A127881 -id:A127881 - cp's OEIS frontend

A127882 Primes of the form 60*(x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1).

163, 977611, 12294697, 37985853397, 49252877161, 137434331779, 830329719061, 1626105882361, 8060524420261, 11467771684597, 13008402510163, 15315610041211, 43633838254429, 71635442712061, 125119099806661

Offset: 1

Artur Jasinski, Feb 04 2007

nonn

Generating polynomial is Schur's polynomial of 5-degree. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A127873, A127874, A127875, A127876, A127877, A127878, A127879, A127880, A127881, A127883.

Filtered(List([1..2000],x->60*(x^5/120+x^4/24+x^3/6+x^2/2+x+1)),IsPrime); # Muniru A Asiru, Apr 30 2018

select(isprime,[seq(60*(x^5/120+x^4/24+x^3/6+x^2/2+x+1),x=1..2000)]); # Muniru A Asiru, Apr 30 2018

a = {}; Do[If[PrimeQ[60 + 60*x + 30*x^2 + 10*x^3 + (5*x^4)/2 + x^5/2], AppendTo[a, 60 + 60*x + 30*x^2 + 10*x^3 + (5*x^4)/2 + x^5/2]], {x, 1, 1000}]; a

A127877 Integers of the form (x^4)/24 + (x^3)/6 + (x^2)/2 + x + 1 with x > 0.

Original entry on oeis.org

7, 115, 297, 1237, 2171, 5527, 8221, 16441, 22335, 38731, 49697, 78445, 96787, 142927, 171381, 240817, 282551, 382051, 440665, 577861, 657387, 840775, 945677, 1184617, 1319791, 1624507, 1795281, 2176861, 2388995, 2859391, 3119077, 3691105, 4004967, 4692307

Offset: 1

Artur Jasinski, Feb 04 2007

Generating polynomial is Schur's polynomial of 4-degree. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A127873, A127874, A127875, A127876, A127878, A127879, A127880, A127881, A127882, A127883.

Filtered(List([0..150],x->(x^4)/24+(x^3)/6+(x^2)/2+x+1),IsInt); # Muniru A Asiru, Apr 30 2018

[(11 +5*(-1)^n +16*(2+(-1)^n)*n +18*(3+(-1)^n)*n^2 +36*(1+(-1)^n)*n^3 +54*n^4)/16: n in [1..30]]; // G. C. Greubel, Apr 29 2018

a = {}; Do[If[IntegerQ[1 + x + x^2/2 + x^3/6 + x^4/24], AppendTo[a, 1 + x + x^2/2 + x^3/6 + x^4/24]], {x, 1, 100}]; a
Select[Table[(x^4)/24+(x^3)/6+(x^2)/2+x+1,{x,100}],IntegerQ] (* Harvey P. Dale, Aug 14 2012 *)

Vec(x*(7+108*x+154*x^2+508*x^3+248*x^4+244*x^5+22*x^6+4*x^7+x^8)/((1-x)^5*(1+x)^4) + O(x^50)) \\ Colin Barker, May 15 2016

From Colin Barker, May 15 2016: (Start)
a(n) = (11 +5*(-1)^n +16*(2+(-1)^n)*n +18*(3+(-1)^n)*n^2 +36*(1+(-1)^n)*n^3 +54*n^4)/16.
a(n) = (27*n^4+36*n^3+36*n^2+24*n+8)/8 for n even.
a(n) = (27*n^4+18*n^2+8*n+3)/8 for n odd.
a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9) for n>9.
G.f.: x*(7+108*x+154*x^2+508*x^3+248*x^4+244*x^5+22*x^6+4*x^7+x^8) / ((1-x)^5*(1+x)^4).
(End)

A127879 Primes of the form x^4 + 4x^3 + 12x^2 + 24*x + 24.

Original entry on oeis.org

3760073, 9853769, 117051593, 181145609, 2517933833, 8999750153, 10486376969, 20852229449, 26640445193, 56713997513, 65555973569, 136653695753, 172008443273, 262819256009, 330127243553, 340704528713, 362619554249

Offset: 1

Artur Jasinski, Feb 04 2007

nonn

Generating polynomial is Schur's polynomial of 4-degree. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A127873, A127874, A127875, A127876, A127877, A127878, A127880, A127881, A127882, A127883.

Filtered(List([1..2000],x->x^4+4*x^3+12*x^2+24*x+24),IsPrime); # Muniru A Asiru, Apr 30 2018

select(isprime,[seq(x^4+4*x^3+12*x^2+24*x+24,x=1..2000)]); # Muniru A Asiru, Apr 30 2018

a = {}; Do[If[PrimeQ[24 + 24 x + 12 x^2 + 4 x^3 + x^4], AppendTo[a, 24 + 24 x + 12 x^2 + 4 x^3 + x^4]], {x, 1, 1000}]; a
Select[Table[x^4+4x^3+12x^2+24x+24,{x,780}],PrimeQ[#]&] (* Harvey P. Dale, Jan 24 2013 *)

A127880 Numbers x for which x^4 + 4x^3 + 12x^2 + 24x + 24 is prime.

Original entry on oeis.org

43, 55, 103, 115, 223, 307, 319, 379, 403, 487, 505, 607, 643, 715, 757, 763, 775, 799, 883, 925, 979, 1063, 1069, 1135, 1147, 1165, 1189, 1279, 1309, 1369, 1543, 1567, 1585, 1627, 1693, 1729, 1783, 1813, 1819, 1855, 1903, 1939, 1945, 2083, 2149, 2155

Offset: 1

Artur Jasinski, Feb 04 2007

nonn

Generating polynomial is Schur's polynomial of 4-degree. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A127873, A127874, A127875, A127876, A127877, A127878, A127879, A127881, A127882, A127883.

Filtered([1..3000],x->IsPrime(x^4+4*x^3+12*x^2+24*x+24)); # Muniru A Asiru, Apr 30 2018

select(x->isprime(x^4+4*x^3+12*x^2+24*x+24),[$1..3000]); # Muniru A Asiru, Apr 30 2018

a = {}; Do[If[PrimeQ[24 + 24 x + 12 x^2 + 4 x^3 + x^4], AppendTo[a, x]], {x, 1, 1000}]; a

isok(x) = isprime(x^4 + 4*x^3 + 12*x^2 + 24*x + 24); \\ Michel Marcus, Apr 30 2018

cp's OEIS Frontend

A127882 Primes of the form 60*(x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1).

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A127877 Integers of the form (x^4)/24 + (x^3)/6 + (x^2)/2 + x + 1 with x > 0.

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A127879 Primes of the form x^4 + 4*x^3 + 12*x^2 + 24*x + 24.

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GAP

Maple

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A127880 Numbers x for which x^4 + 4x^3 + 12x^2 + 24x + 24 is prime.

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GAP

Maple

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A127879 Primes of the form x^4 + 4x^3 + 12x^2 + 24*x + 24.