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.

A178228 Numbers k such that (k^3 - 2, k^3 + 2) is a pair of cousin primes (see A178227).

Original entry on oeis.org

129, 189, 369, 435, 549, 555, 561, 819, 1245, 1491, 1719, 1779, 1839, 1875, 1935, 2175, 2289, 2415, 2451, 2595, 2709, 2769, 3141, 3441, 4401, 4611, 4851, 5655, 5775, 6075, 6099, 6795, 6969, 7125, 7239, 7365, 8109, 8139, 8325, 8361, 8385, 8535, 8685, 9591, 9765
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 23 2010

Keywords

Comments

Necessarily k is an odd multiple of 3, Least significant digit of k is e = 1, 5 or 9 (3^3 - 2, 7^3 + 2 are multiples of 5).

Examples

			189 is a term since 189^3 - 2 = 6751267 = prime(460792), 189^3 + 2 = 6751271 = prime(460793).
12471 is a term since 12471^3 - 2 = 1939562763109 = prime(i), i = 71166976775, 12471^3 + 2 = 1939562763113 = prime(i+1).

Links

Crossrefs

Cf. A023200, A046132, A090121, A164834, A172494, A174370, A176130, A176229, A178227.

Programs

  • Mathematica
    Select[Range[10^4], And @@ PrimeQ[#^3 + {-2, 2}] &] (* Amiram Eldar, Dec 24 2019 *)
  • PARI
    for(n=1,10000,my(p1=n^3-2,p2=n^3+2);if(isprime(p1)&&isprime(p2)&&ispower((p1+p2)/2,3),print1(n,", "))) \\ Hugo Pfoertner, Dec 24 2019

Extensions

Edited by N. J. A. Sloane, May 23 2010
a(1) and a(21) inserted by Amiram Eldar, Dec 24 2019

A174454 The smaller member p of a twin prime pair such that 5*p+6 is a square of a prime number.

Original entry on oeis.org

71, 191, 6551, 35447, 79631, 103391, 207671, 254927, 264959, 421079, 479879, 592367, 700127, 745751, 949607, 986567, 1013399, 1271087, 1456919, 1478591, 1859279, 2085287, 2272727, 2841071, 5204039, 5472671, 6003887, 6202751
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 20 2010

Keywords

Comments

Subsequence of A174370 and of A001359.

Examples

			(71,73) are twin primes, and 5 * 71 + 6 = 19^2 is a square of a prime, which adds 71 to the sequence.
(9767,9769) are twin primes, 5 * 9767 + 6 = 221^2, but 221 = 13 * 17 is not prime, so 9767 is not in the sequence.

Links

Crossrefs

Cf. A001359, A174370.

Programs

  • Mathematica
    Select[Transpose[Select[Partition[Prime[Range[430000]],2,1], Last[#]- First[#] ==2&]][[1]],PrimeQ[Sqrt[5#+6]]&]  (* Harvey P. Dale, Apr 22 2011 *)
Select[(Select[Range[5569], PrimeQ]^2 - 6)/5, And @@ PrimeQ[# + {0, 2}] &] (* Amiram Eldar, Dec 24 2019 *)

Extensions

Definition simplified, cross-references to unrelated sequences removed - R. J. Mathar, Nov 01 2010

A178227 Lesser of a pair (p,p+4) of cousin primes whose arithmetic mean p+2 is a cube.

Original entry on oeis.org

2146687, 6751267, 50243407, 82312873, 165469147, 170953873, 176558479, 549353257, 1929781123, 3314613769, 5079577957, 5630252137, 6219352717, 6591796873, 7245075373, 10289109373, 11993263567, 14084823373, 14724139849, 17474794873, 19880486827, 21230922607, 30988732219
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 23 2010

Keywords

Comments

p = n^3 - 2, p and p+4 are "near(est) cube" primes as n^3 -/+ 1 = (n -/+ 1) * (n^2 +/- n + 1).

Examples

			p = 2146687 is a term, as p + 2 = 129^3 and both p = 129^3 - 2 and p + 4 = 129^3 + 2 are prime.

Links

Crossrefs

Cf. A023200, A046132, A172494, A174370, A176130, A178228, A176229.

Programs

  • Mathematica
    Select[Range[2300]^3 - 2, And @@ PrimeQ[# + {0, 4}] &] (* Amiram Eldar, Dec 24 2019 *)
  • PARI
    isok(p) = isprime(p) && (q=nextprime(p+1)) && (q-p==4) && ispower(p+2, 3); \\ Michel Marcus, Nov 27 2016

Formula

a(n) = A178228(n)^3 - 2. - Amiram Eldar, Dec 24 2019

Extensions

Corrected by D. S. McNeil, Nov 24 2010
More terms from Amiram Eldar, Dec 24 2019

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.

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

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