A069490 -id:A069490 - cp's OEIS frontend

A069489 Primes > 1000 in which every substring of length 3 is also prime.

1013, 1019, 1031, 1097, 1277, 1373, 1499, 1571, 1733, 1811, 1997, 2113, 2239, 2293, 2719, 3079, 3137, 3313, 3373, 3491, 3499, 3593, 3673, 3677, 3733, 3739, 3797, 4013, 4019, 4211, 4337, 4397, 4673, 4877, 4919, 5233, 5419, 5479, 6011, 6073, 6079, 6131

Offset: 1

Amarnath Murthy, Mar 30 2002

Minimum number of digits is taken to be 4 as all 3-digit primes would be trivial members.
Zero may occur only as second digit from left. - Zak Seidov, Dec 28 2020
All the digits after the two first digits from left are necessarily odd. - Bernard Schott, Mar 20 2022

			11317 is a term as the three substrings of length 3 i.e. 113,131 and 317 all are primes.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A069488 and A069490.

Cf. A211685, A010051, A000040.

a069489 n = a069489_list !! (n-1)
a069489_list = filter g $ dropWhile (<= 1000) a000040_list where
   g x = x < 100 || a010051 (x `mod` 1000) == 1 && g (x `div` 10)
-- Reinhard Zumkeller, Apr 07 2014

Do[ If[ Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[ Prime[n]], 3, 1]]]] == {True}, Print[ Prime[n]]], {n, PrimePi[1000] + 1, 10^3}]
Select[Prime[Range[169,800]],AllTrue[FromDigits/@Partition[ IntegerDigits[ #],3,1], PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 05 2019 *)

Edited, corrected and extended by Robert G. Wilson v, Apr 12 2002

A069488 Primes > 100 in which every substring of length 2 is also prime.

Original entry on oeis.org

113, 131, 137, 173, 179, 197, 311, 313, 317, 373, 379, 419, 431, 479, 613, 617, 619, 673, 719, 797, 971, 1117, 1171, 1319, 1373, 1973, 1979, 2311, 2371, 2971, 3119, 3137, 3719, 3797, 4111, 4373, 6113, 6131, 6173, 6197, 6719, 6737

Offset: 1

Amarnath Murthy, Mar 30 2002

Minimum number of digits is taken to be 3 as all two-digit primes would be trivial members.
From Robert G. Wilson v, May 12 2014: (Start)
The number of terms below 10^n: 0, 0, 21, 46, 123, 329, 810, 1733, 3985, 9710, ..., .
The least term with n digits is:  113, 1117, 11113, 111119, ..., see A090534.
The largest term with n digits is: 971, 9719, 97973, 979717, ..., see A242377.
The digits 2, 4, 5, 6 and 8 can only appear at the beginning of the prime and the digit 0 never appears. But the digits 1, 3, 7 and 9 can appear anywhere, yet only 1,1 can appear as a pair.
\10^n
d\  1&2   3    4    5     6     7     8      9     10 Total % @ 10^10
  \
1     0  19   34  146   648  1162  2678   8037  22740   39.188034
2     0   0    3    6    27    18    66    175    449    0.816186
3     0  14   19   63   326   712  1526   3855  11040   19.403018
4     0   3    2   13    54    92   143    384   1031    1.895550
5     0   0    0    9    17    24    45    176    426    0.763995
6     0   4    6    4    24    66   146    233    630    1.224834
7     0  14   20  100   436   907  1980   5442  15421   26.875285
8     0   0    3    6    24    25    37    176    388    0.721797
9     0   9   13   38   157   361   763   1790   5125    9.111301
Total 0  63  100  385  1713  3367  7384  20268  57250  100.00000
(End)

			3719 is a term as the three substrings of length 2, i.e., 37, 71 and 19, are all prime.

Robert G. Wilson v, Table of n, a(n) for n = 1..10101 (first 1000 terms from Reinhard Zumkeller)

Cf. A069489 and A069490.

Cf. A010051, subsequence of zeroless primes: A038618.

a069488 n = a069488_list !! (n-1)
a069488_list = filter f $ dropWhile (<= 100) a038618_list where
   f x = x < 10 || a010051 (x `mod` 100) == 1 && f (x `div` 10)
-- Reinhard Zumkeller, Apr 07 2014

Do[ If[ Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[ Prime[n]], 2, 1]]]] == {True}, Print[ Prime[n]]], {n, PrimePi[100] + 1, 500}]

Edited, corrected and extended by Robert G. Wilson v, Apr 12 2002

A168167 Numbers with d digits (d>0) which have at least 2d distinct primes as substrings.

Original entry on oeis.org

1373, 3137, 3797, 5237, 6173, 11317, 11373, 13733, 13739, 13797, 17331, 19739, 19973, 21137, 21317, 21373, 21379, 22397, 22937, 23117, 23137, 23173, 23371, 23373, 23719, 23797, 23971, 24373, 26173, 26317, 27193, 27197, 29173, 29537

Offset: 1

M. F. Hasler, Nov 28 2009

"Substrings" includes the whole number in itself.
The terms up to 11317 are primes themselves. The subsequence A168169 lists primes which have more than 2d prime substrings.
From Robert Israel, Nov 11 2020: (Start)
Palindromes in the sequence include 1337331, 1375731, and 1793971.
Even numbers in the sequence include 313732, 313792 and 1131712. (End)

			The least number with d digits to have 2d distinct prime substrings is a(1)=1373, with 4 digits and #{3, 7, 13, 37, 73, 137, 373, 1373} = 8.

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A069490, A131648, A012884, A168168, A168169.

filter:= proc(n) local i,j,count,d,S,x,y;
  d:= ilog10(n)+1;
  count:= 0; S:= {};
  for i from 0 to d-1 do
    x:= floor(n/10^i);
    for j from i to d-1 do
      y:= x mod 10^(j-i+1);
      if not member(y,S) and isprime(y) then count:= count+1; S:= S union {y}; if count = 2*d then return true fi fi
  od od;
  false
end proc:
select(filter, [$10..10^5]); # Robert Israel, Nov 11 2020

{for( p=1, 1e6, #prime_substrings(p) >= #Str(p)*2 & print1(p", "))} /* see A168168 for prime_substrings() */

A168169 Primes with d digits (d>0) which have more than 2d distinct primes as substrings.

Original entry on oeis.org

23719, 31379, 52379, 113171, 113173, 113797, 123719, 153137, 179719, 199739, 211373, 213173, 229373, 231197, 231379, 233113, 233713, 236779, 237331, 237619, 237971, 241973, 259397, 291373, 313739, 317971, 327193, 337397, 343373, 353173

Offset: 1

M. F. Hasler, Nov 28 2009

"Substrings" includes the whole number in itself.
This is a subsequence of A168167.
The least palindrome in this sequence is 9179719.

			The least number with d digits to have over 2d distinct prime substrings is the prime a(1)=23719, with 5 digits and #{2, 3, 7, 19, 23, 37, 71, 719, 2371, 3719, 23719} = 11.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A093301, A069490, A131648, A012884, A168167, A168168.

filter:= proc(n) local i,j,count,d,S,x,y;
  if not isprime(n) then return false fi;
  d:= ilog10(n)+1;
  count:= 0; S:= {};
  for i from 0 to d-1 do
    x:= floor(n/10^i);
    for j from i to d-1 do
      y:= x mod 10^(j-i+1);
      if not member(y,S) and isprime(y) then count:= count+1; S:= S union {y}; if count > 2*d then return true fi fi
  od od;
  false
end proc:
select(filter, [seq(i,i=1..10^6,2)]); # Robert Israel, Nov 11 2020

{forprime( p=1, default(primelimit), #prime_substrings(p) > #Str(p)*2 & print1(p", "))} /* see A168168 for prime_substrings() */

cp's OEIS Frontend

A069489 Primes > 1000 in which every substring of length 3 is also prime.

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A069488 Primes > 100 in which every substring of length 2 is also prime.

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A168167 Numbers with d digits (d>0) which have at least 2d distinct primes as substrings.

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Maple

PARI

A168169 Primes with d digits (d>0) which have more than 2d distinct primes as substrings.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI