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 1-4 of 4 results.

A176130 Lesser of a pair (p,p+4) of cousin primes whose arithmetic mean p+2 is a square number.

Original entry on oeis.org

7, 79, 223, 439, 1087, 13687, 56167, 74527, 91807, 95479, 149767, 184039, 194479, 199807, 263167, 314719, 328327, 370879, 651247, 804607, 1071223, 1147039, 1238767, 1306447, 1520287, 1535119, 1718719, 2442967, 2595319, 2614687
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 09 2010

Keywords

Comments

Necessarily p = 9 * (2*m - 1)^2 - 2.

Examples

			(7 + 11)/2 = 3^2, 1st term is prime(4) = 7.
(79 + 83)/2 = 9^2, 2nd term is prime(22) = 79.
m = 173 = prime(40): 21st term is p = 1071223 = prime(83637), p+2 = 3^4 * 5^2 * 23^2.
60th term is p = 27029599 = prime(1684797): p+2 = 3^2 * 1733^2.

References

  • L. E. Dickson, History of the Theory of numbers, vol. 2: Diophantine Analysis, Dover Publications 2005.
  • H. Pieper, Zahlen aus Primzahlen. Eine Einfuehrung in die Zahlentheorie. VEB Deutscher Verlag der Wissenschaften, 2. Aufl., 1984.
  • A. Warusfel, Les nombres et leurs mystères, Edition du Seuil, Paris 1980.

Links

Crossrefs

Cf. A023200, A046132, A174454.

Programs

  • Mathematica
    Select[Range[1617]^2 - 2, And @@ PrimeQ[# + {0, 4}] &] (* Amiram Eldar, Dec 24 2019 *)
  • PARI
    isok(n) = isprime(n) && isprime(n+4) && issquare(n+2) \\ Michel Marcus, Jul 22 2013
  • PARI
    forstep(n=3,1e4,2,if(isprime(n^2-2)&&isprime(n^2+2),print1(n^2-2", "))) \\ Charles R Greathouse IV, Jul 23 2013

Extensions

Edited by D. S. McNeil, Nov 18 2010

A176603 Smallest prime p of three consecutive primes (p,q,r) with p + q + r equal to a lower twin prime.

Original entry on oeis.org

11, 17, 19, 83, 101, 281, 347, 349, 379, 401, 547, 641, 701, 839, 1103, 1151, 1171, 1187, 1279, 1303, 1409, 1439, 1489, 1823, 2089, 2243, 2857, 2861, 2927, 2999, 3083, 3203, 3347, 3359, 3467, 4639, 5087, 5233, 5861, 5879, 5881, 5923, 5953, 6007, 6299, 6491
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 21 2010

Keywords

Comments

The sequence is constructed by intersecting A034961 and A001359, then printing the smallest of the three primes that sum to A034961.

Examples

			11+13+17 = 41 = prime(13), 43 = prime(14), 11 is 1st term.
17+19+23 = 59 = prime(17), 61 = prime(18), 17 is 2nd term.
Detailed list:
11+13+17 = 41, 17+19+23 = 59, 19+23+29 = 71, 83+89+97 = 269,
101+103+107 = 311, 281+283+293 = 857, 347+349+353 = 1049,
349+353+359 = 1061, 379+383+389 = 1151, 401+409+419 = 1229,
547+557+563 = 1667, 641+643+647 = 1931, 701+709+719 = 2129,
839+853+857 = 2549, 1103+1109+1117 = 3329, 1151+1153+1163 = 3467,
1171+1181+1187 = 3539, 1187+1193+1201 = 3581, 1279+1283+1289 = 3851,
1303+1307+1319 = 3929, 1409+1423+1427 = 4259, 1439+1447+1451 = 4337,
1489+1493+1499 = 4481, 1823+1831+1847 = 5501, 2089+2099+2111 = 6299,
2243+2251+2267 = 6761, 2857+2861+2879 = 8597, 2861+2879+2887 = 8627,
2927+2939+2953 = 8819, 2999+3001+3011 = 9011.

References

  • Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005.
  • Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, 1909.

Links

Crossrefs

Cf. A001359, A034961, A174454.

Programs

Extensions

keyword:base removed, and sequence extended by R. J. Mathar, Apr 23 2010

A178033 Lesser of a twin prime pair (p,p+2) such that permuting the digits of p and those of p+2 gives a different twin prime pair (q, q+2).

Original entry on oeis.org

281, 461, 641, 821, 1031, 1091, 1229, 1277, 1301, 1319, 1427, 1697, 1721, 1787, 1877, 2081, 2129, 2381, 2687, 2711, 2801, 3119, 3251, 3257, 3371, 3467, 3527, 3581, 3821, 3851, 4091, 4127, 4157, 4217, 4241, 4271, 4421, 4517, 4637, 4649, 4721, 4787, 4931, 4967, 5231, 5417, 5477, 5651
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 17 2010

Keywords

Comments

Permutations with initial zeros are disallowed, so that 101 is not a member (101,103 and 11,13); equivalently, we require that p is a permutation of the digits of q as well.

Examples

			281 is a term as 281 is the lesser of the twin prime pair 281,283, and after permuting 821, 823 is also a twin prime pair.
1229 is a term as (1229,1231) is a twin prime pair and after permuting (2129, 2131) is also a twin prime pair.

Links

Crossrefs

Cf. A001359, A172271, A172494, A173255, A174454

Programs

  • Mathematica
    perm@n_ :=
 Select[FromDigits@# & /@
   DeleteCases[Rest@Permutations@IntegerDigits@n, _?(First@# == 0 &)],
   PrimeQ];
Cases[{#, perm@# & /@ #} & /@
  Cases[6*# + {-1, 1} & /@
    Range@2000, {?PrimeQ ..}], {{x, }, {{__, a_, _}, {_, b_, _}} /; b - a == 2} :> x] (* Hans Rudolf Widmer, Oct 04 2024 *)

Extensions

Corrected and edited by D. S. McNeil, Nov 23 2010
More terms from Hans Rudolf Widmer, Oct 04 2024

A176185 Numbers n with property that concatenation (2*n+1)//n of the decimals is a square.

Original entry on oeis.org

29, 76, 2289, 3796, 6369, 8756, 16736, 19696, 24900, 28484, 77529, 83761, 94169, 222889, 887556, 22228889, 88875556, 112594641, 368762025, 651177616
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 11 2010

Keywords

Comments

Sequence is infinite; two infinite "families" of such numbers n are:
(a) n = 8_(k)75_(k)6, 2 * n + 1 = 17_(k)51_(k)3, N = 2 * 6_(k+1)16_(k-1)7,
(b) n = 2_(k+1)8_(k)9, 2 * n + 1 = 4_(k)57_(k)9, N = 6_(k)76_(k)7, (k = 1, 2, ...)
List of (2*n+1)//n = N^2:
59//29 = 7^2 x 11^2, 153//76 = 2^4 x 31^2, 4579//2289 = 67^2 x 101^2,
7593//3796 = 2^2 x 4357^2, 12739//6369 = 11287^2, 17513//8756 = 2^2 x 13^2 x 509^2,
33473//16736 = 2^18 x 113^2, 39393//19696 = 2^4 x 13^2 x 17^2 x 71^2, 49801//24900,
56969//28484 = 2^2 x 13^2 x 2903^2, 155059//77529 = 7^2 x 17789^2, 167523//83761 = 347^2 x 373^2,
188339//94169 = 19^2 x 31^2 x 233^2, 445779//222889 = 7^2 x 11^2 x 13^2 x 23^2 x 29^2,
1775113//887556 = 2^2 x 666167^2, 44457779//22228889 = 59^2 x 73^2 x 113^2 x 137^2,
177751113//88875556 = 2^2 x 66661667 ^ 2, 225189283//112594641 = 23^2 x 83^2 x 331^2 x 751^2,
737524051//368762025 = 5^2 x 2161^2 x 79481^2, 1302355233//651177616 = 2^4 x 285301949^2

Examples

			n = 29 is a term: 2 * n + 1 = 59, 5929 = 59//29 = 77^2 is a perfect square.
n = 6369 is a term: 2 * n + 1 = 12739. 12739//6369 = 11287^2 is a perfect square.

References

  • J. Buchmann, U. Vollmer: Binary Quadratic Forms, Springer, Berlin, 2007
  • L. E. Dickson: History of the Theory of numbers, vol. 2: Diophantine Analysis, Dover Publications, 2005

Crossrefs

Cf. A174370, A174454, A174926, A176130

Programs

  • Maple
    isA176185 := proc(n)
    digcat2(2*n+1,n) ; # of oeis.org/transforms.txt
    issqr(%)  ;
end proc:
for n from 1 do
    if isA176185(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, May 21 2025
  • Mathematica
    Select[Range[6512*10^5],IntegerQ[Sqrt[(2 #+1)10^IntegerLength[#]+#]]&] (* Harvey P. Dale, Mar 05 2022 *)
Showing 1-4 of 4 results.

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