A163418 -id:A163418 - cp's OEIS frontend

A162652 Primes p such that there are positive integers m and n and a prime q such that p = m^2+m-q = n^2+n+q.

7, 13, 31, 43, 73, 211, 241, 421, 463, 1123, 1723, 2551, 2971, 4831, 5701, 6163, 8011, 8191, 9901, 11131, 12433, 14281, 17293, 19183, 20023, 23563, 24181, 28393, 30103, 31153, 35911, 37831, 43891, 46441, 53593, 60271, 77563, 83233, 86143, 95791

Offset: 1

Daniel Tisdale, Jul 08 2009

nonn

To test if a prime p is a member, p = n^2+n+q gives a finite list of possible pairs (n,q), and, for each value of q, m^2+m = p+q determines a putative value of m. - N. J. A. Sloane, Jul 17 2009

Also, primes of the form (p^2+3)/4 with p odd prime. - Zak Seidov, May 10 2014

			7 = 1^2+1+5 = 3^2+3-5.

Jean-François Alcover, Table of n, a(n) for n = 1..77

Cf. A163418. - R. J. Mathar, Feb 05 2010

isA002378 := proc(n) if n >= 0 then if issqr(4*n+1) then RETURN(type( sqrt(4*n+1),'odd')) ; else false; fi; else false; fi; end: # primes p such there is a prime qA162652 := proc(p) local j,q; if isprime(p) then for j from 1 do q := ithprime(j) ; if q >= p then break; fi; if isA002378(p+q) and isA002378(p-q) then RETURN(true) ; fi; od: false ; else false; fi; end: for n from 1 to 4000 do if isA162652(ithprime(n)) then printf("%d,",ithprime(n)) ; fi; od; # R. J. Mathar, Jul 17 2009

sol[p_] := m^2 + m - p /. Solve[m>0 && n>0 && 2p == m + m^2 + n + n^2, {m, n}, Integers];
Reap[For[p = 2, p < 10^6, p = NextPrime[p], qsel = Select[sol[p], PrimeQ]; If[qsel != {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Mar 25 2020 *)

Definition revised by N. J. A. Sloane, Jul 17 2009
More terms from R. J. Mathar, Jul 17 2009
Extended beyond a(31) by R. J. Mathar, Feb 05 2010

A163419 Primes of the form ((p+1)/2)^2+((p-1)/2), where p is prime.

Original entry on oeis.org

5, 11, 19, 41, 89, 109, 239, 271, 379, 461, 599, 929, 991, 2069, 2969, 3079, 4159, 4421, 4969, 5851, 9311, 10099, 13109, 13339, 14519, 16001, 20021, 23869, 25439, 28729, 30449, 32579, 34039, 38219, 39799, 48619, 50849, 53591, 57839, 59779, 60761

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Subsequence of A002327. - Charles R Greathouse IV, Aug 11 2009

			((3+1)/2)^2+((3-1)/2) = 4+1 = 5;
((5+1)/2)^2+((5-1)/2) = 9+2 = 11;
((7+1)/2)^2+((7-1)/2) = 16+3 = 19.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A002327, A162652, A163418.

[a: p in PrimesInInterval(3, 600) | IsPrime(a) where a is (p^2 + 4*p - 1) div 4]; // Vincenzo Librandi, Sep 17 2016

f[n_]:=((p+1)/2)^2+((p-1)/2); lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,6!}];lst
Select[((#+1)/2)^2+(#-1)/2&/@Prime[Range[500]],PrimeQ] (* Harvey P. Dale, Nov 25 2012 *)

lista(nn) = forprime(p=3, nn, if(isprime(P=(p^2+4*p-1)/4), print1(P, ", "))); \\ Altug Alkan, Sep 17 2016

A163420 Primes p such that p+(p^2-1)/4 is also prime.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 29, 31, 37, 41, 47, 59, 61, 89, 107, 109, 127, 131, 139, 151, 191, 199, 227, 229, 239, 251, 281, 307, 317, 337, 347, 359, 367, 389, 397, 439, 449, 461, 479, 487, 491, 569, 587, 601, 617, 659, 661, 677, 701, 719, 727, 769, 809, 839, 911, 941

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

			3 is in the sequence because 3+(3^2-1)/4=5 is a prime number.
5 is in the sequence because 5+(5^2-1)/4=11 is a prime number.

J. Mulder, Table of n, a(n) for n = 1..50000

Cf. A162652, A163418, A165350, A163419.

[p: p in PrimesInInterval(3,1000) | IsPrime(p+(p^2-1) div 4)]; // Vincenzo Librandi, Apr 08 2013

f[n_]:=((p+1)/2)^2+((p-1)/2); lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst, p]],{n,6!}];lst
Select[Range[700], PrimeQ[#] && PrimeQ[# + (#^2 - 1)/4] &] (* Vincenzo Librandi, Apr 08 2013 *)
Select[Prime[Range[200]],PrimeQ[#+(#^2-1)/4]&] (* Harvey P. Dale, Jun 18 2014 *)

A163419(n) = a(n)+( a(n)^2-1 )/4. [R. J. Mathar, Aug 17 2009]

{A000040(k): A000040(k)+A024701(k-1) in A000040}.

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

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

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

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,..

J. Mulder, Table of n, a(n) for n = 1..10000

Cf. A162652, A163418, A163419, A163420.

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

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

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

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

			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.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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

[p: p in PrimesUpTo(1000) | IsPrime((p^3-3*p^2+7*p+3) div 8)]; // Vincenzo Librandi, Apr 10 2013

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 *)

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

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

			(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.

G. C. Greubel, Table of n, a(n) for n = 1..1100

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

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 *)

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

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

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

			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.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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

[p: p in PrimesInInterval(3, 2000) | IsPrime((p-1)^3 div 8 + (p+1)^2 div 4)]; // Vincenzo Librandi, Apr 08 2013

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 *)

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

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

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

G. C. Greubel, Table of n, a(n) for n = 1..1500

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

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

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

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

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

			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.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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

[p: p in PrimesInInterval(3, 1200) | IsPrime((p+1)^3 div 8+(p-1) div 2)]; // Vincenzo Librandi, Apr 09 2013

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 *)

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

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

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

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

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

Robert Israel, Table of n, a(n) for n = 1..10000

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

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

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

lista(nn) = forprime(p=3, nn, if (isprime(q=((p+1)/2)^3 + ((p-1)/2)^2), print1(q, ", "))); \\ Michel Marcus, Oct 11 2016

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

cp's OEIS Frontend

A162652 Primes p such that there are positive integers m and n and a prime q such that p = m^2+m-q = n^2+n+q.

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A163419 Primes of the form ((p+1)/2)^2+((p-1)/2), where p is prime.

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A163420 Primes p such that p+(p^2-1)/4 is also prime.

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

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

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

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

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