A024770 Right-truncatable primes: every prefix is prime.

2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797, 5939, 7193, 7331, 7333, 7393, 23333, 23339, 23399, 23993, 29399, 31193

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• base
• easy
• fini
• full
• nice
• tabf

Comments

Primes in which repeatedly deleting the least significant digit gives a prime at every step until a single-digit prime remains. The sequence ends at a(83) = 73939133 = A023107(10).

The subsequence which consists of the following "chain" of consecutive right truncatable primes: 73939133, 7393913, 739391, 73939, 7393, 739, 73, 7 yields the largest sum, compared with other chains formed from subsets of this sequence: 73939133 + 7393913 + 739391 + 73939 + 7393 + 739 + 73 + 7 = 82154588. - Alexander R. Povolotsky, Jan 22 2008

Can also be seen as a table whose n-th row lists the n-digit terms; row lengths (0 for n >= 9) are given by A050986. The sequence can be constructed starting with the single-digit primes and appending, for each p in the list, the primes within 10*p and 10(p+1), formed by appending a digit to p. - M. F. Hasler, Nov 07 2018

References

• Roozbeh Hazrat, Mathematica: A Problem-Centered Approach, Springer London 2010, pp. 86-89.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 112-113.

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..83 (The full list of terms, taken from link below)
• Jens Kruse Andersen, Right-truncatable primes
• I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
• Patrick De Geest, The list of 4260 left-truncatable primes
• R. Schroeppel, HAKMEM item 33; "Russian Doll Primes", but with a slightly different definition.
• Eric Weisstein's World of Mathematics, Truncatable Prime
• Index entries for sequences related to truncatable primes

Crossrefs

Supersequence of A085823, A202263. Subsequence of A012883, A068669. - Jaroslav Krizek, Jan 28 2012

Supersequence of A239747.

Cf. A033664, A024785 (left-truncatable primes), A032437, A020994, A052023, A052024, A052025, A050986, A050987, A069866, A077390 (left-and-right-truncatable primes), A137812 (left-or-right truncatable primes), A254751, A254753.

Cf. A237600 for the base-16 analog.

Programs

Haskell

import Data.List (inits)
a024770 n = a024770_list !! (n-1)
a024770_list = filter (\x ->
   all (== 1) $ map (a010051 . read) $ tail $ inits $ show x) a038618_list
-- Reinhard Zumkeller, Nov 01 2011

Maple

s:=[1,3,7,9]: a:=[[2],[3],[5],[7]]: l1:=1: l2:=4: do for j from l1 to l2 do for k from 1 to 4 do d:=[s[k],op(a[j])]: if(isprime(op(convert(d, base, 10, 10^nops(d)))))then a:=[op(a), d]: fi: od: od: l1:=l2+1: l2:=nops(a): if(l1>l2)then break: fi: od: seq(op(convert(a[j], base, 10, 10^nops(a[j]))),j=1..nops(a)); # Nathaniel Johnston, Jun 21 2011

Mathematica

max = 100000; truncate[p_] := If[PrimeQ[q = Quotient[p, 10]], q, p]; ok[p_] := FixedPoint[ truncate, p] < 10; p = 1; A024770 = {}; While[ (p = NextPrime[p]) < max, If[ok[p], AppendTo[ A024770, p]]]; A024770 (* Jean-François Alcover, Nov 09 2011, after Pari *)
eppQ[n_]:=AllTrue[FromDigits/@Table[Take[IntegerDigits[n],i],{i, IntegerLength[ n]-1}], PrimeQ]; Select[Prime[Range[3400]],eppQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 14 2015 *)

PARI

{fileO="b024770.txt";v=vector(100);v[1]=2;v[2]=3;v[3]=5;v[4]=7;j=4;j1=1; write(fileO,"1 2");write(fileO,"2 3");write(fileO,"3 5");write(fileO,"4 7"); until(0,if(j1>j,break);new=1;for(i=j1,j,if(new,j1=j+1;new=0);for(k=1,9, z=10*v[i]+k;if(isprime(z),j++;v[j]=z;write(fileO,j," ",z);))));} \\ Harry J. Smith, Sep 20 2008

PARI

for(n=2, 31193, v=n; while(isprime(n), c=n; n=(c-lift(Mod(c, 10)))/10); if(n==0, print1(v, ", ")); n=v); \\ Arkadiusz Wesolowski, Mar 20 2014

PARI

A024770=vector(9, n, p=concat(apply(t->primes([t, t+1]*10), if(n>1, p)))) \\ The list of n-digit terms, 1 <= n <= 9. Use concat(%) to "flatten" it. - M. F. Hasler, Nov 07 2018

Python

from sympy import primerange
p = lambda x: list(primerange(x, x+10)); A024770 = p(0); i=0
while iA024770): A024770+=p(A024770[i]*10); i+=1 # M. F. Hasler, Mar 11 2020