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 1-6 of 6 results.

A224888 Primes of the form p^2 + (q-p)^2, where p and q are consecutive primes.

Original entry on oeis.org

5, 13, 29, 293, 997, 6257, 11897, 18773, 19421, 52457, 73477, 109597, 120413, 167381, 192737, 218233, 249017, 292717, 333029, 361237, 398261, 466553, 502781, 546137, 552113, 591377, 635353, 683933, 687341, 704117, 737897, 885517, 966353, 982117, 1018097, 1079621
Offset: 1

Views

Author

Thomas Ordowski, Jul 24 2013

Keywords

Comments

Primes of the form A000040(n)^2 + A001223(n)^2.
Primes of the form A134735(2n-1)^2 + A134735(2n)^2.
Conjecture: a(n) ~ A093343(n).
There are 20421247 members of this sequence below 10^20. - Charles R Greathouse IV, Jul 29 2013

Examples

			3 and 5 are consecutive primes and 3^2 + (5-3)^2 = 9 + 4 = 13 is prime, so 13 is in the sequence.

Links

Crossrefs

Cf. A093343.

Programs

  • Mathematica
    Select[Table[Prime[n]^2 + (Prime[n + 1] - Prime[n])^2, {n, 200}], PrimeQ] (* Alonso del Arte, Jul 29 2013 *)
  • PARI
    p=2;forprime(q=3,1e4,if(isprime(t=p^2+(q-p)^2),print1(t", "));p=q) \\ Charles R Greathouse IV, Jul 24 2013

Formula

c(x) is O( sqrt(x/log x) / log x ), where c(x) is the counting function, the number of terms less than x.

Extensions

a(5), a(9)-a(36) from Charles R Greathouse IV, Jul 24 2013

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 1-6 of 6 results.

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