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 1-10 of 10 results.

A163421 Primes of the form ((p-1)/2)^3+((p+1)/2), p are prime numbers.

Original entry on oeis.org

3, 11, 31, 131, 223, 521, 739, 3391, 5851, 9283, 24419, 27031, 59359, 68963, 85229, 110641, 148931, 157519, 175673, 328579, 405299, 571871, 857471, 1561013, 1728121, 2248223, 2460511, 3112283, 3581731, 3724031, 4741801, 5735519
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Keywords

Comments

Subsequence of A095692.
((3-1)/2)^3+((3+1)/2)=1+2=3, ((5-1)/2)^3+((5+1)/2)=8+3=11, ((7-1)/2)^3+((7+1)/2)=27+4=31,..

Links

Crossrefs

Cf. A162652, A163418, A163419, A163420.

Programs

  • Mathematica
    f[n_]:=((p-1)/2)^3+((p+1)/2); lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,6!}];lst

Extensions

Comment from Charles R Greathouse IV, Aug 11 2009

A163422 Primes p such that A071568((p-1)/2) is also prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 43, 59, 61, 79, 83, 89, 97, 107, 109, 113, 139, 149, 167, 191, 233, 241, 263, 271, 293, 307, 311, 337, 359, 373, 383, 439, 443, 479, 487, 491, 523, 557, 617, 641, 647, 659, 673, 683, 701, 733, 757, 811, 829, 853, 857, 859, 877
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Keywords

Comments

Primes p such that (p-1)^3/8+(p+1)/2 is also prime, i.e., in A095692.

Examples

			p=3 is in the sequence because (3-1)^3/8+(3+1)/2=3 is prime.
p=5 is in the sequence because (5-1)^3/8+(5+1)/2=11 is prime.

Links

Crossrefs

Cf. A162652, A163418, A163419, A163420, A163421.

Programs

  • Magma
    [p: p in PrimesUpTo(1000) | IsPrime((p^3-3*p^2+7*p+3) div 8)]; // Vincenzo Librandi, Apr 10 2013
  • Mathematica
    f[n_]:=((n-1)/2)^3+((n+1)/2); lst={}; Do[p=Prime[n]; If[PrimeQ[f[p]], AppendTo[lst, p]], {n,6!}]; lst
Select[Prime[Range[180]], PrimeQ[(#-1)^3/8+(#+1)/2]&]  (* Harvey P. Dale, Jan 05 2011 *)

Extensions

Definition rewritten by R. J. Mathar, Aug 17 2009

A163424 Primes of the form (p-1)^3/8 + (p+1)^2/4 where p is prime.

Original entry on oeis.org

5, 17, 43, 593, 829, 2969, 3631, 12743, 27961, 44171, 60919, 127601, 278981, 578843, 737281, 950993, 980299, 1455893, 1969001, 2424329, 2763881, 3605293, 5767739, 7801993, 9305521, 11290049, 12220361, 12704093, 16452089, 22987529, 35720189
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Keywords

Examples

			(3-1)^3/8 + (3+1)^2/4 = 1 + 4 = 5;
(5-1)^3/8 + (5+1)^2/4 = 8 + 9 = 17;
(7-1)^3/8 + (7+1)^2/4 = 27 + 16 = 43.

Links

Crossrefs

Subsequence of A100662.
For the corresponding primes p, see A163425.
Cf. A162652, A163418, A163419, A163420, A163421, A163422.

Programs

  • Mathematica
    f[n_]:=((p-1)/2)^3+((p+1)/2)^2; lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,7!}];lst
Select[(#-1)^3/8+(#+1)^2/4&/@Prime[Range[150]],PrimeQ] (* Harvey P. Dale, Oct 05 2018 *)
  • PARI
    list(lim)=my(v=List(),t); forprime(p=3,, t=((p-1)/2)^3 + ((p+1)/2)^2; if(t>lim, break); if(isprime(t), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Dec 23 2016

A163425 Primes p such that (p-1)^3/8+(p+1)^2/4 is also prime.

Original entry on oeis.org

3, 5, 7, 17, 19, 29, 31, 47, 61, 71, 79, 101, 131, 167, 181, 197, 199, 227, 251, 269, 281, 307, 359, 397, 421, 449, 461, 467, 509, 569, 659, 691, 709, 811, 859, 919, 937, 997, 1031, 1087, 1151, 1217, 1231, 1249, 1277, 1279, 1301, 1307, 1361, 1409, 1447, 1451
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Keywords

Comments

The associated (p-1)^3/8+(p+1)^2/4 are in A163424.

Examples

			p=3 is in the sequence because (3-1)^3/8+(3+1)^2/4=1+4=5 is also prime.
p=5 is in the sequence because (5-1)^3/8+(5+1)^2/4=17 is also prime.

Links

Crossrefs

Cf. A162652, A163418, A163419, A163420, A163421, A163422.

Programs

  • Magma
    [p: p in PrimesInInterval(3, 2000) | IsPrime((p-1)^3 div 8 + (p+1)^2 div 4)]; // Vincenzo Librandi, Apr 08 2013
  • Mathematica
    f[n_]:=((p-1)/2)^3+((p+1)/2)^2; lst={};Do[p=Prime[n];If[PrimeQ[f[p]], AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]], PrimeQ[(# - 1)^3 / 8 + (# + 1)^2 / 4]&] (* Vincenzo Librandi, Apr 08 2013 *)

Extensions

Comment turned into examples by R. J. Mathar, Sep 02 2009

A163426 Primes of the form ((p+1)/2)^3 + ((p-1)/2), p is prime.

Original entry on oeis.org

29, 67, 349, 1009, 3389, 4111, 9281, 19709, 46691, 132701, 140659, 166429, 658589, 884831, 1000099, 1405039, 1520989, 1601729, 1728119, 2146817, 2460509, 2685757, 4574461, 7078079, 7880797, 10077911, 14887181, 23149409, 23393941, 27000299
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Keywords

Examples

			((5+1)/2)^3 + ((5-1)/2) = 27 + 2 = 29;
((7+1)/2)^3 + ((7-1)/2) = 64 + 3 = 67.

Links

Crossrefs

Cf. A162652, A163418, A163419, A163420, A163421, A163422, A163424, A163425.

Programs

  • Mathematica
    f[n_]:=((p+1)/2)^3+((p-1)/2); lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,6!}];lst

Extensions

Checked by Charles R Greathouse IV, Aug 11 2009

A163427 Primes p such that (p+1)^3/8+(p-1)/2 is also prime.

Original entry on oeis.org

5, 7, 13, 19, 29, 31, 41, 53, 71, 101, 103, 109, 173, 191, 199, 223, 229, 233, 239, 257, 269, 277, 331, 383, 397, 431, 491, 569, 571, 599, 619, 631, 719, 733, 751, 757, 761, 823, 857, 859, 863, 887, 907, 937, 967, 971, 977, 1009, 1019, 1063, 1069, 1123, 1163
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Keywords

Comments

Primes A000040(k) such that (A006254(k-1))^3+ A005097(k-1) is also prime.

Examples

			For p=5, (5+1)^3/8+(5-1)/2=27+2=29, prime, which adds p=5 to the sequence.
For p=7, (7+1)^3/8+(7-1)/2=67, prime, which adds p=7 to the sequence.

Links

Crossrefs

Cf. A162652, A163418, A163419, A163420, A163421, A163422, A163424, A163425, A163426.

Programs

  • Magma
    [p: p in PrimesInInterval(3, 1200) | IsPrime((p+1)^3 div 8+(p-1) div 2)]; // Vincenzo Librandi, Apr 09 2013
  • Mathematica
    f[n_]:=((p+1)/2)^3+((p-1)/2); lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst, p]],{n,6!}];lst
Select[Prime[Range[100]], PrimeQ[(# + 1)^3 / 8 + (# - 1) / 2]&] (* Vincenzo Librandi, Apr 09 2013 *)

Formula

(a(n)+1)^3/8+(a(n)-1)/2 = A163426(n).

Extensions

Edited by R. J. Mathar, Aug 24 2009

A163428 Primes of the form ((p+1)/2)^3 + ((p-1)/2)^2 where p is prime.

Original entry on oeis.org

31, 73, 241, 379, 3571, 9661, 20359, 47881, 51949, 65521, 119953, 135151, 291721, 427351, 736921, 761671, 921889, 1202041, 1494313, 1533871, 1742161, 1785961, 2478331, 2533681, 3197839, 3820441, 3894229, 4044643, 4855033, 6573799
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Keywords

Comments

Primes of the form k^3 + k^2 - 2k + 1 where 2k-1 is prime.

Examples

			((5+1)/2)^3 + ((5-1)/2)^2 = 27 + 4 = 31, ((7+1)/2)^3 + ((7-1)/2)^2 = 64 + 9 = 73

Links

Crossrefs

Cf. A162652, A163418, A163419, A163420, A163421, A163422, A163424, A163425, A163426, A163427

Programs

  • Maple
    res:= NULL:
count:= 0:
p:= 2
while count < 100 do
  p:= nextprime(p);
  r:=  ((p+1)/2)^3 + ((p-1)/2)^2;
  if isprime(r) then
     res:= res, r;
     count:= count+1;
  fi
od:
res; # Robert Israel, Oct 10 2016
  • Mathematica
    f[n_]:=((p+1)/2)^3+((p-1)/2)^2; lst={}; Do[p=Prime[n]; If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,6!}]; lst
  • PARI
    lista(nn) = forprime(p=3, nn, if (isprime(q=((p+1)/2)^3 + ((p-1)/2)^2), print1(q, ", "))); \\ Michel Marcus, Oct 11 2016

Extensions

Description and edits by Charles R Greathouse IV, Oct 05 2009

A163442 Primes of the form floor((p/3)^3), where p is prime.

Original entry on oeis.org

181, 1103, 40471, 143329, 212419, 266261, 468493, 14586401, 20948491, 48894061, 53298877, 86546399, 136061111, 150851969, 189448891, 227353303, 249650309, 256855171, 328033129, 361451309, 507533053, 710528249, 815653171, 1172016731
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Keywords

Examples

			(17/3)^3=181.963 -> 181, (31/3)^3=1103.37 -> 1103, (103/3)^3=40471.4 -> 40471

Links

Crossrefs

Cf. A162652, A163418, A163419, A163420, A163421, A163422, A163424, A163425, A163426, A163427, A163428, A163429, A163430, A163431.

Programs

  • Mathematica
    f[n_]:=IntegerPart[(p/3)^3]; lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,7!}];lst
Select[Table[Floor[(p/3)^3],{p,Prime[Range[800]]}],PrimeQ] (* Harvey P. Dale, Dec 16 2017 *)
  • PARI
    forprime(p=2,1e3,n=p^3\27;if(isprime(n),print1(n",")))

Extensions

Program and editing by Charles R Greathouse IV, Nov 09 2009

A165350 Primes p such that floor((p^2-1)/4)+p is not prime.

Original entry on oeis.org

13, 23, 43, 53, 67, 71, 73, 79, 83, 97, 101, 103, 113, 137, 149, 157, 163, 167, 173, 179, 181, 193, 197, 211, 223, 233, 241, 257, 263, 269, 271, 277, 283, 293, 311, 313, 331, 349, 353, 373, 379, 383, 401, 409, 419, 421, 431, 433, 443, 457, 463, 467, 499
Offset: 1

Views

Author

Vincenzo Librandi, Sep 16 2009

Keywords

Links

Crossrefs

Cf. A163419, A163420.

Programs

  • Magma
    [p: p in PrimesUpTo(500)| not IsPrime(Floor((p^2-1)/4)+p)]; // Vincenzo Librandi, Sep 12 2013
  • Mathematica
    Select[Prime[Range[2, 200]], ! PrimeQ[(#^2 - 1) / 4 + #]&] (* Vincenzo Librandi, Sep 12 2013 *)

Extensions

Edited (but not checked) by N. J. A. Sloane, Sep 25 2009

A163443 Primes p such that floor(p^3/27) is prime.

Original entry on oeis.org

17, 31, 103, 157, 179, 193, 233, 733, 827, 1097, 1129, 1327, 1543, 1597, 1723, 1831, 1889, 1907, 2069, 2137, 2393, 2677, 2803, 3163, 3257, 3433, 3617, 3797, 4261, 4999, 5233, 5237, 5309, 5449, 5701, 5939, 6079, 6173, 6637, 6781, 6961, 7069, 7321, 7879
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Keywords

Examples

			p=17 is in the sequence because [(17/3)^3] = [181.963] = 181 is prime.
p=31 is in the sequence because [(31/3)^3] = [1103.37] = 1103 is prime.

Links

Crossrefs

Cf. A162652, A163418, A163419, A163420, A163421, A163422, A163424, A163425, A163426, A163427, A163428, A163429, A163430, A163431, A163442

Programs

  • Mathematica
    f[n_]:=IntegerPart[(p/3)^3]; lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst, p]],{n,7!}];lst

Extensions

Introduced standard terminology in the definition - R. J. Mathar, Aug 02 2009
Showing 1-10 of 10 results.

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