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 1-7 of 7 results.

A234297 Squares t^2 = (p+q+r)/3 which are the arithmetic mean of three consecutive primes such that p < t^2 < q < r.

Original entry on oeis.org

47961, 123201, 131769, 826281, 870489, 2486929, 3294225, 5239521, 5294601, 5774409, 6215049, 6335289, 6848689, 9308601, 10634121, 16072081, 17164449, 17732521, 18896409, 19298449, 22667121, 24413481, 25391521, 25836889, 30769209, 32569849, 33535681
Offset: 1

Views

Author

K. D. Bajpai, Dec 22 2013

Keywords

Examples

			47961 is in the sequence because 47961 = 219^2 = (47951+47963+47969)/3, the arithmetic mean of three consecutive primes.
131769 is in the sequence because 131769 = 363^2 = (131759+131771+131777)/3, the arithmetic mean of three consecutive primes.

Links

Crossrefs

Cf. A000290 (squares: a(n) = n^2).
Cf. A062703 (squares: sum of two consecutive primes).
Cf. A069495 (squares: arithmetic mean of two consecutive primes).
Cf. A234240 (cubes: arithmetic mean of three consecutive primes).

Programs

  • Maple
    with(numtheory):KD := proc() local a,b,d,e,f; a:=n^2; b:=prevprime(a); d:=nextprime(a); e:=nextprime(d); f:=(b+d+e)/3; if a=f then RETURN (a); fi; end: seq(KD(), n=2..10000);
  • Mathematica
    amQ[{a_,b_,c_}]:=Module[{m=Mean[{a,b,c}]},IntegerQ[Sqrt[m]]&&aHarvey P. Dale, Mar 14 2014 *)
  • PARI
    list(lim)=my(v=List(),p=2,q=3,t); forprime(r=5, nextprime(nextprime(lim+1)+1), t=(p+q+r)/3; if(denominator(t)==1 && issquare(t) && t < q, listput(v, t)); p=q;q=r); Vec(v) \\ Charles R Greathouse IV, Jan 03 2014

Extensions

Definition corrected by Michel Marcus and Charles R Greathouse IV, Jan 03 2014

A234358 Cubes t^3 = (p+q+r+s)/4 which are the arithmetic mean of four consecutive primes such that p < t^3 < q < r < s.

Original entry on oeis.org

25934336, 194104539, 320013504, 332812557, 428661064, 8072216216, 8640364608, 11239424000, 16290480375, 17738739712, 26730899000, 44136677304, 46850670125, 68117264704, 114366627864, 119168121961
Offset: 1

Views

Author

K. D. Bajpai, Dec 24 2013

Keywords

Examples

			25934336 is in the sequence because 25934336 = 296^3 = (25934303+25934341+25934347+25934353)/4, the arithmetic mean of four consecutive primes.
320013504 is in the sequence because 320013504 = 684^3 = (320013479+320013509+320013511+320013517)/4, the arithmetic mean of four consecutive primes.

Links

Crossrefs

Cf. A000578 (cubes: a(n) = n^3).
Cf. A062703 (squares: sum of two consecutive primes).
Cf. A069495 (squares: arithmetic mean of two consecutive primes).
Cf. A234240 (cubes: arithmetic mean of two consecutive primes).
Cf. A234256 (cubes: arithmetic mean of three consecutive primes).

Programs

  • Maple
    KD := proc() local a,b,d,e,f,g; a:=n^3; b:=prevprime(a); d:=nextprime(a); e:=nextprime(d); f:=nextprime(e); g:=(b+d+e+f)/4;  if a=g then RETURN (a);  fi; end: seq(KD(), n=2..10000);

Extensions

Definition corrected by K. D. Bajpai, Jan 07 2014

A234256 Cubes t^3 = (p+q+r)/3 which are the arithmetic mean of three consecutive primes such that p < t^3 < q < r.

Original entry on oeis.org

5735339, 10503459, 73560059, 253636137, 393832837, 761048497, 791453125, 1064332261, 1829276567, 2014698447, 2487813875, 2893640625, 4533086375, 7845011803, 14437662875, 45998156287, 55611739513, 62429032063, 63378025803, 72877493233, 87115050737, 104154702625
Offset: 1

Views

Author

K. D. Bajpai, Dec 22 2013

Keywords

Examples

			5735339 is in the sequence because 5735339 = 179^3 = (5735291+5735357+5735369)/3, the arithmetic mean of three consecutive primes.
10503459 is in the sequence because 10503459 = 219^3 = (10503443+10503461+10503473)/3, the arithmetic mean of three consecutive primes.

Links

Crossrefs

Cf. A000578 (cubes: n^3).
Cf. A062703 (squares: sum of two consecutive primes).
Cf. A069495 (squares: arithmetic mean of two consecutive primes).
Cf. A234240 (cubes: arithmetic mean of two consecutive primes).

Programs

  • Maple
    with(numtheory):KD := proc() local a,b,d,e,f; a:=n^3; b:=prevprime(a); d:=nextprime(a); e:=nextprime(d);  f:=(b+d+e)/3; if  a=f then RETURN (a);  fi;  end: seq(KD(), n=2..10000);
  • PARI
    list(lim)=my(v=List(), p=2, q=3, t); forprime(r=5, nextprime(nextprime(lim\3+1)+1), t=(p+q+r)/3; if(denominator(t)==1 && ispower(t,3) && t < q, listput(v, t)); p=q; q=r); Vec(v) \\ Charles R Greathouse IV, Jan 03 2014

Extensions

Definition corrected by Michel Marcus and Charles R Greathouse IV, Jan 03 2014

A234318 Squares t^2 = (p+q+r+s)/4 which are the arithmetic mean of four consecutive primes such that p < t^2 < q < r < s.

Original entry on oeis.org

15876, 35721, 59049, 65025, 488601, 828100, 1144900, 3857296, 4822416, 4901796, 5107600, 5322249, 5856400, 6100900, 6760000, 10536516, 11716929, 12503296, 13468900, 14197824, 14638276, 15163236, 18748900, 21455424, 22127616, 22638564, 24049216, 24098281, 24108100
Offset: 1

Views

Author

K. D. Bajpai, Dec 23 2013

Keywords

Examples

			15876 is in the sequence because 15876 = 126^2 = (15859+15877+15881+15887)/4, the arithmetic mean of four consecutive primes.
35721 is in the sequence because 35721 = 189^2 = (35677+35729+35731+35747)/4, the arithmetic mean of four consecutive primes.

Links

Crossrefs

Cf. A000290 (squares: a(n) = n^2).
Cf. A062703 (squares: sum of two consecutive primes).
Cf. A069495 (squares: arithmetic mean of two consecutive primes).
Cf. A234240 (cubes: arithmetic mean of three consecutive primes).
Cf. A234297 (squares: arithmetic mean of three consecutive primes).

Programs

  • Maple
    KD := proc() local a,b,d,e,f,g; a:=n^2;b:=prevprime(a); d:=nextprime(a); e:=nextprime(d); f:=nextprime(e); g:=(b+d+e+f)/4; if a=g then RETURN (a); fi; end: seq(KD(), n=2..10000);
  • Mathematica
    fcpQ[{a_,b_,c_,d_}]:=Module[{m=Mean[{a,b,c,d}]},IntegerQ[ Sqrt[ m]] && a< m< b]; Mean/@Select[Partition[Prime[Range[1600000]],4,1],fcpQ] (* Harvey P. Dale, Apr 24 2017 *)
  • PARI
    list(lim)=my(v=List(), p=2, q=3, r=5, t); forprime(s=7, nextprime(nextprime(lim+1)+1), t=(p+q+r+s)/4; if(denominator(t)==1 && issquare(t) && t < q, listput(v, t)); p=q; q=r; r=s); Vec(v) \\ Charles R Greathouse IV, Jan 03 2014

Extensions

Definition corrected by Michel Marcus and Charles R Greathouse IV, Jan 03 2014

A234531 Pentagonal numbers which are the arithmetic mean of two consecutive primes.

Original entry on oeis.org

12, 176, 376, 532, 590, 3015, 4510, 4676, 7315, 7526, 7957, 8855, 12650, 15555, 17120, 19437, 20126, 22265, 25676, 29330, 30175, 40755, 48510, 54626, 78547, 82017, 91390, 97410, 101270, 102051, 102835, 105205
Offset: 1

Views

Author

K. D. Bajpai, Dec 27 2013

Keywords

Comments

The n-th pentagonal number is (3*n^2 - n)/2 = n*(3*n - 1)/2.

Examples

			376 is in the sequence because 376 = 16*(3*16 - 1)/2 = (373 + 379)/2, the arithmetic mean of two consecutive primes.
532 is in the sequence because 532 = 19*(3*19 - 1)/2 = (523 + 541)/2, the arithmetic mean of two consecutive primes.

Links

Crossrefs

Cf. A000326 (pentagonal numbers: n * (3 * n - 1 ) / 2).
Cf. A069495 (squares: arithmetic mean of two consecutive primes).
Cf. A234240 (cubes: arithmetic mean of three consecutive primes).

Programs

  • Maple
    KD := proc() local a,b,d,g;  a:= n*(3*n-1)/2; b:=prevprime(a); d:=nextprime(b); g:=(b+d)/2;  if a=g then RETURN (a);  fi; end: seq(KD(), n = 2..500);
  • Mathematica
    Select[PolygonalNumber[5,Range[300]],!PrimeQ[#]&&#==(NextPrime[ #]+ NextPrime[ #,-1])/2&] (* Harvey P. Dale, Dec 26 2022 *)
  • PARI
    lista(nn) = for (n=1, nn, pn = n*(3*n-1)/2; if (pn > 2, precp = precprime(pn-1); if (pn == (precp + nextprime(precp+1))/2, print1(pn, ", ")))) \\ Michel Marcus, Dec 30 2013

Extensions

Erroneous term 5 removed by Michel Marcus, Dec 30 2013

A234532 Pentagonal numbers penta(n) = (p + q + r)/3 which are the arithmetic mean of three consecutive primes such that p < penta(n) < q < r.

Original entry on oeis.org

9087, 29751, 291501, 602617, 1505505, 1778337, 1941997, 2137857, 3032415, 4629695, 5016947, 5038917, 7837551, 8030737, 9328807, 11935651, 19158427, 35616757, 40964001, 41073817, 42594697, 44289817, 56141827, 59267551
Offset: 1

Views

Author

K. D. Bajpai, Dec 27 2013

Keywords

Comments

The n-th pentagonal number is (3*n^2 - n)/2 = n*(3*n - 1)/2.

Examples

			9087 is in the sequence because 9087 = 78 *(3*78 - 1)/2 = (9067 + 9091 + 9103)/3, the arithmetic mean of three consecutive primes.
29751 is in the sequence because 29751 = 141*(3*141 - 1)/2 = (29741 + 29753 + 29759)/3, the arithmetic mean of three consecutive primes.

Links

Crossrefs

Cf. A000326 (pentagonal numbers: n * (3*n - 1)/2).
Cf. A069495 (squares: arithmetic mean of two consecutive primes).
Cf. A234240 (cubes: arithmetic mean of three consecutive primes).

Programs

  • Maple
    KD := proc() local a,b,d,e,g; a:= n*(3*n-1)/2; b:=prevprime(a); d:=nextprime(a); e:=nextprime(d); g:=(b+d+e)/3;  if a=g then RETURN (a); fi; end: seq(KD(), n=2..10000);

A234539 Pentagonal numbers P = (p+q+r+s)/4 which are the arithmetic mean of four consecutive primes such that p < P < q < r < s.

Original entry on oeis.org

852, 172212, 272001, 341055, 777240, 1451892, 1710402, 2361910, 2922526, 3950382, 4794522, 5414050, 6272015, 7989142, 10580176, 10780301, 11325882, 12173777, 12483395, 14432055, 14665630, 15890910, 16581775, 16962972, 17124772
Offset: 1

Views

Author

K. D. Bajpai, Dec 27 2013

Keywords

Comments

The n-th pentagonal number is (3*n^2 - n)/2 = n*(3*n - 1)/2.

Examples

			852 is in the sequence because 852 = 24*(3*24-1)/2 = (839 + 853 + 857 + 859)/4, the arithmetic mean of four consecutive primes.
172212 is in the sequence because 172212 = 339*(3*339-1)/2 = (172199 +172213+ 172217 + 172219)/4, the arithmetic mean of four consecutive primes.

Links

Crossrefs

Cf. A000326 (pentagonal numbers: n*(3*n-1)/2).
Cf. A069495 (squares: arithmetic mean of two consecutive primes).
Cf. A234240 (cubes: arithmetic mean of three consecutive primes).

Programs

  • Maple
    KD := proc() local a,b,d,e,f,g; a:= n*(3*n-1)/2; b:=prevprime(a); d:=nextprime(a); e:=nextprime(d); f:=nextprime(e);  g:=(b+d+e+f)/4; if a=g then RETURN (a); fi; end: seq(KD(), n=2..5000);
  • Mathematica
    Mean/@Select[Partition[Prime[Range[2*10^6]],4,1],With[{m=Mean[#]},IntegerQ[(1+Sqrt[1+24m])/6]&&#[[1]]Harvey P. Dale, May 12 2025 *)

Extensions

Typo in definition corrected by N. J. A. Sloane, May 13 2025 at the suggestion of Harvey P. Dale.
Showing 1-7 of 7 results.

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

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