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.

A272382 Primes p == 1 (mod 3) for which A261029(14*p) = 3.

Original entry on oeis.org

13, 19, 31, 37, 43, 61, 67, 97, 157
Offset: 1

Views

Author

Vladimir Shevelev, Apr 28 2016

Keywords

Comments

Peter J. C. Moses did not find any term > 157. The author proved that the sequence is full. Moreover, he proved the following more general result.
Theorem. If p,q == 1 (mod 3) are prime and A261029(2*q*p) > 2, then sqrt(q)/2 < p < 4*q^2.
In this sequence q=7, so a(n) < 196.
Proof of Theorem is similar to proof of the theorem in A272384.

Links

Crossrefs

Cf. A261029, A272381, A272384, A272404, A272406, A272407, A272409.

Programs

  • Mathematica
    r[n_] := Reduce[0 <= x <= y <= z && z >= x + 1 && n == x^3 + y^3 + z^3 - 3 x y z, {x, y, z}, Integers];
a29[n_] := Which[rn = r[n]; rn === False, 0, rn[[0]] === And, 1, rn[[0]] === Or, Length[rn], True, Print["error ", rn]];
Select[Select[Range[1, 1000, 3], PrimeQ], a29[14 #] == 3&] (* Jean-François Alcover, Nov 21 2018 *)

A272404 Primes p == 1 (mod 3) for which A261029(26*p) = 3.

Original entry on oeis.org

7, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 223, 241, 277, 349, 463, 601
Offset: 1

Views

Author

Vladimir Shevelev, Apr 29 2016

Keywords

Comments

By theorem in A272382, case q=13, the sequence is finite with a(n) <= 676.

Links

Crossrefs

Cf. A261029, A272381, A272382, A272384.

Programs

  • Mathematica
    r[n_] := Reduce[0 <= x <= y <= z && z >= x + 1 && n == x^3 + y^3 + z^3 - 3 x y z, {x, y, z}, Integers];
a29[n_] := Which[rn = r[n]; rn === False, 0, rn[[0]] === And, 1, rn[[0]] === Or, Length[rn], True, Print["error ", rn]];
Select[Range[1, 997, 3], PrimeQ[#] && a29[26#] == 3&] (* Jean-François Alcover, Nov 06 2018 *)

Extensions

All terms (after first author's ones) were calculated by Peter J. C. Moses, Apr 29 2016

A272406 Primes p == 1 (mod 3) for which A261029(34*p) = 2.

Original entry on oeis.org

7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 367, 373, 397, 409, 421, 439, 457, 487, 571, 709, 787, 877
Offset: 1

Views

Author

Vladimir Shevelev, Apr 29 2016

Keywords

Comments

By theorem in A272384, case q=17, the sequence is finite with a(n)<1156.

Links

Crossrefs

Cf. A261029, A272381, A272382, A272384, A272404.

Programs

  • Mathematica
    r[n_] := Reduce[0 <= x <= y <= z && z >= x+1 && n == x^3 + y^3 + z^3 - 3 x y z, {x, y, z}, Integers];
a29[n_] := Which[rn = r[n]; rn === False, 0, rn[[0]] === And, 1, rn[[0]] === Or, Length[rn], True, Print["error ", rn]];
Select[Select[Range[7, 997, 3], PrimeQ], a29[34 #] == 2&] (* Jean-François Alcover, Dec 01 2018 *)

Extensions

All terms (after first author's ones) were calculated by Peter J. C. Moses, Apr 29 2016

A272407 Primes p == 1 (mod 3) for which A261029(38*p) = 3.

Original entry on oeis.org

7, 13, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 433, 463, 499, 523, 541, 577, 601, 607, 619, 661, 757, 853, 937, 1123, 1129
Offset: 1

Views

Author

Vladimir Shevelev, Apr 29 2016

Keywords

Comments

By theorem in A272382, case q=19, the sequence is finite with a(n)<1444.

Links

Crossrefs

Cf. A261029, A272381, A272382, A272384, A272404, A272406.

Programs

  • Mathematica
    r[n_] := Reduce[0 <= x <= y <= z && z >= x+1 && n == x^3+y^3+z^3 - 3 x y z, {x, y, z}, Integers];
a261029[n_] := Which[rn = r[n]; rn === False, 0, rn[[0]] === And, 1, rn[[0]] === Or, Length[rn], True, Print["error ", rn]];
Select[Select[Range[1, 1171, 3], PrimeQ], a261029[38 #] == 3&] (* Jean-François Alcover, Dec 04 2018 *)

Extensions

All terms (after first author's ones) were calculated by Peter J. C. Moses, Apr 29 2016

A272409 Primes p == 1 (mod 3) for which A261029(46*p) = 2.

Original entry on oeis.org

7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 613, 631, 643, 673, 709, 733, 739, 787, 811, 829, 859, 877, 907, 1009, 1063, 1093, 1117, 1279, 1297, 1381, 1483, 1489, 1723
Offset: 1

Views

Author

Vladimir Shevelev, Apr 29 2016

Keywords

Comments

By theorem in A272384, case q=23, the sequence is finite with a(n)<2116.

Links

Crossrefs

Cf. A261029, A272381, A272382, A272384, A272404, A272406, A272407.

Programs

  • Mathematica
    r[n_] := Reduce[0 <= x <= y <= z && z >= x+1 && n == x^3 + y^3 + z^3 - 3 x y z, {x, y, z}, Integers];
a29[n_] := Which[rn = r[n]; rn === False, 0, rn[[0]] === And, 1, rn[[0]] === Or, Length[rn], True, Print["error ", rn]];
Select[Select[Range[1, 2002, 3], PrimeQ], a29[ 46 # ] == 2&] (* Jean-François Alcover, Dec 06 2018 *)

Extensions

All terms (after first author's ones) were calculated by Peter J. C. Moses, Apr 29 2016

A268665 Number of primes p==1 (mod 3) for which A261029(2*prime(n)*p) is 4-i if prime(n)==i (mod 3), where i=1 or 2.

Original entry on oeis.org

6, 9, 22, 26, 44, 52, 73, 111, 122, 164, 201, 214, 254, 311, 374, 398, 465, 521, 542, 617, 684, 774, 899, 969, 1005, 1064, 1100, 1181, 1441, 1548, 1658, 1694, 1918, 1977, 2114, 2255, 2376, 2537, 2684, 2727, 3019, 3068, 3181, 3238, 3611, 3985, 4114, 4182, 4313
Offset: 3

Views

Author

Vladimir Shevelev and Peter J. C. Moses, May 07 2016

Keywords

Examples

			Let n=3, then prime(n)=5. Since 5==2(mod 3), then i=2. So a(3) is the number of primes p==1(mod 3) for which A261029(10*p)=4-2=2. So it is number of terms in A272381, i.e., a(3)=6.
Let n=4, then prime(n)=7. Since 7==1(mod 3), then i=1. So a(4) is the number of primes p==1(mod 3) for which A261029(14*p)=4-1=3. So it is number of terms in A272382, i.e., a(4)=9.

Crossrefs

Cf. A261029, A272381, A272382, A272384, A272404, A272406, A272407, A272409.

A272856 Greatest length of a chain of consecutive primes p==1 (mod 3) for which A261029 (2*prime(n)*p) is 4-i if prime(n) == i (mod 3), where i=1,2.

Original entry on oeis.org

5, 7, 16, 19, 32, 35, 50, 76, 81, 108, 140, 139, 171, 206, 254, 259, 305, 346, 349, 404, 449, 504, 582, 634, 645, 699, 707, 772, 930, 1006, 1078, 1097, 1258, 1271, 1362, 1448, 1529, 1633, 1737, 1752, 1951, 1970, 2064, 2082, 2310, 2550, 2659, 2672, 2783, 2917
Offset: 3

Views

Author

Vladimir Shevelev and Peter J. C. Moses, May 08 2016

Keywords

Examples

			Let n=3; then prime(n)=5. Since 5 == 2 (mod 3), i=2. So a(3) is the greatest length of a chain of consecutive primes p == 1 (mod 3) for which A261029(10*p) = 4 - 2 = 2. So these primes are in A272381. The first term is 7, and we have the chain of consecutive primes == 1 (mod 3): {7, 13, 19, 31, 37}. Since the following prime 43 == 1 (mod 3) is not in A272381, the chain ends and its length is 5. The second chain is the singleton {71}. So a(3)=5.

Crossrefs

Cf. A261029, A268665, A272381, A272382, A272384, A272404, A272406, A272407, A272409.

Programs

  • Mathematica
    a261029[n_]:=a261029[n]={x,y,z}/.{ToRules[Reduce[x^3+y^3+z^3-3 x y z==n&&0<=x<=y<=z&&z>=x+1,Integers]]}/.{x,y,z}->{};
data={};
Do[p=Prime[n];
primes=Select[Prime[Range[1+PrimePi[(2p)^2]]],Mod[#,3]==1&];
tmp=Map[{#,Length[a261029[2 # p]]}&,primes];
AppendTo[data,{{n,2p,1+Mod[2p,3]},{{Length[#],Max[Map[Length,Select[Split[Differences[Flatten[Map[Position[primes,#,1,1]&,#]]]],#[[1]]==1&]]+1]},#}&[Map[#[[1]]&,Select[tmp,#[[2]]==(1+Mod[2p,3])&]]]}];Print[Last[data]],{n,3,10}]
Map[Length[a261029[#]]&,Range[0,20]] (* A261029 *)
Last[Last[data[[1]]]] (* A272381 *)
Last[Last[data[[2]]]] (* A272382 *)
Last[Last[data[[3]]]] (* A272384 *)
Last[Last[data[[4]]]] (* A272404 *)
Last[Last[data[[5]]]] (* A272406 *)
Last[Last[data[[6]]]] (* A272407 *)
Last[Last[data[[7]]]] (* A272409 *)
Map[#[[2]][[1]][[1]]&,data] (* A268665 *)
Map[#[[2]][[1]][[2]]&,data] (* A272856 *)
Showing 1-7 of 7 results.

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

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