A127882 -id:A127882 - cp's OEIS frontend

A127883 a(n) = 60*(n^5/120 + n^4/24 + n^3/6 + n^2/2 + n + 1).

163, 436, 1104, 2572, 5485, 10788, 19786, 34204, 56247, 88660, 134788, 198636, 284929, 399172, 547710, 737788, 977611, 1276404, 1644472, 2093260, 2635413, 3284836, 4056754, 4967772, 6035935, 7280788, 8723436, 10386604, 12294697

Offset: 1

Artur Jasinski, Feb 04 2007

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).

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A127873-A127882.

[n^4*(n+5)/2+10*(n^3+3*n^2+6*n+6): n in [1..30]]; // Bruno Berselli, Apr 03 2012

A127883:=n->60*(n^5/120 + n^4/24 + n^3/6 + n^2/2 + n + 1); seq(A127883(n), n=1..40); # Wesley Ivan Hurt, Mar 27 2014

Table[1/2 (120+x (120+x (60+x (20+x (5+x))))), {x,40}] (* Harvey P. Dale, Mar 12 2011 *)
CoefficientList[Series[(163 - 542 x + 933 x^2 - 772 x^3 + 338 x^4 - 60 x^5)/(1 - x)^6, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 28 2014 *)

G.f.: x*(163-542*x+933*x^2-772*x^3+338*x^4-60*x^5)/(1-x)^6. - Colin Barker, Apr 02 2012

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

A127881 Integers of the form x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1 with x > 0.

Original entry on oeis.org

241231, 7057861, 21166951, 52066891, 216295321, 654480151, 1619368381, 2411089396, 3486017011, 6776093041, 12182173471, 20592045301, 26260194241, 33113005531, 51096161161, 76160729191, 110218336621, 131302849486

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, A127882, A127883, A127884.

a = {}; Do[If[IntegerQ[1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120], AppendTo[a, 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120]], {x, 1, 1000}]; a
Select[Table[ x^5/120+x^4/24+x^3/6+x^2/2+x+1,{x,450}],IntegerQ] (* Harvey P. Dale, Jan 20 2019 *)

for(x=1,500,y=x^5+5*x^4+20*x^3+60*x^2+120*x+120;if(y%120==0,print1(y/120, ", "))) \\ Michael B. Porter, Jan 29 2010

isA127881(n)={local(r);r=0;fordiv(120*n-120,x,if(x^5/120+x^4/24+x^3/6+x^2/2+x+1==n,r=1));r} \\ Michael B. Porter, Jan 29 2010

cp's OEIS Frontend

A127883 a(n) = 60*(n^5/120 + n^4/24 + n^3/6 + n^2/2 + n + 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

Mathematica

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

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Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

A127881 Integers of the form x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1 with x > 0.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

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