^floor(sqrt(8*n-7)-1)/2), for n>0.
a(n) <= 2*(4^n - 1)/3, n>0.
a(n+1) <= 4*a(n) + 2.
EXAMPLE
a(1) = 2 = 2_4, since 2 is the least number with 1 prime substring in base-4 representation.
a(2) = 7 = 13_4, since 7 is the least number with 2 prime substrings in base-4 representation (3_4=3 and 13_4=7).
a(3) = 11 = 23_4, since 11 is the least number with 3 prime substrings in base-4 representation (2_4, 3_4, and 23_4).
a(5) = 43 = 223_4, since 43 is the least number with 5 prime substrings in base-4 representation (2 times 2_4, 3_4, 23_4=11, and 223_4=43).
a(7) = 151 = 2113_4, since 151 is the least number with 7 prime substrings in base-4 representation (2 times 2_4, 3_4, 11_4=5, 13_4=7, 113_4=23, and 2113_4=151).
Minimal natural number (in decimal representation) with n prime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).
+10
2
1, 2, 7, 13, 37, 88, 67, 192, 317, 932, 942, 1567, 4663, 4692, 8442, 23317, 23442, 36067, 102217, 114192, 180337, 192317, 511087, 901682, 582942, 2495443, 2555436, 2536067, 5289942, 12321061, 12680337, 12301692, 26461592, 61508461, 61508462, 63885918
COMMENTS
The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=5*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 5^j = (5^n - 1)/2 or m(n)=1, 2, 22, 222, 2222, 22222,…, (in base-5) for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-5 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be a prime number.
No term is divisible by 5.
FORMULA
a(n) > 5^floor(sqrt(8*n-7)-1)/2), for n>0.
a(n) <= (5^n - 1)/2, n>0.
EXAMPLE
a(1) = 2 = 2_5, since 2 is the least number with 1 prime substring in base-5 representation.
a(2) = 7 = 12_5, since 7 is the least number with 2 prime substrings in base-5 representation (2_5 and 12_5=7).
a(3) = 13 = 23_5, since 13 is the least number with 3 prime substrings in base-5 representation (2_5, 3_5, and 23_5).
a(4) = 37 = 122_5, since 37 is the least number with 4 prime substrings in base-5 representation (2 times 2_5, 12_5=7, and 122_5=37).
a(7) = 192 = 1232_5, since 192 is the least number with 7 prime substrings in base-5 representation (2 times 2_5, 3_5, 12_5=7, 23_5=13, 32_5=17, and 232_5=67).
CROSSREFS
Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685, A035244, A079397, A213300- A213321, A217302- A217309.
Minimal natural number (in decimal representation) with n prime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).
+10
2
1, 2, 11, 17, 47, 83, 269, 263, 479, 839, 1559, 1579, 2999, 5039, 9355, 9479, 14759, 56131, 56135, 61343, 56879, 336791, 341351, 336815, 341279, 341275, 2020727, 2020895, 2047651, 2020891, 4055159, 12098587, 12125347, 12285907, 15737755, 19128523, 39190247
COMMENTS
The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=6*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 6^j = 2*(6^n - 1)/5 or m(n)=1, 2, 22, 222, 2222, 22222, …, (in base-6) for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-6 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.
No term is divisible by 6.
FORMULA
a(n) > 6^floor(sqrt(8*n-7)-1)/2), for n>0.
a(n) <= 2*(6^n - 1)/5, n>0.
a(n+1) <= 6*a(n)+2.
EXAMPLE
a(1) = 2 = 2_6, since 2 is the least number with 1 prime substring in base-6 representation.
a(2) = 11 = 15_6, since 11 is the least number with 2 prime substrings in base-6 representation (5_6=5 and 15_6=11).
a(3) = 17 = 25_6, since 17 is the least number with 3 prime substrings in base-6 representation (2_6, 5_6, and 25_6).
a(4) = 47 = 115_6, since 47 is the least number with 4 prime substrings in base-6 representation (5_6, 11_6=7, 15_6=11, and 115_6=47).
a(8) = 479 = 2115_6, since 479 is the least number with 8 prime substrings in base-6 representation (2_6, 5_6, 11_6=7, 15_6=11, 21_6=13, 115_6=47, 211_6=79, and 2115_6=479).
Minimal natural number (in decimal representation) with n prime substrings in base-7 representation (substrings with leading zeros are considered to be nonprime).
+10
2
1, 2, 16, 17, 115, 121, 509, 821, 3251, 4721, 5749, 22760, 25301, 41673, 142950, 173819, 291714, 920561, 1222716, 2041709, 4450031, 8559017, 9350687, 14295199, 31150219, 50568439, 71502954, 100066398, 218051538, 353979075, 500526787, 702815371, 1512442643
COMMENTS
The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=7*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 7^j = (7^n - 1)/3 or m(n)=1, 2, 22, 222, 2222, 22222,…, (in base-7) for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-7 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.
No term is divisible by 7.
FORMULA
a(n) > 7^floor(sqrt(8*n-7)-1)/2), for n>0.
a(n) <= (7^n - 1)/3, n>0.
a(n+1) <= 7*a(n) + 2.
EXAMPLE
a(1) = 2 = 2_7, since 2 is the least number with 1 prime substring in base-7 representation.
a(2) = 16 = 22_7, since 16 is the least number with 2 prime substrings in base-7 representation (2 times 2_7=2).
a(3) = 17 = 23_7, since 17 is the least number with 3 prime substrings in base-7 representation (2_7, 3_7, and 23_7).
a(5) = 121 = 232_7, since 121 is the least number with 5 prime substrings in base-7 representation (2 times 2_7, 3_7, 23_7=17, and 32_7=23).
a(6) = 509 = 1325_7, since 509 is the least number with 6 prime substrings in base-7 representation (2_7, 3_7, 5_7, 25_7=19, 32_7=23, and 1325_7=509).
Primes that have both prime digits (2,3,5,7) and nonprime digits (1,4,6,8,9), without digits "0".
+10
2
13, 17, 29, 31, 43, 47, 59, 67, 71, 79, 83, 97, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 179, 193, 197, 211, 229, 239, 241, 251, 263, 269, 271, 281, 283, 293, 311, 313, 317, 331, 347, 349, 359, 367, 379, 383, 389, 397, 421, 431, 433, 439
MATHEMATICA
pdnpdQ[n_]:=Module[{idn=IntegerDigits[n], p, z=DigitCount[n, 10, 0]}, p=Count[ idn, _?PrimeQ]; p>0&&z==0&&Length[idn]>p]; Select[ Prime[ Range[ 150]], pdnpdQ] (* Harvey P. Dale, Oct 01 2013 *)
CROSSREFS
Cf. A018252, A019546, A034844, A038618, A087363, A092621, A092626, A152312, A152313, A152426, A152427.
First prime > 10^n in which every substring of length n is prime.
+10
1
23, 113, 1013, 10139, 100379, 1000037, 10000379, 100000193, 1000001237, 10000000097, 100000000193, 1000000000193, 10000000001777, 100000000001831, 1000000000036931, 10000000000001873, 100000000000000691
MATHEMATICA
Do[k = 10^n; While[ !PrimeQ[k] || Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[k], n, 1]]]] != {True}, k++ ]; Print[k], {n, 1, 25}]
Primes with exactly two nonprime digits.
+10
1
11, 19, 41, 61, 89, 103, 107, 113, 131, 139, 151, 163, 167, 179, 193, 197, 211, 241, 269, 281, 311, 349, 389, 421, 431, 439, 443, 463, 467, 479, 487, 509, 541, 569, 599, 607, 613, 617, 631, 643, 647, 659, 683, 701, 709, 719, 761, 769, 821, 829, 839, 859, 863
EXAMPLE
11 is prime and it has two nonprime digits, twice 1;
2269 is prime and it has two nonprime digits, 6 and 9.
MAPLE
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_neprnd:=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) = 2) then ans:=[ op(ans), i ]: tren:=tren+1; fi od; RETURN(ans) end: ts_pr_neprnd(4000);
MATHEMATICA
npd2Q[n_]:=Count[IntegerDigits[n], _?(!PrimeQ[#]&)]==2; Select[Prime[ Range[ 200]], npd2Q] (* Harvey P. Dale, May 12 2015 *)
Primes with exactly three nonprime digits.
+10
1
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
EXAMPLE
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.
MAPLE
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);
MATHEMATICA
dgQ[n_]:=Count[IntegerDigits[n], _?(!PrimeQ[#]&)]==3; Select[Prime[ Range[300]], dgQ] (* Harvey P. Dale, Oct 11 2011 *)
Primes whose digits are primes and reverse is prime.
+10
1
2, 3, 5, 7, 37, 73, 337, 353, 373, 727, 733, 757, 3257, 3373, 3527, 3733, 7253, 7523, 7577, 7757, 32233, 32257, 32323, 32353, 32377, 32537, 33223, 33533, 35227, 35257, 35323, 35327, 35353, 35537, 35753, 37273, 37573, 72227, 72253, 72337, 72353
MAPLE
listtoint:= proc(L) local i; add(L[i]*10^(i-1), i=1..nops(L)) end proc:
f:= proc(L) local s;
s:= listtoint(L);
if isprime(s) and isprime(listtoint(ListTools:-Reverse(L))) then s fi
end proc:
Cands:= [[3], [7]]:
A:= 2, 3, 5, 7:
for m from 2 to 6 do
Cands:= map(t -> seq([op(t), j], j=[2, 3, 5, 7]), Cands);
A:= A, op(sort(map(f, Cands)));
od:
MATHEMATICA
okQ[p_] := PrimeQ[IntegerReverse[p] && AllTrue[IntegerDigits[p], PrimeQ]];
PROG
(Magma) [p: p in PrimesUpTo(2*10^5) | Set(Intseq(p)) subset [2, 3, 5, 7] and IsPrime(Seqint(Reverse(Intseq(p))))]; // Vincenzo Librandi, Dec 04 2015
Numbers > 100 such that all the substrings of length = 2 are primes.
+10
1
111, 113, 117, 119, 131, 137, 171, 173, 179, 197, 231, 237, 297, 311, 313, 317, 319, 371, 373, 379, 411, 413, 417, 419, 431, 437, 471, 473, 479, 531, 537, 597, 611, 613, 617
COMMENTS
Only numbers > 100 are considered, since all 2-digit primes are trivial members. See A069488 for the sequence with prime terms > 100. The sequence is infinite (for example, consider the continued concatenation of ‘11’ or of ‘13’: 111, 1111, 11111, ..., 131, 1313, 13131, ... are members).
Infinitely many terms are palindromic.
EXAMPLE
a(2)=113, since all substrings of length = 2 are primes (11 and 13).
a(10)=197, since all substrings of length = 2 (19, 97) are primes.
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