A024770 -id:A024770 - cp's OEIS frontend

A173057 Partial sums of A024770.

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.

A085823 Numbers in which all substrings are primes.

Original entry on oeis.org

2, 3, 5, 7, 23, 37, 53, 73, 373

Offset: 1

Zak Seidov, Jul 04 2003

The definition implies that the number itself must be prime.
Apparently there are no such primes > 373.
From Jean-Marc Falcoz, Jan 11 2009: (Start)
This is correct.
There can't be any more terms, because they must necessarily be of the form
23737373733737... but the substring 237 is composite
or 273737373... but 273 is composite
or 5373737373... but 537 is composite
or 5737373737... but 573 is composite
or 37373737373... but 3737 is composite
or 7373737373... but 737 is composite
No other form is possible, otherwise, if the digit 2 or 5 is anywhere inside or at the end of the number, one substring-number is even or divisible by 5, and furthermore, there can't be twin digits, because one substring-number would then be divisible by 11.
Obviously, the digits 0, 1, 4, 6, 8, 9 can't appear anywhere in a term of the sequence. (End)
Subsequence of A024770 (right-truncatable primes), A068669 (noncomposite numbers in which all substrings are noncomposite). Supersequence of A202263 (primes in which all substrings and reversal substrings are primes). - Jaroslav Krizek, Jan 28 2012.
From Hieronymus Fischer, Apr 20 2012: (Start)
A more general definition is "Numbers such that all substrings of length <= 3 are primes". Proof: For numbers < 1000 this is plainly true. Suppose that there are such n >= 1000. We recognize that n must contain the string 373, as this is the only valid prime substring with the length 3. It follows, that there are substrings x37 or 73x, with any digit x. Evidently, neither x37 nor 73x are valid prime substrings, independent from the digit x. Thus, there is no number >= 1000 such that all substrings of length <= 3 are primes.
Also, numbers such that all substrings of length <= 2 are primes and the number of prime substrings of length = 3 is greater than m-3 for n <= 37373 and is greater than min(m-4,floor((m-1)/2) else; where m=floor(log_10(a(n)))+1 = number of digits. (End)

			Example : 373 is in the sequence, because 3, 7, 37, 73 and 373 are prime, but 733 is not in the sequence, because 33 is not prime.

Onno M. Cain, Prime-bounded subwords, arXiv:1912.08598 [math.HO], 2019.
NRICH, Strange Numbers
Henri Picciotto's Math Education Page, "Super-slimes" in Infinity, Unit 1

Cf. A085822, A166504, A213300.

Select[Prime@ Range[10^3], AllTrue[FromDigits /@ Rest@ Subsequences@ IntegerDigits@ #, PrimeQ] &] (* Michael De Vlieger, Jul 30 2018 *)

Thanks to Mark Underwood for pointing out misprints in the first version of this sequence.

Edited by N. J. A. Sloane, Jun 20 2009 at the suggestion of Lekraj Beedassy

A024785 Left-truncatable primes: every suffix is prime and no digits are zero.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 937, 947, 953, 967, 983, 997, 1223

Offset: 1

David W. Wilson

Last term is a(4260) = 357686312646216567629137 (Angell and Godwin). - Eric W. Weisstein, Dec 11 1999

Can be seen as table whose rows list n-digit terms, 1 <= n <= 25. Row lengths are A050987. - M. F. Hasler, Nov 07 2018

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.

N. J. A. Sloane, Table of n, a(n) for n = 1..4260 (The full list, based on the De Geest web site)
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
James Grime and Brady Haran, 357686312646216567629137, Numberphile video (2018).
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Rosetta Code, Programs for finding truncatable primes
Eric Weisstein's World of Mathematics, Truncatable Prime
Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.
Index entries for sequences related to truncatable primes

Supersequence of A240768.

Cf. A033664, A032437, A020994, A024770 (right-truncatable primes), A052023, A052024, A052025, A050986, A050987, A077390 (left-and-right truncatable primes), A137812 (left-or-right truncatable primes), A254753.

import Data.List (tails)
a024785 n = a024785_list !! (n-1)
a024785_list = filter (\x ->
   all (== 1) $ map (a010051 . read) $ init $ tails $ show x) a038618_list
-- Reinhard Zumkeller, Nov 01 2011

a:=[[2],[3],[5],[7]]: l1:=1: l2:=4: for n from 1 to 3 do for k from 1 to 9 do for j from l1 to l2 do d:=[op(a[j]),k]: 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
# second Maple program:
T:= proc(n) option remember; `if`(n=0, "", sort(select(isprime,
      map(x-> seq(parse(cat(j, x)), j=1..9), [T(n-1)])))[])
    end:
seq(T(n), n=1..4);  # Alois P. Heinz, Sep 01 2021

max = 2000; truncate[p_] := If[id = IntegerDigits[p]; FreeQ[id, 0] && (Last[id] == 3 || Last[id] == 7) && PrimeQ[q = FromDigits[ Rest[id]]], q, p]; ok[n_] := FixedPoint[ truncate, n] < 10;p = 5; A024785 = {2, 3, 5}; While[(p = NextPrime[p]) < max, If[ok[p], AppendTo[A024785, p]]]; A024785 (* Jean-François Alcover, Nov 09 2011 *)
d[n_]:=IntegerDigits[n]; ltrQ[n_]:=And@@PrimeQ[NestList[FromDigits[Drop[d[#],1]]&,n,Length[d[n]]-1]]; Select[Range[1225],ltrQ[#]&] (* Jayanta Basu, May 29 2013 *)
FullList=Sort[Flatten[Table[FixedPointList[Select[Flatten[Table[Range[9]*10^Length@IntegerDigits[#[[1]]] + #[[i]], {i, Length[#]}]], PrimeQ] &, {i}], {i, {2, 3, 5, 7}}]]] (* Fabrice Laussy, Nov 10 2019 *)

v=vector(4260);v[1]=2;v[2]=3;v[3]=5;v[4]=7;i=0;j=4; until(i>=j,i++;p=v[i];P10=10^(1+log(p)\log(10)); for(k=1,9,z=k*P10+p;if(isprime(z),j++;v[j]=z;))); s=vector(4260);s=vecsort(v);for(i=1,j,write("b024785.txt",i," ",s[i]);); \\

is_A024785(n,t=1)={until(t>10*p,isprime(p=n%t*=10)||return);n==p} \\ M. F. Hasler, Apr 17 2014

A024785=vector(25,n,p=vecsort(concat(apply(p->select(isprime, vector(9,i, i*10^(n-1)+p)),if(n>1,p))))); \\ Yields the list of rows (n-digit terms, n = 1..25). Use concat(%) to flatten. There are faster variants using matrices (vectorv(9,i,1)*p+[1..9]~*10^(n-1)*vector(#p,i,1)) and/or predefined vectors, but they are less concise and this takes less than 0.1 sec. - M. F. Hasler, Nov 07 2018

from sympy import isprime
def alst():
  primes, alst = [2, 3, 5, 7], []
  while len(primes) > 0:
    alst += sorted(primes)
    candidates = set(int(d+str(p)) for p in primes for d in "123456789")
    primes = [c for c in candidates if isprime(c)]
  return alst
print(alst()) # Michael S. Branicky, Apr 11 2021

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

A020994 Primes that are both left-truncatable and right-truncatable.

Original entry on oeis.org

2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397

Offset: 1

Mario Velucchi (mathchess(AT)velucchi.it)

Two-sided primes: deleting any number of digits at left or at right, but not both, leaves a prime.
Primes in which every digit string containing the most significant digit or the least significant digit is prime. - Amarnath Murthy, Sep 24 2003
Intersection of A024785 and A024770. - Robert Israel, Mar 23 2015

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, p. 178 (Rev. ed. 1997).

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
Index entries for sequences related to truncatable primes

Cf. A033664, A024785, A032437, A024770, A052023, A052024, A052025, A050986, A050987, A254751, A254753.

tspQ[n_] := Module[{idn=IntegerDigits[n], l}, l=Length[idn]; Union[PrimeQ/@(FromDigits/@ Join[Table[Take[idn, i], {i, l}], Table[Take[idn, -i], {i, l}]])]=={True}] Select[Prime[Range[PrimePi[740000]]], tspQ]

Corrected by David W. Wilson

Additional comments from Harvey P. Dale, Jul 10 2002

A050986 Number of n-digit right-truncatable primes.

Original entry on oeis.org

4, 9, 14, 16, 15, 12, 8, 5, 0

Offset: 1

Eric W. Weisstein

Right-truncatable means that the integer part of successive divisions by 10 always yields primes (or zero). - M. F. Hasler, Nov 07 2018

Jens Kruse Andersen, Right-truncatable primes
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Eric Weisstein's World of Mathematics, Truncatable Prime.
Index entries for sequences related to truncatable primes
Index entries for sequences related to numbers of primes in various ranges

Cf. A033664, A024785, A032437, A020994, A024770, A052023, A052024, A052025, A050987.

A050986=vector(9, n, #p=concat(apply(t->primes([t, t+1]*10), if(n>1, p)))) \\ M. F. Hasler, Nov 07 2018

Edited by Ray Chandler, Mar 13 2007

a(9) = 0 added by M. F. Hasler, Nov 07 2018

A050987 Number of n-digit left-truncatable primes.

Original entry on oeis.org

4, 11, 39, 99, 192, 326, 429, 521, 545, 517, 448, 354, 276, 212, 117, 72, 42, 24, 13, 6, 5, 4, 3, 1, 0

Offset: 1

Eric W. Weisstein

The sequence is well defined for any positive integer, with a(n) = 0 for n >= 25. But it makes sense to consider it to be full & finite. - M. F. Hasler, Nov 07 2018

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
P. De Geest, The list of 4260 left-truncatable primes
Eric Weisstein's World of Mathematics, Truncatable Prime.
Index entries for sequences related to numbers of primes in various ranges
Index entries for sequences related to truncatable primes

Cf. A033664, A024785, A032437, A020994, A024770, A052023, A052024, A052025, A050986.

A050987=vector(25, n, #p=concat(apply(p->select(isprime, vector(9, i, i*10^(n-1)+p)), if(n>1, p)))) \\ M. F. Hasler, Nov 07 2018

from sympy import isprime
def alst():
  primes, alst = [2, 3, 5, 7], [4]
  while len(primes) > 0:
    candidates = set(int(d+str(p)) for p in primes for d in "123456789")
    primes = [c for c in candidates if isprime(c)]
    alst.append(len(primes))
  return alst
print(alst()) # Michael S. Branicky, Apr 11 2021

Edited by Ray Chandler, Mar 13 2007

a(25) = 0 added by M. F. Hasler, Nov 07 2018

A052025 Every prefix (or suffix) of palindromic prime a(n) is prime (right/left-truncatable).

Original entry on oeis.org

2, 3, 5, 7, 313, 373, 797

Offset: 1

G. L. Honaker, Jr. and Patrick De Geest, Nov 15 1999

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
C. K. Caldwell, Left and Right truncatable primes.
P. De Geest, The list of 4260 left-truncatable primes
Eric Weisstein's World of Mathematics, Prime strings
Index entries for sequences related to truncatable primes

Cf. A033664, A024785, A032437, A020994, A024770, A052023, A052024, A050986, A050987.

A076586 Total number of right truncatable primes in base n.

Original entry on oeis.org

0, 4, 7, 14, 36, 19, 68, 68, 83, 89, 179, 176, 439, 373, 414, 473, 839, 1010, 1577, 2271, 2848, 1762, 3376, 5913, 6795, 6352, 10319, 5866, 14639, 13303, 19439, 29982, 38956, 39323, 58857, 41646, 68371, 80754, 128859, 81453, 175734, 161438, 228543, 396274, 538797

Offset: 2

Martin Renner, Oct 20 2002, Sep 24 2007

Seth A. Troisi, Table of n, a(n) for n = 2..100 (terms n=2..53 from Martin Renner)
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Index entries for sequences related to truncatable primes

Cf. A024763, A024764, A024765, A024766, A024767, A024768, A024769, A024770, A076623.

from sympy import isprime, primerange
from sympy.ntheory.digits import digits
def fromdigits(digs, base):
    return sum(d*base**i for i, d in enumerate(digs))
def a(n):
    prime_lists, an = [(p,) for p in primerange(1, n)], 0
    digits = 1
    while len(prime_lists) > 0:
        an += len(prime_lists)
        candidates = set((d,)+p for p in prime_lists for d in range(1, n))
        prime_lists = [c for c in candidates if isprime(fromdigits(c, n))]
        digits += 1
    return an
print([a(n) for n in range(2, 27)]) # Michael S. Branicky, May 03 2022

A237600 Right-truncatable primes in base 16.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 37, 41, 43, 47, 53, 59, 61, 83, 89, 113, 127, 179, 181, 191, 211, 223, 593, 599, 601, 607, 659, 661, 691, 701, 757, 761, 853, 857, 859, 863, 947, 953, 977, 983, 991, 1427, 1429, 1433, 1439, 1811, 1823, 2039, 2879, 2897, 2903, 2909, 3061

Offset: 1

Stanislav Sykora, Feb 15 2014

Numbers with these properties: (i) a(n) is a prime and (ii) its image under the function E(k) = k\16 = floor(k/16) is zero or has the same properties. [Corrected by M. F. Hasler, Nov 07 2018]
The sequence has 414 nonzero members.
Otherwise said, integers p > 0 such that floor(p/16^k) is prime or zero for all k >= 0. One might relax to p >= 0, i.e., include an initial term 0, corresponding to an empty string of digits. The recursive definition can also be used to produce all of the terms, starting with the primes < 16, and adding, for each term of the list, the primes made from appending a digit to that term, i.e., the primes between 16 x that term and 16 more. The sequence can also be seen as a table whose n-th row yields the terms with n digits in base 16: row lengths are A237601 and the last term of row n is A237602(n). - M. F. Hasler, Nov 07 2018

			a(414) = 16778492037124607, in hexadecimal notation 3B9BF319BD51FF, belongs to a(n) because each of its hexadecimal prefixes (including itself) is a prime. Being the largest of such numbers, it is also a member of A023107.

Stanislav Sykora, Table of n, a(n) for n = 1..414
Stanislav Sykora, PARI/GP scripts for genetic threads

Cf. A023107, A024770 (base 10), A237601, A237602, A254756.

Select[Range@ 3600, AllTrue[Most[DeleteDuplicates@ FixedPointList[f, #]], PrimeQ] &] (* Michael De Vlieger, Mar 07 2015, Version 10 *)

GT_Trunc1(nmax,prop,b=10) = { \\ See the link for details
  my (n=0,v=vector(nmax),g=1,lgs=1,lge,an,c);
  for (k=1,b-1,if (prop(k),v[n++]=k));
  lge=n; c=lge-lgs+1;
  while (c, g++;for (k=lgs,lge,for (m=0,b-1, an=b*v[k]+m;
    if (prop(an), v[n++]=an;if (n>=nmax,return (v)));););
    lgs=lge+1; lge=n; c=lge-lgs+1;);
  if (n, return (v[1..n]));
  print("No solution");}
v = GT_Trunc1(1000000,isprime,16)

isok(n)={ while(n, if(!isprime(n),return(0));n\=16); 1} \\ Joerg Arndt, Mar 07 2015

my(A=primes([0,15]),i=1); until(#AA237600=A \\ M. F. Hasler, Nov 07 2018

from gmpy2 import is_prime
A237600_list = []
for n in range(1,10**9):
    if is_prime(n):
        s = format(n,'x')
        for i in range(1,len(s)):
            if not is_prime(int(s[:-i],16)):
                break
        else:
            A237600_list.append(n) # Chai Wah Wu, Apr 16 2015

from sympy import primerange
p = lambda x: list(primerange(x,x+16)); A237600 = p(0); i=0
while iA237600): A237600+=p(A237600[i]*16); i+=1 # M. F. Hasler, Mar 11 2020

cp's OEIS Frontend

A173057 Partial sums of A024770.

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A085823 Numbers in which all substrings are primes.

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A024785 Left-truncatable primes: every suffix is prime and no digits are zero.

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A033664 Every suffix is prime.

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A020994 Primes that are both left-truncatable and right-truncatable.

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A050986 Number of n-digit right-truncatable primes.

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PARI

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A050987 Number of n-digit left-truncatable primes.

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PARI