A069866 -id:A069866 - cp's OEIS frontend

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

David W. Wilson

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

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.

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

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.

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

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

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 *)

{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

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

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

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

A033664 Every suffix is prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 137, 167, 173, 197, 223, 283, 307, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 503, 523, 547, 607, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 907, 937, 947

Offset: 1

N. J. A. Sloane

Primes in which repeatedly deleting the most significant digit gives a prime at every step until a single-digit prime remains.

Every digit string containing the least significant digit is prime. - Amarnath Murthy, Sep 24 2003

Alois P. Heinz, Table of n, a(n) for n = 1..66973 (first 8779 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Truncatable Prime.

Cf. A024785, A032437, A020994, A024770, A052023, A052024, A052025, A050986, A050987, A069866, A038680, A010051, A000040.

a033664 n = a033664_list !! (n-1)
a033664_list = filter (all ((== 1) . a010051. read) .
                           init . tail . tails . show) a000040_list
-- Reinhard Zumkeller, Jul 10 2013

T:= proc(n) option remember; `if`(n=0, "", select(isprime, [seq(seq(
      seq(parse(cat(j, 0$(n-i), p)), p=[T(i-1)]), i=1..n), j=1..9)])[])
    end:
seq(T(n), n=1..4);  # Alois P. Heinz, Sep 01 2021

h8pQ[n_]:=And@@PrimeQ/@Most[NestWhileList[FromDigits[Rest[ IntegerDigits[ #]]]&, n,#>0&]]; Select[Prime[Range[1000]],h8pQ] (* Harvey P. Dale, May 26 2011 *)

fileO="b033664.txt";lim=8779;v=vector(lim);v[1]=2;v[2]=3;v[3]=5;v[4]=7;j=4; write(fileO,"1 2");write(fileO,"2 3");write(fileO,"3 5");write(fileO,"4 7"); p10=1;until(0,p10*=10;j0=j;for(k=1,9,k10=k*p10; for(i=1,j0,if(j==lim,break(3));z=k10+v[i]; if(isprime(z),j++;v[j]=z;write(fileO,j," ",z);)))) \\ Harry J. Smith, Sep 20 2008

from sympy import isprime, primerange
def ok(p): # does prime p satisfy the property
    s = str(p)
    return all(isprime(int(s[i:])) for i in range(1, len(s)))
print(list(filter(ok, primerange(1, 1000)))) # Michael S. Branicky, Sep 01 2021

# alternate for going to large numbers
def agen(maxdigits):
    yield from [2, 3, 5, 7]
    primestrs, digits, d = ["2", "3", "5", "7"], "0123456789", 1
    while len(primestrs) > 0 and d < maxdigits:
        cands = set(d+p for p in primestrs for d in "0123456789")
        primestrs = [c for c in cands if c[0] == "0" or isprime(int(c))]
        yield from sorted(map(int, (p for p in primestrs if p[0] != "0")))
        d += 1
print([p for p in agen(11)]) # Michael S. Branicky, Sep 01 2021

More terms from Erich Friedman

A069867 Primes in which repeatedly deleting the least significant digit then the most significant digit gives a prime at every step until a single-digit prime remains.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 131, 137, 139, 173, 179, 233, 239, 373, 379, 431, 433, 439, 479, 673, 677, 733, 739, 839, 971, 977, 1319, 1373, 1733, 2237, 2239, 2293, 2297, 2711, 2713, 2719, 3313, 3319, 3371, 3373, 3533, 3539, 3593, 3733

Offset: 1

Amarnath Murthy, Apr 21 2002

Michael De Vlieger, Table of n, a(n) for n = 1..440 (Primes p <= 5 * 10^7).

Cf. A033664, A024770, A069866.

Select[Prime@ Range@ 512, AllTrue[FromDigits /@ Rest@ Fold[Append[#1, Delete[Last[#1], 1 - 2 Boole[OddQ@ #2]]] &, {#}, Range[Length@ # - 1]] &@ IntegerDigits[#], PrimeQ] &] (* Michael De Vlieger, Jan 20 2018 *)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Sep 24 2002

A173057 Partial sums of A024770.

Original entry on oeis.org

2, 5, 10, 17, 40, 69, 100, 137, 190, 249, 320, 393, 472, 705, 944, 1237, 1548, 1861, 2178, 2551, 2930, 3523, 4122, 4841, 5574, 6313, 7110, 9443, 11782, 14175, 16574, 19513, 22632, 25769, 29502, 33241, 37034, 40831, 46770, 53963, 61294, 68627

Offset: 1

Jonathan Vos Post, Feb 08 2010

Partial sums of right-truncatable primes, primes whose every prefix is prime (in decimal representation). The sequence has 83 terms. The subsequence of prime partial sums of right-truncatable primes begins: 2, 5, 17, 137, 1237, 1861, 2551, 199483. What is the largest value in the subsubsequence of right-truncatable prime partial sums of right-truncatable primes?

			a(50) = 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 + 31379.

Cf. A000040, A024770, A033664, A024785 (left-truncatable primes), A032437, A020994, A052023, A052024, A052025, A050986, A050987, A069866.

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A024770 Right-truncatable primes: every prefix is prime.

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A033664 Every suffix is prime.

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A069867 Primes in which repeatedly deleting the least significant digit then the most significant digit gives a prime at every step until a single-digit prime remains.

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Mathematica

Extensions

A173057 Partial sums of A024770.

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