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 1-9 of 9 results.

A030469 Primes which are concatenations of three consecutive primes.

Original entry on oeis.org

5711, 111317, 171923, 313741, 414347, 8997101, 229233239, 239241251, 263269271, 307311313, 313317331, 317331337, 353359367, 359367373, 383389397, 389397401, 401409419, 409419421, 439443449, 449457461
Offset: 1

Views

Author

Patrick De Geest

Keywords

Comments

a(n) = "p(k) p(k+1) p(k+2)" where p(k) is k-th prime
It is conjectured that sequence is infinite. - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009

Examples

			(1) 5=p(3), 7=p(4), 11=p(5) gives a(1).
(2) 7=p(4), 11=p(5), 13=p(6), but 71113 = 7 x 10159

References

  • Richard E. Crandall, Carl Pomerance: Prime Numbers, Springer 2005 - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009
  • John Derbyshire: Prime obsession, Joseph Henry Press, Washington, DC 2003 - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009
  • Marcus du Sautoy: Die Musik der Primzahlen. Auf den Spuren des groessten Raetsels der Mathematik, Beck, Muenchen 2004

Links

Crossrefs

Cf. A030461, A167517, A132903, A068655, A030997, A030473, A086041, A099727.

Programs

  • Mathematica
    Select[Table[FromDigits[Flatten[IntegerDigits/@{Prime[n],Prime[n+1],Prime[n+2]}]],{n,11000}],PrimeQ] (* Zak Seidov, Oct 16 2009 *)
concat[{a_,b_,c_}]:=FromDigits[Flatten[IntegerDigits/@{a,b,c}]]; Select[ concat/@ Partition[ Prime[ Range[200]],3,1],PrimeQ] (* Harvey P. Dale, Sep 06 2017 *)
  • PARI
    for(i=1,999, isprime(p=eval(Str(prime(i),prime(i+1),prime(i+2)))) & print1(p," ")) \\ M. F. Hasler, Nov 10 2009

Formula

A132903 INTERSECT A000040. - R. J. Mathar, Nov 11 2009

A030997 Smallest prime which is a concatenation of n consecutive primes.

Original entry on oeis.org

2, 23, 5711, 2357, 711131719, 113127131137139149, 29313741434753, 107109113127131137139149, 211223227229233239241251257, 691701709719727733739743751757, 2329313741434753596167
Offset: 1

Views

Author

Patrick De Geest and Warut Roonguthai

Keywords

Examples

			a(5) = 711131719 is the smallest prime which is the concatenation of five consecutive primes 7, 11, 13, 17 and 19.

Links

Crossrefs

Cf. A030996, A068655, A030473, A086041, A099727, A167517.
Cf. A030461 (primes that are concatenations of two primes), A030469 (three primes), A030473 (four primes), A086041 (five primes).

Programs

  • PARI
    for(k=1,19, for(i=0,1e9, isprime( eval( p=concat( vector( k,j,Str( prime( i+j )))))) & break); print1(p,", ")) \\ M. F. Hasler, Nov 10 2009

A030473 Primes which are concatenations of 4 consecutive primes.

Original entry on oeis.org

2357, 67717379, 838997101, 139149151157, 149151157163, 383389397401, 503509521523, 557563569571, 577587593599, 587593599601, 601607613617, 613617619631, 727733739743, 937941947953, 1181118711931201
Offset: 1

Views

Author

Patrick De Geest

Keywords

Links

Crossrefs

Cf. A030461, A030469, A086040, A086041.

Programs

  • Maple
    P:= select(isprime,[2,seq(p,p=3..10^4,2)]):
select(isprime,[seq(P[i+3]+10^(1+ilog10(P[i+3]))*P[i+2] + 10^(2+ilog10(P[i+3])+ilog10(P[i+2]))*P[i+1] + 10^(3+ilog10(P[i+3])+ilog10(P[i+2])+ilog10(P[i+1]))*P[i], i=1..nops(P)-3)]); # Robert Israel, Apr 14 2016

Extensions

Edited by Charles R Greathouse IV, Apr 23 2010

A086040 Prime p concatenated with next 4 primes is also prime.

Original entry on oeis.org

7, 47, 53, 67, 97, 101, 149, 401, 431, 479, 487, 757, 827, 887, 1061, 1171, 1409, 1429, 1543, 1721, 1789, 1811, 1889, 1987, 2099, 2113, 2137, 2273, 2689, 2719, 2857, 3203, 3571, 3613, 3623, 3761, 3853, 3917, 3929, 4007, 4217, 4441, 4943, 5039, 5209, 5281, 5449
Offset: 1

Views

Author

Chuck Seggelin, Jul 07 2003

Keywords

Examples

			a(1) = 7 because 7, 11, 13, 17 and 19 concatenated together yield 711131719, which is prime.

Links

Crossrefs

Cf. A030459, A030468, A030472, A086041.

Programs

  • Mathematica
    Select[Partition[Prime[Range[800]],5,1],PrimeQ[FromDigits[Flatten[IntegerDigits/@#]]]&][[;;,1]] (* Harvey P. Dale, May 25 2025 *)
  • Python
    from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    plst = [2, 3, 5, 7, 11]
    slst = list(map(str, plst))
    while True:
        if isprime(int("".join(slst))):
            yield plst[0]
        plst = plst[1:] + [nextprime(plst[-1])]
        slst = slst[1:] + [str(plst[-1])]
print(list(islice(agen(), 50))) # Michael S. Branicky, Jan 26 2023

Extensions

Offset corrected by Arkadiusz Wesolowski, May 10 2012
a(15) and beyond from Michael S. Branicky, Jan 26 2023

A099727 Concatenations of six consecutive primes forming a prime.

Original entry on oeis.org

113127131137139149, 569571577587593599, 727733739743751757, 733739743751757761, 739743751757761769, 102110311033103910491051, 105110611063106910871091, 110911171123112911511153, 118111871193120112131217, 138113991409142314271429
Offset: 1

Views

Author

Ray G. Opao, Nov 07 2004

Keywords

Examples

			The prime 113127131137139149 is a concatenation of the consecutive primes 113, 127, 131, 137, 139 and 149.

Links

Crossrefs

Cf. A030461, A030469, A030473, A086040, A086041.

Programs

  • Maple
    select(isprime, [seq(parse(cat([seq(ithprime(i), i=n+0..n+5)][])), n=1..500)])[]; # K. D. Bajpai, Mar 24 2014
  • Mathematica
    Select[FromDigits[Flatten[IntegerDigits/@#]]&/@Partition[Prime[Range[ 300]],6,1],PrimeQ] (* Harvey P. Dale, Apr 30 2020 *)

A239789 Primes which are a concatenation of prime(k), prime(k+2) and prime(k+4) for some k.

Original entry on oeis.org

172331, 233141, 717989, 137149157, 191197211, 197211227, 223229239, 229239251, 257269277, 331347353, 353367379, 359373383, 467487499, 521541557, 617631643, 619641647, 647659673, 677691709, 733743757, 787809821, 797811823, 103310491061, 106110691091, 109711091123
Offset: 1

Views

Author

K. D. Bajpai, Mar 26 2014

Keywords

Examples

			172331 is a prime and appears in the sequence because it is the concatenation of prime(7), prime(7+2) and prime(7+4).
233141 is a prime and appears in the sequence because it is the concatenation of prime(9), prime(9+2) and prime(9+4).

Links

Crossrefs

Cf. A000040, A030461, A030469, A030473, A086040, A086041.

Programs

  • Maple
    with(StringTools): KD := proc() local a,b,d,e; a:=ithprime(n); b:=ithprime(n+2); d:=ithprime(n+4);
e:= parse(cat(a,b,d)); if isprime(e) then RETURN (e); fi; end: seq(KD(), n=1..500);
  • Mathematica
    Select[Table[FromDigits[Flatten[{IntegerDigits[Prime[n]], IntegerDigits[Prime[n+2]], IntegerDigits[Prime[n+4]]}]], {n,1,500}],PrimeQ]

A239974 Primes which are a concatenation of prime(k+4), prime(k+2) and prime(k) for some k.

Original entry on oeis.org

1373, 433729, 615343, 797161, 837367, 897971, 149137127, 193181173, 227211197, 337317311, 367353347, 401389379, 443433421, 557541521, 577569557, 587571563, 757743733, 811797773, 823811797, 10191009991, 10211013997, 116311511123, 120111871171, 130713011291
Offset: 1

Views

Author

K. D. Bajpai, Mar 30 2014

Keywords

Comments

All the terms in the sequence are primes which are a reverse concatenation of prime(k), prime(k+2) and prime(k+4) for some k.

Examples

			1373 is a prime and appears in the sequence because it is the concatenation of prime(2+4), prime(2+2) and prime(2).
433729 is a prime and appears in the sequence because it is the concatenation of prime(10+4), prime(10+2) and prime(10).

Links

Crossrefs

Cf. A000040, A030461, A030469, A030473, A086040, A086041, A239789.

Programs

  • Maple
    with(StringTools): KD := proc() local a, b, d, e; a:=ithprime(n+4); b:=ithprime(n+2); d:=ithprime(n);  e:= parse(cat(a, b, d)); if isprime(e) then RETURN (e); fi; end: seq(KD(), n=1..500);
  • Mathematica
    Select[Table[FromDigits[Flatten[{IntegerDigits[Prime[n+4]],IntegerDigits[Prime[n+2]], IntegerDigits[Prime[n]]}]], {n,1,500}], PrimeQ]

A244186 Primes which are the concatenation of five consecutive primes p, q, r, s, t while the sum (p + q + r + s + t) is another prime.

Original entry on oeis.org

711131719, 5359616771, 6771737983, 149151157163167, 401409419421431, 479487491499503, 757761769773787, 14091423142714291433, 18111823183118471861, 21132129213121372141, 26892693269927072711, 27192729273127412749, 36133617362336313637, 37613767376937793793
Offset: 1

Views

Author

K. D. Bajpai, Jun 21 2014

Keywords

Comments

Subsequence of A086041.
Numbers: Concatenation of 5 consecutive primes at A132905.

Examples

			711131719 is in the sequence because the concatenation of [7, 11, 13, 17, 19] = 711131719 which is prime. The sum [7 + 11 + 13 + 17 + 19] = 67 is another prime.
5359616771 is in the sequence because the concatenation of [53, 59, 61, 67, 71] = 5359616771 which is prime. The sum [53 + 59 + 61 + 67 + 71] = 311 is another prime.

Links

Crossrefs

Cf. A000040, A086041, A034961, A034962, A132903, A244163.

Programs

  • Mathematica
    FromDigits[Flatten[IntegerDigits/@#]]&/@Select[Partition[Prime[Range[ 1000]],5,1],AllTrue[{Total[#],FromDigits[Flatten[ IntegerDigits/@ #]]}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 24 2014 *)

A173448 Smallest prime(k) such that the concatenation prime(k)//prime(k+1)//...//prime(k+n-1) represents an emirp.

Original entry on oeis.org

13, 151, 353, 139, 101, 70451, 97, 15193, 3821, 9319, 7717, 103619, 10883, 18353, 108821, 701, 10091, 99251, 78497, 3559, 930043, 99787, 18671, 12251, 711751, 9293, 10861, 121921, 103099, 986189, 74287, 796567, 323003, 108707, 365779, 192377, 393901, 380251, 98479, 114343, 329729
Offset: 1

Views

Author

Lekraj Beedassy, Feb 18 2010

Keywords

Examples

			a(5) = 101 because 101103107109113 = A086041(6) is the smallest emirp formed by concatenating 5 consecutive primes (101, 103, 107, 109, 113).

Crossrefs

Cf. A006567, A167518, A030996.

Extensions

Keyword:base added and definition reworded by R. J. Mathar, Feb 24 2010
More terms from Sean A. Irvine, Nov 14 2010
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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