A163426 -id:A163426

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A163427

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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.

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 *)

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

EXTENSIONS

Edited by R. J. Mathar, Aug 24 2009

STATUS

approved

A163428

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

EXAMPLE

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

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;

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

PROG

(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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

EXTENSIONS

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

STATUS

approved

A163442

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

EXAMPLE

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

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 *)

PROG

(PARI) forprime(p=2, 1e3, n=p^3\27; if(isprime(n), print1(n", ")))

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

EXTENSIONS

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

STATUS

approved

A163443

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

EXTENSIONS

Introduced standard terminology in the definition - R. J. Mathar, Aug 02 2009

STATUS

approved

A165945

Primes of the form p+(p+1)^3, where p is also prime.

+10
1

29, 67, 5849, 3375149, 7078079, 7762589, 11852579, 17173769, 42144539, 46656359, 80621999, 87528827, 91125449, 102503699, 132651509, 142237169, 173741669, 264609929, 287496659, 320014187, 567664379, 686129849, 700227959, 851972339

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Generated by p=2, 3, 17, 149, 191, 197, 227, 257 etc.

LINKS

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

MATHEMATICA

Select[Table[p + (p + 1)^3, {p, Prime[Range[300]]}], PrimeQ] (* Vincenzo Librandi, Oct 12 2012 *)

PROG

(Magma) [a: p in PrimesInInterval(1, 1000) | IsPrime(a) where a is p + (p + 1)^3]; // Vincenzo Librandi, Ovt 12 2012

CROSSREFS

Cf. A163426. [R. J. Mathar, Oct 28 2009]

KEYWORD

nonn,easy

AUTHOR

Claudio Meller, Oct 01 2009

EXTENSIONS

Definition shortened by R. J. Mathar, Oct 28 2009

STATUS

approved

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