A056578 -id:A056578 - cp's OEIS frontend

A056579 1+2n+3n^2+4n^3+5n^4.

1, 15, 129, 547, 1593, 3711, 7465, 13539, 22737, 35983, 54321, 78915, 111049, 152127, 203673, 267331, 344865, 438159, 549217, 680163, 833241, 1010815, 1215369, 1449507, 1715953, 2017551, 2357265, 2738179, 3163497, 3636543

Offset: 0

Henry Bottomley, Jun 29 2000

			For n>5 this is 54321 translated from base n to base 10

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Note: 1+2x+3x^2+4x^3+5x^4 is derivative of 1+x+x^2+x^3+x^4 +x^5, i.e. A053700. Cf. A000012, A005408, A056109, A056578.

Join[{1},Table[Total[Table[i n^(i-1),{i,5}]],{n,30}]] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,15,129,547,1593},30] (* Harvey P. Dale, Sep 20 2017 *)

a(n)=1+2*n+3*n^2+4*n^3+5*n^4 \\ Charles R Greathouse IV, Oct 07 2015

a(n) =(A053700(n+1)-A053700(n-1))/2-10n^2-4n-2.

G.f.: -(3*x^4+42*x^3+64*x^2+10*x+1) / (x-1)^5. - Colin Barker, May 04 2013

A123059 Primes of the form 1 + 2k + 3k^2 + 4*k^3.

Original entry on oeis.org

313, 7369, 11593, 24337, 44089, 57073, 90217, 160753, 570649, 964969, 1060993, 1916617, 3349033, 4532113, 5360521, 6614137, 7308289, 9252409, 11035081, 12006433, 14680513, 15852457, 16461121, 22654417, 29318833, 34083913, 39339193, 41583937, 42737641, 51416353

Offset: 1

Zak Seidov, Sep 26 2006

nonn

Corresponding values of k are in A123076.

Jason Bard, Table of n, a(n) for n = 1..10000

Cf. A056578, A123076.

[ a: n in [0..300] | IsPrime(a) where a is 1+2*n+3*n^2+4*n^3 ]; // Vincenzo Librandi, Dec 17 2010

Select[Total/@Table[(Range[4]n^Range[0,3]),{n,250}],PrimeQ]  (* Harvey P. Dale, Jan 18 2011 *)

a(n) = A056578(A123076(n)). - Elmo R. Oliveira, Apr 20 2025

A059045 Square array T(n,k) read by antidiagonals where T(0,k) = 0 and T(n,k) = 1 + 2k + 3k^2 + ... + n*k^(n-1).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 17, 7, 1, 0, 1, 15, 49, 34, 9, 1, 0, 1, 21, 129, 142, 57, 11, 1, 0, 1, 28, 321, 547, 313, 86, 13, 1, 0, 1, 36, 769, 2005, 1593, 586, 121, 15, 1, 0, 1, 45, 1793, 7108, 7737, 3711, 985, 162, 17, 1, 0, 1, 55, 4097, 24604, 36409

Offset: 0

Henry Bottomley, Dec 18 2000

			   0,   0,   0,    0,     0,      0,      0,      0,       0, ...
   1,   1,   1,    1,     1,      1,      1,      1,       1, ...
   1,   3,   5,    7,     9,     11,     13,     15,      17, ...
   1,   6,  17,   34,    57,     86,    121,    162,     209, ...
   1,  10,  49,  142,   313,    586,    985,   1534,    2257, ...
   1,  15, 129,  547,  1593,   3711,   7465,  13539,   22737, ...
   1,  21, 321, 2005,  7737,  22461,  54121, 114381,  219345, ...
   1,  28, 769, 7108, 36409, 131836, 380713, 937924, 2054353, ...

Cf. A000004, A000012, A005408, A056109, A056578, A056579 (rows 1-6).

Cf. A057427, A000217, A000337, A014915, A014916, A014917, A014918, A014920 (columns 1-7).

A059045 := proc(n,k)
    if k = 1 then
        n*(n+1) /2 ;
    else
        (1+n*k^(n+1)-k^n*(n+1))/(k-1)^2 ;
    end if;
end proc: # R. J. Mathar, Mar 29 2013

T(n,k) = n*k^(n-1)+T(n-1, k) = (n*k^(n+1)-(n+1)*k^n+1)/(k-1)^2.

A113630 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8n^7 + 9n^8.

Original entry on oeis.org

1, 45, 4097, 83653, 757305, 4272461, 17736745, 59409477, 169826513, 429794605, 987654321, 2098573445, 4178995657, 7879732173, 14181546905, 24517448581, 40926266145, 66242446637, 104327377633, 160347899205, 241108033241

Offset: 0

Jonathan Vos Post, Jan 14 2006

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7*x^6 + 8*x^7 + 9*x^8 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 = (x^10 - 1)/(x-1).

			a(3) = 1 + 2*3 + 3*3^2 + 4*3^3 + 5*3^4 + 6*3^5 + 7*3^6 + 8*3^7 + 9*3^8 = 83653 is prime.
a(5) = 1 + 2*5 + 3*5^2 + 4*5^3 + 5*5^4 + 6*5^5 + 7*5^6 + 8*5^7 + 9*5^8 = 4272461 is prime.
a(8) = 1 + 2*8 + 3*8^2 + 4*8^3 + 5*8^4 + 6*8^5 + 7*8^6 + 8*8^7 + 9*8^8 = 169826513 is prime.
a(23) = 1 + 2*23 + 3*23^2 + 4*23^3 + 5*23^4 + 6*23^5 + 7*23^6 + 8*23^7 + 9*23^8 = 733113789893 is prime.

Chai Wah Wu, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000012, A005408, A056109, A056578, A056579.

Cf. A249951 (primes), A068475.

a113630 n = sum $ zipWith (*) [1..9] $ iterate (* n) 1
-- Reinhard Zumkeller, Nov 22 2014

[1+2*n+3*n^2+4*n^3+5*n^4+6*n^5+7*n^6+8*n^7+9*n^8: n in [0..20]]; // Vincenzo Librandi, Nov 09 2014

CoefficientList[Series[(5 x^8 + 1548 x^7 + 31360 x^6 + 129620 x^5 + 148266 x^4 + 48316 x^3 + 3728 x^2 + 36 x + 1) / (1 - x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 09 2014 *)
With[{c=Total[Table[k n^(k-1),{k,9}]]},Table[c,{n,0,30}]] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{1,45,4097,83653,757305,4272461,17736745,59409477,169826513},30] (* Harvey P. Dale, Jul 18 2017 *)

vector(100,n,1 + 2*(n-1)+ 3*(n-1)^2 + 4*(n-1)^3 + 5*(n-1)^4 + 6*(n-1)^5 + 7*(n-1)^6 + 8*(n-1)^7 + 9*(n-1)^8) \\ Derek Orr, Nov 09 2014

A113630_list, m = [1], [362880, -1229760, 1607760, -1011480, 309816, -40752, 1584, -4, 1]
for _ in range(10**3):
    for i in range(8):
        m[i+1]+= m[i]
    A113630_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8.

G.f.: -(5*x^8 +1548*x^7 +31360*x^6 +129620*x^5 +148266*x^4 +48316*x^3 +3728*x^2 +36*x +1) / (x -1)^9. - Colin Barker, May 08 2013

A123076 Numbers k such that p = 1 + 2k + 3k^2 + 4k^3 is prime.

Original entry on oeis.org

4, 12, 14, 18, 22, 24, 28, 34, 52, 62, 64, 78, 94, 104, 110, 118, 122, 132, 140, 144, 154, 158, 160, 178, 194, 204, 214, 218, 220, 234, 258, 262, 270, 272, 290, 294, 312, 314, 322, 344, 368, 370, 372, 382, 388, 424, 430, 440, 442, 454, 482, 494, 498, 518, 542

Offset: 1

Zak Seidov, Sep 27 2006

Corresponding p's are in A123059.

			For k=4, 1 + 2k + 3k^2 + 4k^3 = 313 which is prime.

James C. McMahon, Table of n, a(n) for n = 1..10000

Cf. A056578, A123059.

Select[Range[542],PrimeQ[1+2#+3#^2+4#^3]&] (* James C. McMahon, Nov 15 2024 *)

lista(m) = {for (n=1, m, if (isprime(1 + 2*n + 3*n^2 + 4*n^3), print1(n, ", ")););} \\ Michel Marcus, Apr 19 2013

A131464 a(n) = 4n^3 - 3n^2 + 2*n - 1.

Original entry on oeis.org

2, 23, 86, 215, 434, 767, 1238, 1871, 2690, 3719, 4982, 6503, 8306, 10415, 12854, 15647, 18818, 22391, 26390, 30839, 35762, 41183, 47126, 53615, 60674, 68327, 76598, 85511, 95090, 105359, 116342, 128063, 140546, 153815, 167894, 182807, 198578, 215231, 232790

Offset: 1

Mohammad K. Azarian, Jul 26 2007

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

Cf. A084849, A056109, A056105, A056578.

[4*n^3-3*n^2+2*n-1: n in [1..40]]; // Vincenzo Librandi, Feb 12 2013

I:=[2, 23, 86, 215]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 12 2013

CoefficientList[Series[(2 + 15 x + 6 x^2 + x^3)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2013 *)
Table[4n^3-3n^2+2n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{2,23,86,215},40] (* Harvey P. Dale, May 05 2018 *)

From Vincenzo Librandi, Feb 12 2013: (Start)
G.f.: x*(2 + 15*x + 6*x^2 + x^3)/(1 - x)^4.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). (End)
E.g.f.: 1 - exp(x)*(1 - 3*x - 9*x^2 - 4*x^3). - Stefano Spezia, Dec 06 2024

A113531 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6*n^5.

Original entry on oeis.org

1, 21, 321, 2005, 7737, 22461, 54121, 114381, 219345, 390277, 654321, 1045221, 1604041, 2379885, 3430617, 4823581, 6636321, 8957301, 11886625, 15536757, 20033241, 25515421, 32137161, 40067565, 49491697, 60611301, 73645521

Offset: 0

Jonathan Vos Post, Jan 12 2006

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 = (n^7 - 1)/(n-1). a(6) = 1 + 2*6 + 3*6^2 + 4*6^3 + 5*6^4 + 6*6^5 = 54121 is prime, the smallest prime in the sequence. The next is a(a(1)) = a(21) = 1 + 2*21 + 3*21^2 + 4*21^3 + 5*21^4 + 6*21^5 = 25515421. Then a(24) = 49491697.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A000012, A005408, A056109, A056578, A056579.

With[{eq=Total[# n^(#-1)&/@Range[6]]},Table[eq,{n,0,30}]] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,21,321,2005,7737,22461},30] (* Harvey P. Dale, Nov 02 2011 *)

for(n=0,50, print1(1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5, ", ")) \\ G. C. Greubel, Mar 15 2017

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5.
O.g.f.: 2064/(-1+x)^4+3/(-1+x)+2040/(-1+x)^5+132/(-1+x)^2+720/(-1+x)^6+872/(-1+x)^3 . - R. J. Mathar, Feb 26 2008
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6), a(0)=1, a(1)=21, a(2)=321, a(3)=2005, a(4)=7737, a(5)=22461. - Harvey P. Dale, Nov 02 2011

Corrected by R. J. Mathar, Feb 26 2008

A113532 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6.

Original entry on oeis.org

1, 28, 769, 7108, 36409, 131836, 380713, 937924, 2054353, 4110364, 7654321, 13446148, 22505929, 36167548, 56137369, 84557956, 124076833, 177920284, 249972193, 344857924, 468033241, 625878268, 825796489, 1076318788

Offset: 0

Jonathan Vos Post, Jan 12 2006

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7*n^6 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 = (x^8 - 1)/(x-1). a(2) = 1 + 2*2 + 3*2^2 + 4*2^3 + 5*2^4 + 6*2^5 + 7*2^6 = 769 is prime. Other primes begin a(6) = 380713, a(12) = 22505929, a(26) = 2236055953, a(38) = 21562615273, a(44) = 51802781449, a(52) = 140712620569.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000012, A005408, A056109, A056578, A056579.

Table[1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6, {n,0,50}] (* or *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 28, 769, 7108, 36409, 131836, 380713}, 50] (* G. C. Greubel, Mar 15 2017 *)

for(n=0,50, print1(1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6, ", ")) \\ G. C. Greubel, Mar 15 2017

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6.

O.g.f.: -12636/(-1+x)^4 -4/(-1+x) -21480/(-1+x)^5 -309/(-1+x)^2 -16920/(-1+x)^6 -3342/(-1+x)^3-5040/(-1+x)^7 . - R. J. Mathar, Feb 26 2008

A123077 Primes of the form (1+2n+3n^2+4n^3)/2.

Original entry on oeis.org

5, 71, 293, 7103, 32213, 40487, 50069, 87623, 161831, 211007, 238949, 337343, 852263, 922037, 1328447, 1421909, 1955399, 2607989, 3061703, 3744551, 4121087, 4318469, 4731941, 5400359, 5879231, 7198421, 9356927, 10400501, 10764863

Offset: 1

Zak Seidov, Sep 27 2006

Corresponding n's are 1, 3, 5, 15, 25, 27, 29, 35, 43, 47, 49, 55, 75, 77, 87, 89, 99, 109, 115, 123, 127, 129, 133, 139, 143, 153, 167, 173, 175, 179, 183, 185, 195, 199, 207, 209, 227, 229, 239, 245, 257, 259, 269, 273, 283, 285, 299, 309, 315, 325, 327, 337, 347, 349, 357, 363, 369, 377, 379, 393, 399, 403, 409, 417, 425, 439, 523, 539, 545, 559, 567, 575, 587, 589, 593, 607, 623, 659, 687, 697, 699.

There are no primes of the form (1+2n+3n^2+4n^3)/3.

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

Cf. A056578, A123059.

[a: n in [0..250] | IsPrime(a) where a is  (1 + 2*n + 3*n^2 + 4*n^3) div 2]; // Vincenzo Librandi, Mar 21 2013

Select[Table[(1 + 2 n + 3 n^2 + 4 n^3)/2, {n, 0, 200}], PrimeQ] (* Vincenzo Librandi, Mar 21 2013 *)

A123100 Primes of the form (1+2n+3n^2+4n^3)/5.

Original entry on oeis.org

2, 197, 354677, 713357, 959597, 1256837, 3676037, 5168717, 7018997, 11945957, 15099437, 18764117, 25303637, 36170597, 42610877, 46099517, 49773557, 71092757, 75979997, 91974917, 110070437, 123365117, 161190317, 306442277

Offset: 1

Zak Seidov, Sep 27 2006

All terms > 2 are congruent to 7 (mod 10).

Corresponding n's are: 1, 6, 76, 96, 106, 116, 166, 186, 206, 246, 266, 286, 316, 356, 376, 386, 396, 446, 456, 486, 516, 536, 586, 726, 736, 746, 766, 796, 846, 866, 906, 916, 1036, 1046, 1076, 1116, 1126, 1156, 1176, 1236, 1296, 1316, 1326, 1406, 1456, 1546, 1586, 1596, 1686, 1706, 1786, 1816, 1896, 1926, 1956, all are congruent to 1 (mod 5).

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

Cf. A056578, A123059.

Select[Table[(1 + 2 n + 3 n^2 + 4 n^3)/5, {n, 0, 200}], PrimeQ] (* Vincenzo Librandi, Mar 21 2013 *)

cp's OEIS Frontend

A056579 1+2n+3n^2+4n^3+5n^4.

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A123059 Primes of the form 1 + 2*k + 3*k^2 + 4*k^3.

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A059045 Square array T(n,k) read by antidiagonals where T(0,k) = 0 and T(n,k) = 1 + 2k + 3k^2 + ... + n*k^(n-1).

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A113630 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8.

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A123076 Numbers k such that p = 1 + 2k + 3k^2 + 4k^3 is prime.

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A131464 a(n) = 4*n^3 - 3*n^2 + 2*n - 1.

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A113531 a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5.

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A123059 Primes of the form 1 + 2k + 3k^2 + 4*k^3.

A113630 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8n^7 + 9n^8.

A131464 a(n) = 4n^3 - 3n^2 + 2*n - 1.

A113531 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6*n^5.

A113532 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6.