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 1-5 of 5 results.

A049054 Numbers k such that 10^k + 3 is prime.

Original entry on oeis.org

1, 2, 5, 6, 11, 17, 18, 39, 56, 101, 105, 107, 123, 413, 426, 2607, 7668, 10470, 11021, 17753, 26927, 60776, 98288, 300476, 509546
Offset: 1

Views

Author

G. L. Honaker, Jr.

Keywords

Comments

A102006 is another version of the same sequence. - N. J. A. Sloane, Jan 28 2010
Verified existing 21 terms. If another term exists, it is > 39456. - Robert Price, Aug 16 2010

Examples

			5 is a term since 10^5 + 3 = 100003 is a prime.
6 is a term since 10^6 + 3 = 1000003 is a prime.

Links

Crossrefs

Cf. A102006, A159352.

Programs

Formula

a(n) = A102006(n) + 1.

Extensions

More terms from Robert G. Wilson v, Jun 15 2002
a(16) from Ray Chandler, Oct 09 2003
a(17)-a(20) from Robert G. Wilson v, Jan 18 2005
a(21) from Jason Earls, Jan 01 2008
a(22) from Robert Price, Jan 09 2011
a(23) from Robert Price, Mar 03 2011
a(24) from Edward A. Trice, Oct 21 2012
a(25) from Paul Bourdelais, Jan 28 2021

A159352 Primes of the form "1 [0]_n 3" - with zeros between 1 and 3.

Original entry on oeis.org

13, 103, 100003, 1000003, 100000000003, 100000000000000003, 1000000000000000003, 1000000000000000000000000000000000000003, 100000000000000000000000000000000000000000000000000000003
Offset: 1

Views

Author

Parthasarathy Nambi, Apr 11 2009

Keywords

Examples

			1000000000000000000000000000000000000003 is a prime with 38 zeros between 1 and 3.

Links

Crossrefs

Cf. A159031.
Cf. A049054 (numbers n such that 10^n + 3 is prime), A102006 (numbers n such that 10*10^n + 3 is prime), A011557 (powers of 10).

Programs

  • Magma
    [p: n in [1..100] | IsPrime(p) where p is 10^n+3 ]; // Klaus Brockhaus, Apr 12 2009
  • Maple
    select(isprime, [seq(10^k+3, k=1..998)]); # Robert Israel, Dec 28 2015

A173836 Natural numbers n such that the concatenation 1331//n^3 is a prime number.

Original entry on oeis.org

21, 27, 29, 41, 101, 119, 141, 171, 173, 177, 191, 197, 219, 243, 267, 291, 309, 327, 333, 369, 371, 383, 411, 417, 1019, 1049, 1059, 1091, 1157, 1163, 1211, 1311, 1337, 1343, 1359, 1371, 1379, 1409, 1461, 1473, 1481, 1503, 1521, 1593, 1599, 1613, 1637
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 26 2010

Keywords

Comments

Given the cube n^3 with k = A111393(n) decimal digits, we have to check whether the concatenation, 11^3 * 10^k + n^3, is a prime.
The number k of digits that 1331=11^3 is shifted is not a multiple of 3,
because the form a^3+b^3 = (a^2+a*b+b^2) * (a - b) cannot construct a prime.

Examples

			21 is in the sequence because 21^3=9261, and the concatenation is 13319261=prime(868687).
27 is in the sequence because 27^3=19683, and the concatenation is 133119683=prime(7545064).

References

  • K. Haase, P. Mauksch: Spass mit Mathe, Urania-Verlag Leipzig, Verlag Dausien Hanau, 2. Auflage 1985

Links

Crossrefs

Cf. A102006, A167535, A168147, A168219, A168274, A173579, A173733

Programs

  • Mathematica
    Select[Range[2000],PrimeQ[FromDigits[Join[{1,3,3,1}, IntegerDigits[ #^3]]]]&] (* Harvey P. Dale, Oct 14 2011 *)

Extensions

Comments sligthly rephrased - R. J. Mathar, Mar 05 2010

A173579 Natural numbers n which give primes when 1331 = 11^3 is prefixed.

Original entry on oeis.org

3, 17, 21, 53, 57, 69, 83, 87, 107, 119, 123, 153, 207, 227, 243, 249, 251, 261, 269, 279, 293, 299, 327, 329, 333, 339, 347, 377, 381, 383, 399, 411, 431, 437, 443, 471, 489, 497, 513, 521, 527, 549, 567, 573, 579, 587, 591, 597, 599, 611, 633, 641, 647, 657
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 22 2010

Keywords

Comments

Concatenation of N = 1331 = 11^3 = palindrome(113) and natural n is a prime. No zeros "between" N and n.
13 = emirp(1) = prime(6), R(13) = 31 = emirp(3) = prime(11).
Necessarily n = 3 * k or n = 3 * k + 2, but not n = 3 * k + 1, because sod(1331) = 8. So no prime twins are terms of the sequence.

Examples

			13313 = prime(1581) => a(1) = 3.
133117 = prime(12425) => a(2) = 17.
133103, 133109 are prime, but "0" included: "03" resp. "09" are no terms of the sequence.

References

  • Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
  • K. Haase, P. Mauksch: Spass mit Mathe, Urania-Verlag Leipzig, Verlag Dausien Hanau, 2. Auflage 1985
  • Theo Kempermann: Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005

Links

Crossrefs

A102006, A167535, A168147, A168219, A168274

Programs

  • Mathematica
    Select[Range[700],PrimeQ[1331*10^IntegerLength[#]+#]&] (* Harvey P. Dale, Jun 25 2020 *)
  • PARI
    isok(n) = isprime(n + 1331*10^(length(Str(n)))); \\ Michel Marcus, Aug 27 2013

A173733 Primes p which give primes when 1331 = 11^3 is prefixed (see A173579).

Original entry on oeis.org

3, 17, 53, 83, 107, 227, 251, 269, 293, 347, 383, 431, 443, 521, 587, 599, 641, 647, 683, 719, 761, 773, 821, 857, 929, 1031, 1097, 1217, 1223, 1301, 1367, 1409, 1433, 1451, 1619, 1637, 1709, 1787, 1973, 2081, 2087, 2129, 2399, 2477, 2591, 2633, 2657, 2693
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 23 2010

Keywords

Comments

N = 1331 = 11^3, p k-digit prime, to check if q = N * 10^k + p is prime
With exception of 3 necessarily p of form 3k+2, as sod(1331 = 8)

Examples

			13313 = prime(1581) => a(1) = prime(2) = 3
133117 = prime(12425) => a(2) = prime(7) = 17
133153 = prime(12427) => a(3) = prime(16) = 53
13311217 = prime(868166) => a(28) = prime(199) = 1217
13311223 = prime(868167) => a(29) = prime(200) = 1223
Note: two consecutive primes P = prime(n), Q = prime(n+1) yield consecutive prime concatenations "N P" = prime(m) and "N Q" = prime(m+1)

References

  • K. Haase, P. Mauksch: Spass mit Mathe, Urania-Verlag Leipzig, Verlag Dausien Hanau, 2. Auflage 1985
  • Helmut Kracke, Mathe-musische Knobelisken, Duemmler Bonn, 2. Auflage 1983

Links

Crossrefs

Cf. A102006, A167535, A168147, A168219, A168274, A173579

Programs

  • Mathematica
    Select[Prime[Range[400]],PrimeQ[FromDigits[Join[{1,3,3,1}, IntegerDigits[ #]]]]&] (* Harvey P. Dale, Jun 09 2015 *)

Extensions

Edited and extended by Charles R Greathouse IV, Apr 24 2010
Showing 1-5 of 5 results.

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