A127873 -id:A127873 - cp's OEIS frontend

A127874 Prime numbers of the form (x^3)/2+(3x^2)/2+3x+3.

19, 71, 269, 379, 683, 883, 4663, 6949, 9883, 12239, 16433, 21491, 45631, 66403, 92683, 125119, 186733, 211051, 228383, 256121, 286019, 296479, 352619, 389483, 562589, 578971, 683983, 721619, 842759, 930619, 1150183, 1230391, 1372211

Offset: 1

Artur Jasinski, Feb 04 2007

nonn

Generating polynomial is Schur's polynomial of degree 3. 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).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A127873.

a = {}; Do[If[PrimeQ[3 + 3 x + (3 x^2)/2 + x^3/2], AppendTo[a, 3 + 3 x + (3 x^2)/2 + x^3/2]], {x, 1, 300}]; a
Select[Table[x^3/2+(3x^2)/2+3x+3,{x,150}],PrimeQ] (* Harvey P. Dale, Apr 30 2018 *)

A127876 Integers of the form (x^3)/6 + (x^2)/2 + x + 1.

Original entry on oeis.org

1, 13, 61, 172, 373, 691, 1153, 1786, 2617, 3673, 4981, 6568, 8461, 10687, 13273, 16246, 19633, 23461, 27757, 32548, 37861, 43723, 50161, 57202, 64873, 73201, 82213, 91936, 102397, 113623, 125641, 138478, 152161, 166717, 182173, 198556, 215893, 234211

Offset: 1

Artur Jasinski, Feb 04 2007

Generating polynomial is Schur's polynomial of degree 3. 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).

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=3, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=4, a(n-2)=-coeff(charpoly(A,x),x^(n-3)). - Milan Janjic, Jan 27 2010

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

Cf. A127873, A127874, A127875.

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

[(9*n^3-18*n^2+15*n-4)/2: n in [1..30]]; // G. C. Greubel, Apr 29 2018

a = {}; Do[If[IntegerQ[1 + x + x^2/2 + x^3/6], AppendTo[a, 1 + x + x^2/2 + x^3/6]], {x, 1, 300}]; a
Select[Table[x^3/6 + x^2/2 + x + 1, {x, 0, 200}], IntegerQ] (* Harvey P. Dale, Jan 06 2011 *)

Vec(x*(1+2*x)*(1+7*x+x^2)/(1-x)^4 + O(x^50)) \\ Colin Barker, May 15 2016

From Colin Barker, May 15 2016: (Start)
a(n) = (9*n^3-18*n^2+15*n-4)/2.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: x*(1+2*x)*(1+7*x+x^2) / (1-x)^4.
(End)
E.g.f.: 2 + (9*x^3 + 9*x^2 + 6*x - 4)*exp(x)/2. - G. C. Greubel, Apr 29 2018

a(1) = 1 added by Harvey P. Dale, Jan 06 2011

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

Original entry on oeis.org

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

A127875 Numbers x for which (x^3)/2+(3x^2)/2+3x+3 is prime.

Original entry on oeis.org

2, 4, 7, 8, 10, 11, 20, 23, 26, 28, 31, 34, 44, 50, 56, 62, 71, 74, 76, 79, 82, 83, 88, 91, 103, 104, 110, 112, 118, 122, 131, 134, 139, 140, 142, 148, 152, 163, 170, 175, 176, 179, 199, 202, 206, 226, 227, 235, 238, 239, 242, 244, 266, 271, 274, 278, 296, 299

Offset: 1

Artur Jasinski, Feb 04 2007

nonn

Generating polynomial is Schur's polynomial of degree 3. 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).

Cf. A127873, A127874.

a = {}; Do[If[PrimeQ[3 + 3 x + (3 x^2)/2 + x^3/2], AppendTo[a, x]], {x, 1, 300}]; a

A127878 a(n) = n^4 + 4n^3 + 12n^2 + 24*n + 24.

Original entry on oeis.org

24, 65, 168, 393, 824, 1569, 2760, 4553, 7128, 10689, 15464, 21705, 29688, 39713, 52104, 67209, 85400, 107073, 132648, 162569, 197304, 237345, 283208, 335433, 394584, 461249, 536040, 619593, 712568, 815649, 929544, 1054985, 1192728

Offset: 0

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).

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A127873-A127883.

List([0..40],n->n^4+4*n^3+12*n^2+24*n+24); # Muniru A Asiru, Apr 30 2018

[n^4 +4*n^3 +12*n^2 +24*n +24: n in [0..30]]; // G. C. Greubel, Apr 29 2018

seq(n^4+4*n^3+12*n^2+24*n+24,n=0..40); # Muniru A Asiru, Apr 30 2018

Table[24 + 24*n + 12*n^2 + 4*n^3 + n^4, {n, 0, 50}]
LinearRecurrence[{5, -10, 10, -5, 1}, {24, 65, 168, 393, 824}, 50] (* G. C. Greubel, Apr 29 2018 *)

for(n=0,30, print1(n^4 +4*n^3 +12*n^2 +24*n +24, ", ")) \\ G. C. Greubel, Apr 29 2018

Integral representation in terms of incomplete Gamma function : a(n)= Exp[n]Gamma[5,n], where Gamma[5,n]= Integrate[x^4 Exp[ -x], {x, n, +infinity}]. - N-E. Fahssi, Jan 25 2008
G.f.: (24 -55*x +83*x^2 -37*x^3 +9*x^4)/(1-x)^5. - Colin Barker, Apr 02 2012
E.g.f.: (24 + 41*x + 31*x^2 + 10*x^3 + x^4)*exp(x). - G. C. Greubel, Apr 29 2018

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

Original entry on oeis.org

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

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

A127874 Prime numbers of the form (x^3)/2+(3x^2)/2+3x+3.

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A127876 Integers of the form (x^3)/6 + (x^2)/2 + x + 1.

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

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A127875 Numbers x for which (x^3)/2+(3x^2)/2+3x+3 is prime.

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A127878 a(n) = n^4 + 4*n^3 + 12*n^2 + 24*n + 24.

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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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A127878 a(n) = n^4 + 4n^3 + 12n^2 + 24*n + 24.

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