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.

User: Vladimir Joseph Stephan Orlovsky

Vladimir Joseph Stephan Orlovsky's wiki page.

Vladimir Joseph Stephan Orlovsky has authored 1679 sequences. Here are the ten most recent ones:

A242366 Primes p such that p1 = ceiling(p/2) + p is prime and p2 = floor(p1/2) + p is prime.

Original entry on oeis.org

2, 3, 11, 59, 131, 179, 347, 1259, 1571, 1979, 2027, 2411, 2699, 2819, 3251, 3347, 4211, 5051, 5099, 5171, 5531, 6779, 7187, 8747, 10091, 12227, 13259, 13451, 13499, 13931, 14411, 14771, 15131, 15467, 16451, 16691, 17987, 18131, 18539, 18731, 18899, 19211
Offset: 1

Views

Author

Robert Israel and Vladimir Joseph Stephan Orlovsky, May 11 2014

Keywords

Comments

All terms after 2 are congruent to 3 mod 8, as this is needed for p, p1 and p2 to be odd. If p = 3 + 8*k, then p1 = 5 + 12*k and p2 = 5 + 14*k.

Examples

			11 is in the sequence since 11, ceiling(11/2) + 11 = 17 and floor(17/2) + 11 = 19 are all primes.

Links

Crossrefs

Cf. A158714.

Programs

  • Maple
    N:= 100000: # to get all terms <= N
filter:= proc(p) local p1, p2;
if not isprime(p) then return false fi;
p1:= ceil(p/2)+p;
if not isprime(p1) then return false fi;
p2:= floor(p1/2)+p;
isprime(p2);
end;
select(filter,[2, seq(3+8*k, k=0 .. floor((N-3)/8))]);
  • Mathematica
    M = 100000;
filterQ[p_] := Module[{p1, p2},
If[!PrimeQ[p], Return[False]];
p1 = Ceiling[p/2] + p;
If[!PrimeQ[p1], Return[False]];
p2 = Floor[p1/2] + p;
PrimeQ[p2]];
Select[Join[{2}, Table[3+8*k, {k, 0, Floor[(M-3)/8]}]], filterQ] (* Jean-François Alcover, Apr 27 2019, from Maple *)

A207992 Primes p of the form p = prime(n) + prime(n+1) - 5 and p = prime(k) + prime(k+1) + 5.

Original entry on oeis.org

13, 47, 73, 157, 167, 263, 467, 757, 877, 887, 2027, 2593, 3203, 3733, 4273, 4703, 4787, 5087, 5387, 6373, 6637, 7393, 7823, 8893, 9587, 10007, 10163, 12433, 13933, 15083, 15287, 15373, 16333, 17387, 17483, 18013, 18313, 19237, 19477, 20327, 21467, 23567
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 22 2012

Keywords

Comments

n = k+1 or k+2. - Charles R Greathouse IV, Apr 16 2012

Examples

			3+5+5 = 13 = 7+11-5, 23+29-5 = 47 = 19+23+5

Links

Crossrefs

Cf. A072669, A092738, A207990, A207991.

Programs

  • Mathematica
    a1 = Select[Table[Prime[n] + Prime[n + 1] - 5, {n, 2010}], PrimeQ]; a2 = Select[Table[Prime[n] + Prime[n + 1] + 5, {n, 2000}], PrimeQ]; Intersection[a1, a2]
With[{pr=Transpose[#+{5,-5}&/@Total/@Partition[Prime[Range[3000]],2,1]]}, Select[Intersection[pr[[1]],pr[[2]]], PrimeQ]] (* Harvey P. Dale, Mar 13 2013 *)
  • PARI
    p=2;q=3;r=5;forprime(s=7,1e4,if((r==p+10||r+s==p+q+10) && isprime(p+q+5), print1(p+q+5", "));p=q;q=r;r=s) \\ Charles R Greathouse IV, Apr 16 2012

A207991 Primes of the form prime(n) + prime(n+1) + 5.

Original entry on oeis.org

13, 17, 23, 29, 41, 47, 73, 83, 89, 149, 157, 167, 191, 227, 263, 281, 293, 313, 389, 401, 439, 461, 467, 563, 569, 653, 673, 701, 757, 857, 877, 887, 911, 929, 971, 983, 1049, 1069, 1093, 1109, 1153, 1213, 1277, 1289, 1433, 1451, 1487, 1499, 1523, 1637
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 22 2012

Keywords

Links

Crossrefs

Cf. A072669, A092738, A207990.

Programs

  • Mathematica
    Select[Table[Prime[n] + Prime[n + 1] + 5, {n, 200}], PrimeQ]
Select[Total[#]+5&/@Partition[Prime[Range[300]],2,1],PrimeQ] (* Harvey P. Dale, Dec 28 2021 *)
  • PARI
    p=2;forprime(q=3,1e4,if(isprime(t=p+q+5),print1(t", "));p=q) \\ Charles R Greathouse IV, Apr 13 2012

A207990 Primes of the form prime(n) + prime(n+1) - 5.

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 47, 73, 79, 107, 139, 157, 167, 181, 193, 199, 211, 263, 271, 283, 347, 367, 379, 457, 467, 487, 503, 571, 613, 619, 643, 691, 757, 823, 859, 877, 887, 919, 941, 997, 1039, 1187, 1231, 1279, 1307, 1423, 1489, 1579, 1601, 1627, 1663
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 22 2012

Keywords

Links

Crossrefs

Cf. A072669, A092738.

Programs

  • Mathematica
    Select[Table[Prime[n] + Prime[n + 1] - 5, {n, 200}], PrimeQ]
Select[Total/@Partition[Prime[Range[200]],2,1]-5,PrimeQ] (* Harvey P. Dale, Apr 05 2020 *)
  • PARI
    p=2;forprime(q=3,1e4,if(isprime(t=p+q-5),print1(t", "));p=q) \\ Charles R Greathouse IV, Apr 13 2012

A207640 Squares that can be written as a sum of 3 distinct nonzero squares in exactly two ways.

Original entry on oeis.org

225, 361, 625, 900, 1444, 2500, 3600, 5776, 10000, 14400, 23104, 40000, 57600, 92416, 160000, 230400, 369664, 640000, 921600, 1478656, 2560000, 3686400
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 19 2012

Keywords

Comments

k is a term if and only if 4*k is a term. Are 225, 361 and 625 the only terms not divisible by 4? - Robert Israel, Jan 28 2025

Crossrefs

Cf. A207638, A207639.

Programs

  • Maple
    N:= 1000; # for squares up to N^2
V:= Vector(N):
for x from 1 to N-1 do
  for y from 1 to x-1 while x^2 + y^2 < N^2 do
    for z from 1 to y-1 do
      s:= x^2 + y^2 + z^2;
      if s > N^2 then break fi;
      if issqr(s) then
        v:= sqrt(s);
        V[v]:= V[v]+1
      fi
od od od:
map(`^`, select(t -> V[t]=2, [$1..N]),2); # Robert Israel, Jan 28 2025
  • Mathematica
    t = Sort[Select[Flatten[Table[x^2 + y^2 + z^2, {x, 400}, {y, x + 1, 400}, {z, y + 1, 400}]], # < 160006 && IntegerQ[Sqrt[#]] &]];
f1[l_] := Module[{t = {}}, Do[If[l[[n]] != l[[n + 1]] && l[[n]] != l[[n - 1]], AppendTo[t, l[[n]]]], {n, Length[l] - 1}]; t];
f2[l_] := Module[{t = {}}, Do[If[l[[n]] == l[[n + 1]], AppendTo[t, l[[n]]]], {n, Length[l] - 1}]; t];
s1 = Join[{First[t]}, f1[t]];
Complement[t, s1];
t = f2[t];
s2 = Join[{First[t]}, f1[t]]

Extensions

a(15)-a(22) from Robert Israel, Jan 28 2025

A207639 Squares that can be written as a sum of 3 distinct nonzero squares in more than one way.

Original entry on oeis.org

225, 361, 441, 529, 625, 729, 841, 900, 961, 1089, 1225, 1369, 1444, 1521, 1681, 1764, 1849, 2025, 2116, 2209, 2401, 2500, 2601, 2809, 2916, 3025, 3249, 3364, 3481, 3600, 3721, 3844, 3969, 4225, 4356, 4489, 4761, 4900, 5041, 5329, 5476, 5625, 5776, 5929
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 19 2012

Keywords

Programs

  • Mathematica
    t = Sort[Select[Flatten[Table[x^2 + y^2 + z^2, {x, 400}, {y, x + 1, 400}, {z, y + 1, 400}]], # < 160006 && IntegerQ[Sqrt[#]] &]];
f1[l_] := Module[{t = {}}, Do[If[l[[n]] != l[[n + 1]] && l[[n]] != l[[n - 1]], AppendTo[t, l[[n]]]], {n, Length[l] - 1}]; t];
s1 = Join[{First[t]}, f1[t]];
Complement[t, s1]

A207638 Squares that can be written as a sum of 3 distinct nonzero squares in no more than one way.

Original entry on oeis.org

49, 81, 121, 169, 196, 289, 324, 484, 676, 784, 1156, 1296, 1936, 2704, 3136, 4624, 5184, 7744, 10816, 12544, 18496, 20736, 30976, 43264, 50176, 73984, 82944, 123904
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 19 2012

Keywords

Programs

  • Mathematica
    t = Sort[Select[Flatten[Table[x^2 + y^2 + z^2, {x, 400}, {y, x + 1, 400}, {z, y + 1, 400}]], # < 160006 && IntegerQ[Sqrt[#]] &]];
f1[l_] := Module[{t = {}}, Do[If[l[[n]] != l[[n + 1]] && l[[n]] != l[[n - 1]], AppendTo[t, l[[n]]]], {n, Length[l] - 1}]; t];
s1 = Join[{First[t]}, f1[t]]

A207637 Squarefree sums of 3 successive primes.

Original entry on oeis.org

10, 15, 23, 31, 41, 59, 71, 83, 97, 109, 131, 143, 159, 173, 187, 199, 211, 223, 235, 251, 269, 287, 301, 311, 319, 329, 349, 371, 395, 407, 439, 457, 471, 487, 503, 519, 533, 551, 565, 581, 589, 607, 633, 661, 679, 689, 701, 713, 731, 749, 771, 789, 803
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 19 2012

Keywords

Links

Crossrefs

Cf. A206329.

Programs

  • Mathematica
    Select[Table[Prime[n] + Prime[n + 1] + Prime[n + 2], {n, 300}], SquareFreeQ]
  • PARI
    p=2;q=3;forprime(r=5,1e4,if(issquarefree(t=p+q+r),print1(t", "));p=q;q=r) \\ Charles R Greathouse IV, Jun 14 2013

A207650 Squares that can be written as a sum of 3 distinct nonzero squares in 3 or more ways.

Original entry on oeis.org

441, 529, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1764, 1849, 2025, 2116, 2209, 2401, 2601, 2809, 2916, 3025, 3249, 3364, 3481, 3721, 3844, 3969, 4225, 4356, 4489, 4761, 4900, 5041, 5329, 5476, 5625, 5929, 6084, 6241, 6561, 6724, 6889, 7056, 7225, 7396
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 19 2012

Keywords

Crossrefs

Cf. A161992, A206606, A207638, A207639, A207640.

Programs

  • Mathematica
    t = Sort@Select[Flatten[Table[x^2 + y^2 + z^2, {x, 400}, {y, x + 1, 400}, {z, y + 1, 400}]], # < 160006 && IntegerQ[Sqrt[#]] &];
f1[l_] := Module[{t = {}}, Do[If[l[[n]] != l[[n + 1]] && l[[n]] != l[[n - 1]], AppendTo[t, l[[n]]]], {n, Length[l] - 1}]; t];
f2[l_] := Module[{t = {}}, Do[If[l[[n]] == l[[n + 1]], AppendTo[t, l[[n]]]], {n, Length[l] - 1}]; t];
s1 = Join[{First[t]}, f1[t]];
Complement[t, s1];
t = f2[t];
s2 = Join[{First[t]}, f1[t]];
Complement[t, s2]

A207527 Primes that are the sum of three consecutive primes in A034962.

Original entry on oeis.org

131, 251, 337, 503, 743, 929, 1447, 1597, 3023, 3919, 4051, 4157, 5087, 5897, 6133, 7649, 7877, 8269, 9181, 9433, 9721, 11411, 11677, 13151, 13757, 14009, 14533, 18133, 18493, 18743, 20563, 21023, 21247, 22651, 22921, 23057, 23801, 23893, 24359, 24733, 24809
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 18 2012

Keywords

Links

Crossrefs

Cf. A034962.

Programs

  • Mathematica
    t = Select[Table[Prime[n] + Prime[n + 1] + Prime[n + 2], {n, 1000}], PrimeQ]; Select[Table[t[[n]] + t[[n + 1]] + t[[n + 2]], {n, Length[t] - 2}], PrimeQ]

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