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.

A158232 Numbers which yield primes when "13" is prefixed or appended: N natural number is a member of the sequence, if P="13N" (prefixed 13) and A="N13" (appended 13) are prime.

Original entry on oeis.org

1, 19, 21, 27, 61, 103, 121, 127, 147, 159, 177, 183, 187, 217, 241, 259, 267, 327, 331, 337, 367, 381, 411, 477, 523, 553, 567, 577, 591, 633, 681, 687, 693, 709, 723, 759, 807, 829, 873, 903, 931, 997, 1009, 1011, 1041, 1059, 1129, 1149, 1213, 1231, 1251
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 14 2009

Keywords

Comments

It is conjectured and numerically examined that sequences of this type are infinite.
It is also conjectured that an infinite number of primes are terms of the sequence; first 20 primes are: 19, 61, 103, 127, 241, 331, 337, 367, 523, 577, 709, 829, 997, 1009, 1129, 1213, 1381, 1489, 1543, 1627.

Examples

			19: 1319 and 1913 are primes => a(2)=19;
7 is not a term: 137 is prime but 713=23 * 31 is not.
		

References

  • A. Weil, Number theory: an approach through history, Birkhäuser, 1984.
  • Richard E. Crandall, Carl Pomerance, Prime Numbers, Springer 2005.
  • Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996.

Crossrefs

Cf. A157772.

Programs

  • Maple
    A055642 := proc(n) max(1,ilog10(n)+1) ; end proc: cat2 := proc(a,b) a*10^A055642(b)+b ; end proc: A158232 := proc(n) option remember; local a; if n = 1 then 1; else for a from procname(n-1)+1 do if isprime(cat2(13,a)) and isprime(cat2(a,13)) then return a ; end if ; end do ; end if; end proc: seq(A158232(n),n=1..80) ; # R. J. Mathar, Nov 11 2009
  • Mathematica
    Select[Range[1300],And@@PrimeQ[{13 10^IntegerLength[#]+#,100#+13}]&] (* Harvey P. Dale, May 28 2012 *)

A102339 Numbers k such that k*10^3 + 333 is prime.

Original entry on oeis.org

2, 5, 7, 10, 16, 17, 19, 20, 23, 29, 31, 38, 41, 49, 50, 55, 56, 59, 61, 64, 71, 76, 79, 85, 92, 100, 101, 103, 121, 134, 136, 139, 140, 143, 149, 154, 155, 161, 175, 176, 178, 182, 184, 188, 208, 209, 211, 217, 220, 232, 236, 239, 241, 244, 265, 266, 269, 271, 272, 274, 286, 287, 295, 299, 301, 308
Offset: 1

Views

Author

Parthasarathy Nambi, Feb 20 2005

Keywords

Comments

10^3 and 333 are relatively prime, therefore by Dirichlet's theorem there are infinitely many primes in the arithmetic progression n*10^3+333. No term of the sequence is of the form 3*k, because 3*k*10^3+333 = 3*(k*10^3+111) is divisible by 3, violating the requirement of the definition. - Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 27 2009

Examples

			If k=2,  then k*10^3 + 333 =  2333 (prime).
If k=49, then k*10^3 + 333 = 49333 (prime).
If k=92, then k*10^3 + 333 = 92333 (prime).
		

Crossrefs

Programs

  • Magma
    [ n: n in [1..700] | IsPrime(Seqint([3,3,3] cat Intseq(n))) ]; // Vincenzo Librandi, Feb 04 2011
    
  • Magma
    [ n: n in [0..320] | IsPrime(n*10^3+333) ]; // Klaus Brockhaus, May 20 2009
    
  • Mathematica
    Select[Range[400],PrimeQ[FromDigits[Join[IntegerDigits[#],{3,3,3}]]]&] (* Harvey P. Dale, Oct 14 2014 *)
    Select[Range[0, 1000], PrimeQ[1000 # + 333] &] (* Vincenzo Librandi, Jan 19 2013 *)
  • PARI
    is(n)=isprime(1000*n+333) \\ Charles R Greathouse IV, Jun 06 2017

A102343 Numbers k such that k*10^3 + 777 is prime.

Original entry on oeis.org

1, 2, 11, 19, 22, 26, 41, 43, 44, 47, 50, 53, 65, 67, 68, 71, 76, 79, 80, 83, 94, 97, 107, 110, 113, 115, 122, 124, 125, 131, 134, 136, 137, 145, 146, 152, 155, 158, 167, 169, 170, 173, 176, 181, 184, 199, 202, 211, 212, 226, 229, 232, 233, 250, 253, 254, 268, 272, 274, 281, 284, 286, 292, 295, 298, 299
Offset: 1

Views

Author

Parthasarathy Nambi, Feb 20 2005

Keywords

Comments

From Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 27 2009: (Start)
The sequence is infinite by Dirichlet's theorem about primes in arithmetic progression.
No term of the sequence is of form 3k, because the sum of digits of 10^3*3k + 333 = 3*(10^3 + 259) is divisible by 3, violating the requirement of the definition. (End)

Examples

			k=1: 1*10^3 + 777 = 1777 is prime, hence 1 is in the sequence.
k=50: 50*10^3 + 777 = 50777 is prime, hence 50 is in the sequence.
k=97: 97*10^3 + 777 = 97777 is prime, hence 97 is in the sequence.
		

Crossrefs

Programs

  • Magma
    [ n: n in [0..300] | IsPrime(n*10^3+777) ];
    
  • Mathematica
    Select[Range[300],PrimeQ[1000#+777]&] (* Harvey P. Dale, Jun 06 2022 *)
  • PARI
    is(n)=isprime(n*10^3+777) \\ Charles R Greathouse IV, Jun 13 2017

Extensions

Extended by Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 27 2009
Edited by R. J. Mathar, Apr 30 2009
More terms from Vincenzo Librandi, May 01 2009

A158521 Primes which yield primes when "13" is prefixed or appended.

Original entry on oeis.org

19, 61, 103, 127, 241, 331, 337, 367, 523, 577, 709, 829, 997, 1009, 1129, 1213, 1231, 1321, 1381, 1489, 1543, 1627, 1861, 2113, 2137, 2287, 2347, 2383, 2689, 2851, 2953, 2971, 3187, 3499, 3559, 3583, 3673, 3967, 4219, 4243, 4327, 4363, 4513, 4591, 4789
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 20 2009

Keywords

Comments

Primes in A158232.
It is conjectured that this sequence is infinite.

Examples

			Prime p=3 is not a term: "p13"=313 is prime but "13p"=133 = 7*19.
For p=19, both 1319 and 1913 are prime; this is the first prime that meets the requirements of the definition, so a(1)=19.
		

References

  • Richard E. Crandall, Carl Pomerance, Prime Numbers, Springer, 2005.
  • Wladyslaw Narkiewicz, The development of prime number theory, Springer, 2000.

Crossrefs

Programs

  • Maple
    cat2 := proc(a,b) ndigsb := max(ilog10(b)+1,1) ; a*10^ndigsb+b ; end: for i from 1 to 800 do p := ithprime(i) ; if isprime(cat2(13,p)) and isprime(cat2(p,13)) then printf("%d,",p) ; fi; od: # R. J. Mathar, Apr 02 2009
  • Mathematica
    Select[Prime[Range[1000]],AllTrue[{13*10^IntegerLength[#]+#,100#+13}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 17 2015 *)

Formula

Prime p is a term if the concatenations "13p" and "p13" both yield primes.

Extensions

337, 1231, 1321 inserted by R. J. Mathar, Apr 02 2009

A176722 Primes of the form k^3 + 13, k >= 0.

Original entry on oeis.org

13, 229, 1013, 1741, 39317, 64013, 74101, 157477, 438989, 551381, 830597, 1906637, 2000389, 4096013, 7077901, 9261013, 10941061, 15625013, 16003021, 21024589, 24897101, 27000013, 69934541, 74088013, 79507013, 93576677, 122023949
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 25 2010

Keywords

Comments

Necessarily, k = 6 * j or k = 6 * j + 4.
Values of k corresponding to terms of the sequence: 0, 6, 10, 12, 34, 40, 42, 54, 76, 82, 94, 124, 126, 160, 192, 210, 222, 250, 252, 276, 292, 300, 412, 420, 430, 454, 496, 502, 570, 586, 612, 622, 640, 670, 684, 712, 720, 724, 726, 756, 784, 822, 826, 874, 882, 894, 934, 952, 964, 1006, 1056.

Examples

			0^3 + 13 = 13 = prime(6) = a(1);
6^3 + 13 = 229 = prime(50) = a(2);
300^3 + 13 = 27000013 = prime(1683067) = a(22).
		

References

  • H. Rademacher, Topics in Analytic Number Theory, Springer-Verlag Berlin, 1973.

Crossrefs

Programs

  • Magma
    [a: n in [0..500]|IsPrime(a) where a is n^3+13] // Vincenzo Librandi, Dec 22 2010
  • Maple
    select(isprime,[seq(seq((6*j+m)^3+13,m=[0,4]),j=0..1000)]); # Robert Israel, Jun 28 2018
  • Mathematica
    Select[Range[0,1000]^3+13,PrimeQ]  (* Harvey P. Dale, Mar 12 2011 *)

A159942 Duplicate of A102339.

Original entry on oeis.org

2, 5, 7, 10, 16, 17, 19, 20, 23, 29, 31, 38, 41, 49, 50, 55, 56, 59, 61, 64, 71, 76, 79, 85, 92, 100, 101, 103, 121, 134, 136, 139, 140, 143, 149, 154, 155, 161, 175, 176, 178, 182, 184, 188, 208, 209, 211, 217, 220, 232, 236, 239, 241, 244, 265, 266, 269, 271, 272
Offset: 1

Views

Author

Keywords

Crossrefs

A158013 100*a(n)+13 and 100*a(n)+27 are consecutive primes, i.e., a prime gap 14.

Original entry on oeis.org

1, 106, 133, 154, 184, 217, 220, 307, 334, 436, 454, 496, 505, 574, 580, 604, 616, 631, 805, 892, 1009, 1015, 1045, 1132, 1174, 1189, 1198, 1204, 1360, 1408, 1444, 1504, 1510, 1627, 1702, 1708, 1771, 1954, 1984, 2101, 2182, 2218, 2221, 2245, 2260, 2281
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 11 2009

Keywords

Comments

Notes:
1) Necessarily a(n)=3k+1: a(n)=3k => 100*3k+27= 3*(100k+9), divisible by 3 a(n)= 3k+2 => 100*(3k+2)+13=3*(100k+71), divisible by 3.
2) It is conjectured that sequence is infinite.
3) Each sequence 100*b(n)+13 and 100*c(n)+27 includes an infinite number of primes (because of DIRICHLET's theorem).
4) Analogous sequences for investigation of prime gaps are obvious and useful.

Examples

			1) 113=P(30) and 127=P(31) => a(1)=1.
2) 1613=P(255) and 1627=P(258) prime too but 1619=P(256), 1621=P(257) => 1613 and 1627 are not consecutive primes.
3) next: 10613=P(1295), 10627 = P(1296) => a(2)=106.
		

References

  • N. G. Tchudakoff, On the difference between two neighboring prime numbers, Math. Sb. 1, (1936), 799-814.
  • R. K. Guy, Unsolved problems in number theory

Crossrefs

Cf. A157772 (primes ending with "13" ordered in natural growing size).

Programs

  • Mathematica
    fQ[n_] := PrimeQ[ Range[100 n + 13, 100 n + 27, 2]] == {True, False, False, False, False, False, False, True}; Select[ Range@ 2295, fQ@# &] (* Robert G. Wilson v, Mar 13 2009 *)

Formula

p(k+1)=100*a(n)+27 and p(k)=100*a(n)+13 where p(k) is the k-th prime => prime gap p(k+1)-p(k)=14.

Extensions

a(31)-a(46) from Robert G. Wilson v, Mar 13 2009
Showing 1-7 of 7 results.