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 1-8 of 8 results.

A133940 Numbers n such that (prime(n)^2 + prime(n+1)^2 + prime(n+2)^2)/3 is prime (A084951).

Original entry on oeis.org

4, 5, 8, 13, 15, 26, 46, 47, 50, 55, 57, 59, 61, 65, 66, 69, 77, 82, 89, 91, 94, 101, 105, 116, 134, 136, 137, 138, 144, 157, 194, 216, 219, 221, 224, 225, 229, 230, 234, 249, 257, 261, 263, 271, 272, 275, 306, 316, 319, 323
Offset: 1

Views

Author

Artur Jasinski, Sep 30 2007

Keywords

Comments

With the exception of the first two terms, all numbers in A133529 are divisible by 3.

Examples

			a(1)=4 because (prime(4)^2 + prime(5)^2 + prime(6)^2)/3 = 113 is prime.

Links

Crossrefs

Cf. A133529, A084951.

Programs

  • Maple
    select(n -> isprime((ithprime(n)^2 + ithprime(n+1)^2 + ithprime(n+2)^2)/3), [$3 .. 1000]); # Robert Israel, Apr 21 2015
  • Mathematica
    b = {}; a = 2; Do[k = (Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a)/3; If[PrimeQ[k], AppendTo[b, n]], {n, 1, 200}]; b
  • PARI
    is(n)=my(p=prime(n),q=nextprime(p+1),r=nextprime(q+1)); n>3 && isprime((p^2+q^2+r^2)/3) \\ Charles R Greathouse IV, Apr 21 2015

Extensions

Corrected and edited by Zak Seidov, Apr 21 2015

A075893 Average of three successive primes squared, (prime(n)^2+prime(n+1)^2+prime(n+2)^2)/3, n>=3.

Original entry on oeis.org

65, 113, 193, 273, 393, 577, 777, 1057, 1337, 1633, 1913, 2289, 2833, 3337, 3897, 4417, 4953, 5537, 6153, 7017, 8073, 9177, 10073, 10753, 11313, 12033, 13593, 15353, 17353, 18417, 20097, 21441, 23217, 24673, 26369, 28129, 29953, 31577, 33761
Offset: 3

Views

Author

Zak Seidov, Oct 17 2002

Keywords

Comments

Unlike the average of three successive primes, the average of three successive primes (greater than 3) squared is always integral.
A133529(n)/3, n >= 3. - Artur Jasinski, Sep 30 2007

Examples

			a(3)=65 because (prime(3)^2+prime(4)^2+prime(5)^2)/3=(5^2+7^2+11^2)/3=65.

Links

Crossrefs

Cf. A133529, A084951, A133940.

Programs

  • Magma
    [(NthPrime(n)^2+NthPrime(n+1)^2+NthPrime(n+2)^2)/3: n in [3..50]]; // Vincenzo Librandi, Aug 21 2018
  • Mathematica
    b = {}; a = 2; Do[k = (Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a)/3; AppendTo[b, k], {n, 3, 50}]; b (* Artur Jasinski, Sep 30 2007 *)
Mean[#]&/@Partition[Prime[Range[3,50]]^2,3,1] (* Harvey P. Dale, Jun 09 2013 *)

Formula

a(n) = (prime(n)^2+prime(n+1)^2+prime(n+2)^2)/3, n>=3.

Extensions

Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar

A084952 Middle q of three consecutive primes p,q,r such that (p^2 + q^2 + r^2)/3 is prime.

Original entry on oeis.org

11, 13, 23, 43, 53, 103, 211, 223, 233, 263, 271, 281, 293, 317, 331, 349, 397, 431, 463, 479, 499, 557, 577, 643, 761, 773, 787, 797, 829, 929, 1187, 1327, 1373, 1399, 1427, 1429, 1451, 1453, 1483, 1583, 1627, 1667, 1693, 1747, 1753, 1783, 2027, 2099, 2129
Offset: 1

Views

Author

Hugo Pfoertner, Jun 14 2003

Keywords

Examples

			a(3)=23 because (19^2 + 23^2 + 29^2)/3 = 1731/3 = 577 is prime.

Links

Crossrefs

Cf. A075893, A084951.

Programs

  • Maple
    q:= 5: r:= 7:
Res:= NULL: count:= 0:
while count < 100 do
  p:= q;
  q:= r;
  r:= nextprime(r);
  if isprime((p^2+q^2+r^2)/3) then count:= count+1; Res:= Res,q fi
od:
Res; # Robert Israel, Aug 20 2018
  • Mathematica
    Select[Partition[Prime[Range[400]],3,1],PrimeQ[Total[#^2]/3]&][[;;,2]] (* Harvey P. Dale, Sep 08 2023 *)

A234364 Primes which are the arithmetic mean of the squares of four consecutive primes.

Original entry on oeis.org

157, 337, 673, 1213, 1777, 2137, 11677, 20773, 27259, 32803, 80407, 84787, 89227, 105397, 120097, 165313, 176461, 181513, 250543, 417337, 453667, 463807, 576883, 610867, 791317, 804757, 853873, 935167, 949687, 1087903
Offset: 1

Views

Author

K. D. Bajpai, Dec 25 2013

Keywords

Examples

			157 is in the sequence because (7^2 + 11^2 + 13^2 + 17^2)/4 = 157 which is prime.
1213 is in the sequence because (29^2 + 31^2 + 37^2 + 41^2)/4 = 1213 which is prime.

Links

Crossrefs

Cf. A084951: primes of the form (prime(k)^2 + prime(k+1)^2 + prime(k+2)^2)/3.
Cf. A093343: primes of the form (prime(k)^2 + prime(k+1)^2)/2.

Programs

  • Maple
    KD := proc() local a,b,d,e,f,g; a:=ithprime(n); b:=ithprime(n+1); d:=ithprime(n+2); e:=ithprime(n+3); g:=(a^2+b^2+d^2+e^2)/4; if g=floor(g) and isprime(g) then RETURN (g);  fi;  end: seq(KD(), n=1..500);
  • Mathematica
    Select[Table[Mean[Prime[Range[n, n + 3]]^2], {n, 250}], PrimeQ] (* Alonso del Arte, Dec 26 2013 *)
Select[Mean/@(Partition[Prime[Range[200]],4,1]^2),PrimeQ] (* Harvey P. Dale, Oct 08 2014 *)

A234370 Primes which are the arithmetic mean of the squares of five consecutive primes.

Original entry on oeis.org

2723401, 13036537, 52774873, 78972121, 116515177, 123179113, 235236049, 242120017, 834990721, 850037521, 943067353, 943804801, 1302156313, 1582432681, 1659047497, 1830419449, 1999538809, 2025774697, 2609800657
Offset: 1

Views

Author

K. D. Bajpai, Dec 25 2013

Keywords

Examples

			2723401 is in the sequence because (1627^2 + 1637^2 + 1657^2 + 1663^2 + 1667^2)/5 = 2723401 which is prime.
52774873 is in the sequence because (7243^2 + 7247^2 + 7253^2 + 7283^2 + 7297^2)/5 = 52774873 which is prime.

Links

Crossrefs

Cf. A084951: primes of the form (prime(k)^2 + prime(k+1)^2 + prime(k+2)^2)/3.
Cf. A093343: primes of the form (prime(k)^2 + prime(k+1)^2)/2.

Programs

  • Maple
    KD := proc() local a,b,d,e,f,g; a:=ithprime(n); b:=ithprime(n+1); d:=ithprime(n+2); e:=ithprime(n+3); f:=ithprime(n+4); g:=(a^2+b^2+d^2+e^2+f^2)/5; if g=floor(g) and isprime(g) then RETURN (g); fi; end: seq(KD(), n=1..10000);
  • Mathematica
    Select[Mean/@Partition[Prime[Range[6000]]^2,5,1],PrimeQ] (* Harvey P. Dale, Aug 01 2020 *)

A234432 Primes which are the arithmetic mean of the squares of six consecutive primes.

Original entry on oeis.org

9413, 25673, 38237, 43573, 81553, 106453, 136273, 145513, 257857, 294013, 325753, 430433, 497257, 599273, 702413, 907733, 948173, 1238893, 2053553, 2185577, 2883457, 3972113, 4226077, 4375177, 4494577, 4728313, 6106141
Offset: 1

Views

Author

K. D. Bajpai, Dec 26 2013

Keywords

Examples

			9413 is in the sequence because (83^2 + 89^2 + 97^2 + 101^2 + 103^2 + 107^2)/6 = 9413 which is prime.
25673 is in the sequence because (149^2 + 151^2 + 157^2 + 163^2 + 167^2 + 173^2)/6 = 25673 which is prime.

Links

Crossrefs

Cf. A084951: primes of the form (prime(k)^2 + prime(k+1)^2 + prime(k+2)^2)/3.
Cf. A093343: primes of the form (prime(k)^2 + prime(k+1)^2)/2.

Programs

  • Maple
    KD := proc() local a,b,d,e,f,g,h; a:=ithprime(n); b:=ithprime(n+1); d:=ithprime(n+2); e:=ithprime(n+3); f:=ithprime(n+4);h:=ithprime(n+5); g:=(a^2+b^2+d^2+e^2+f^2+h^2)/6; if g=floor(g) and isprime(g) then RETURN (g); fi; end: seq(KD(), n=2..1000);

A234433 Primes which are the arithmetic mean of the cubes of three consecutive primes.

Original entry on oeis.org

3659642149, 7045360877, 13980508481, 43207190581, 55176987287, 67967949209, 85126672391, 146447402879, 263994755239, 296875570279, 344620720019, 382820725229, 400485072139, 476566488179, 527319634151, 663284454649
Offset: 1

Views

Author

K. D. Bajpai, Dec 26 2013

Keywords

Examples

			3659642149 is in the sequence because (1531^3 + 1543^3 + 1549^3)/3 = 3659642149 which is prime.
7045360877 is in the sequence because (1907^3 + 1913^3+  1931^3)/3 = 7045360877 which is prime.

Links

Crossrefs

Cf. A084951: primes of the form (prime(k)^2 + prime(k+1)^2 + prime(k+2)^2)/3.
Cf. A093343: primes of the form (prime(k)^2 + prime(k+1)^2)/2.
Cf. A234358: cubes which are the arithmetic mean of four consecutive primes.

Programs

  • Maple
    KD := proc() local a,b,d,e,f,g; a:=ithprime(n); b:=ithprime(n+1); d:=ithprime(n+2); g:=(a^3+b^3+d^3)/3; if g=floor(g) and isprime(g) then RETURN (g); fi; end: seq(KD(), n=2..2000);

A234469 Primes which are the arithmetic mean of the cubes of four consecutive primes.

Original entry on oeis.org

2077681, 16244203, 904456921, 2500135411, 2762662109, 10064833601, 65794585811, 122098559279, 144790176847, 245198071093, 268215631223, 2038246966633, 2782403547799, 3022844332973, 3593531892947
Offset: 1

Views

Author

K. D. Bajpai, Dec 26 2013

Keywords

Examples

			2077681 is in the sequence because (113^3 + 127^3 + 131^3 + 137^3)/4 = 2077681 which is prime.
16244203 is in the sequence because (241^3 + 251^3 + 257^3 + 263^3)/4 = 16244203 which is prime.

Links

Crossrefs

Cf. A084951: primes of the form (prime(k)^2 + prime(k+1)^2 + prime(k+2)^2)/3.
Cf. A093343: primes of the form (prime(k)^2 + prime(k+1)^2)/2.
Cf. A234358: cubes which are the arithmetic mean of four consecutive primes.

Programs

  • Maple
    KD := proc() local a,b,d,e,g; a:=ithprime(n); b:=ithprime(n+1); d:=ithprime(n+2); e:=ithprime(n+3); g:=(a^3+b^3+d^3+e^3)/4; if g=floor(g) and isprime(g) then RETURN (g); fi; end: seq(KD(), n=1..5000);
  • Mathematica
    Select[Mean/@Partition[Prime[Range[2000]]^3,4,1],PrimeQ] (* Harvey P. Dale, Oct 12 2020 *)
Showing 1-8 of 8 results.

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

Content is available under The OEIS End-User License Agreement.