cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A024770 Right-truncatable primes: every prefix is prime.

Original entry on oeis.org

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

Views

Author

David W. Wilson

Keywords

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

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

A202259 Right-truncatable nonprimes: every prefix is nonprime number.

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 40, 42, 44, 45, 46, 48, 49, 60, 62, 63, 64, 65, 66, 68, 69, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 120, 121, 122, 123, 124, 125, 126, 128, 129, 140, 141, 142
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2011

Keywords

Comments

Supersequence of A202260, A202265. - Barry Carter, Sep 16 2016

Links

Crossrefs

Cf. A012883 (right-truncatable noncomposites), A024770 (right-truncatable primes), A202260 (right-truncatable composites).

Programs

  • Maple
    filter:= proc(n) option remember; not isprime(n) and procname(floor(n/10)) end proc:
for i from 0 to 9 do filter(i):= not isprime(i) od:
select(filter, [$0..1000]); # Robert Israel, Nov 01 2016

A202260 Right-truncatable composites: every decimal prefix is a composite number.

Original entry on oeis.org

4, 6, 8, 9, 40, 42, 44, 45, 46, 48, 49, 60, 62, 63, 64, 65, 66, 68, 69, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 400, 402, 403, 404, 405, 406, 407, 408, 420, 422, 423, 424, 425, 426, 427, 428, 429, 440, 441, 442, 444, 445, 446, 447, 448
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2011

Keywords

Comments

Subsequence of A202259.

Links

Crossrefs

Cf. A012883 (right-truncatable noncomposites), A202259 (right-truncatable nonprimes), A024770 (right-truncatable primes).
Cf. A254750, A254752, A254754, A254755 (left-truncatable composites).

Programs

  • PARI
    isComposite(n) = (n>2)&&(!isprime(n));
isRightTruncatableComposite(n,b=10) = {my(k=b);if(!isComposite(n),return(0););while(n\k>0,if(!isComposite(n\k),return(0););k*=b);return(1);} \\ Stanislav Sykora, Feb 15 2015

A012884 Numbers such that every prefix and suffix is 1 or a prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 71, 73, 113, 131, 137, 173, 197, 311, 313, 317, 373, 797, 1373, 1997, 3137, 3797, 7331, 73331, 739397
Offset: 1

Views

Author

Larry Calmer (larry(AT)wri.com), Simon Plouffe

Keywords

Comments

Last term is 739397 (confirmed by David W. Wilson).
Intersection of A012883 and A143390. [Reinhard Zumkeller, Aug 13 2008]

Crossrefs

Cf. A068669.

Programs

  • Mathematica
    prQ[n_] := n == 1 || PrimeQ[n];
okQ[n_] := Module[{dd, nd}, dd = IntegerDigits[n]; nd = Length[dd]; AllTrue[Range[nd], prQ@ FromDigits@ Take[dd, #]&] && AllTrue[Range[nd-1], prQ@ FromDigits@ Drop[dd, #]&]];
{1}~Join~Select[Prime@Range[60000], okQ] (* Jean-François Alcover, Nov 20 2019 *)

A143390 Numbers in which every suffix (in base 10) is 1 or a prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 83, 97, 101, 103, 107, 113, 131, 137, 167, 173, 197, 211, 223, 241, 271, 283, 307, 311, 313, 317, 331, 337, 347, 353, 367, 373, 383, 397, 401, 431, 443, 461, 467, 503, 523, 541, 547, 571, 601
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 13 2008

Keywords

Comments

Subsequence of A042986 apart from first term; a(n+1)=A042986(n) for n<25.

Links

Crossrefs

Cf. A012883, A012884.

Programs

  • Mathematica
    prQ[n_] := n == 1 || PrimeQ[n];
okQ[n_] := Module[{dd = IntegerDigits[n]}, AllTrue[Range[Length[dd]-1], prQ@ FromDigits@ Drop[dd, #]&]];
{1}~Join~Select[ Prime@Range[1000], okQ] (* Jean-François Alcover, Nov 20 2019 *)
  • PARI
    is(n)=my(d=digits(n,10)); for(i=1,#d-1, if(!isprime(fromdigits(d[i..#d],10)), return(0))); isprime(d[#d]) || d[#d]==1 \\ Charles R Greathouse IV, Nov 26 2016

A202263 Primes in which all substrings and reversal substrings are primes.

Original entry on oeis.org

2, 3, 5, 7, 37, 73, 373
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2011

Keywords

Comments

Sequence is finite with 7 terms.
Subsequence of A085823, A068669, A024770, A012883.

Examples

			All substrings and reversal substrings of 373 are primes:3, 7, 37, 73, 373.

Crossrefs

Cf. A202264 (noncomposite numbers in which all substrings and reversal substrings are noncomposite), A202265 (nonprimes in which all substrings and reversal substrings are nonprimes), A202266 (composite numbers in which all substrings and reversal substrings are composites).

A202264 Noncomposite numbers in which all substrings and reversal substrings are noncomposites.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 113, 131, 311, 313, 373
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2011

Keywords

Comments

Sequence is finite with 17 terms.
Supersequence of A202263, A085823.
Subsequence of A068669, A012883, A024770, A012883.

Examples

			All substrings and reversal substrings of 311 are noncomposites: 1, 3, 11, 13, 31, 113, 311.

Crossrefs

Cf. A202263 (primes in which all substrings and reversal substrings are primes), A202265 (nonprimes in which all substrings and reversal substrings are nonprimes), A202266 (composite numbers in which all substrings and reversal substrings are composites).
Showing 1-7 of 7 results.

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