A008455 -id:A008455 - cp's OEIS frontend

A123095 Sum of first n 11th powers.

0, 1, 2049, 179196, 4373500, 53201625, 415998681, 2393325424, 10983260016, 42364319625, 142364319625, 427675990236, 1170684360924, 2962844754961, 7012409924625, 15662165784000, 33254351828416, 67526248136049, 131794658215281, 248284917113500, 453084917113500

Offset: 0

Zerinvary Lajos, Sep 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Sequences of the form Sum_{j=0..n} j^m : A000217 (m=1), A000330 (m=2), A000537 (m=3), A000538 (m=4), A000539 (m=5), A000540 (m=6), A000541 (m=7), A000542 (m=8), A007487 (m=9), A023002 (m=10), this sequence (m=11), A123094 (m=12), A181134 (m=13).
Cf. A008455, A215083.
Cf. A008275, A008277.

[(&+[j^11: j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jul 21 2021

[seq(add(i^11, i=1..n), n=0..20)];
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^11 od: seq(a[n], n=0..13); # Zerinvary Lajos, Feb 22 2008

Table[Sum[k^11, {k, n}], {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Range[0,20]^11] (* Harvey P. Dale, Sep 17 2021 *)

A123095_list, m = [0], [39916800, -199584000, 419126400, -479001600, 322494480, -129230640, 29607600, -3498000, 171006, -2046, 1, 0 , 0]
for _ in range(10**2):
    for i in range(12):
        m[i+1]+= m[i]
    A123095_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

[(bernoulli_polynomial(n+1, 12) - bernoulli(12))/12  for n in (0..30)] # G. C. Greubel, Jul 21 2021

a(n) = n*A023002(n) - Sum_{i=0..n-1} A023002(i). - Bruno Berselli, Apr 27 2010
a(n) = n^2*(n+1)^2*(2*n^8 +8*n^7 +4*n^6 -16*n^5 -5*n^4 +26*n^3 -3*n^2 -20*n +10)/24. - Bruno Berselli, Oct 03 2010
G.f.: x*(x^10 +2036*x^9 +152637*x^8 +2203488*x^7 +9738114*x^6 +15724248*x^5 +9738114*x^4 +2203488*x^3 +152637*x^2 +2036*x +1)/(1-x)^13. - Colin Barker, May 27 2012
a(n) = (-1)*Sum_{j=1..11} j*Stirling1(n+1,n+1-j)*Stirling2(n+11-j,n). - Mircea Merca, Jan 25 2014
a(n) = 1728*A006542(n+2)^2 + 216*A288876(n-2) + 96*A006542(n+2) + A000537(n). - Yasser Arath Chavez Reyes, May 25 2024

A220653 a(n) = n^11 + 11*n + 11^n.

Original entry on oeis.org

1, 23, 2191, 178511, 4208989, 48989231, 364568683, 1996813991, 8804293561, 33739007399, 125937424711, 570623341343, 3881436747541, 36314872538111, 383799398753059, 4185897925275191, 45967322049616753, 505481300395601591, 5559981581902310911, 61159206938673444719

Offset: 0

Jonathan Vos Post, Dec 17 2012

This is to A220425 as 11 is to 2, to A220509 as 11 is to 3, to A220511 as 11 is to 5, and to A220528 as 11 is to 7.

The subsequence of primes begins: 23, 4185897925275191, see A220787 for the associated n.

			a(1) = 1^11 + 11*1 + 11^1 = 23.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A008455, A008593, A001020, A220425, A220509, A220511, A220528, A220787.

Table[n^11 + 11*n + 11^n, {n, 0, 30}] (* T. D. Noe, Dec 17 2012 *)

A220653(n):=n^11+11*n+11^n$ makelist(A220653(n),n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n) = A008455(n) + A008593(n) + A001020(n).

G.f.: (131*x^12 +21186*x^11 +1682460*x^10 +24070936*x^9 +104942001*x^8 +163196604*x^7 +91422264*x^6 +14484216*x^5 -518211*x^4 -131726*x^3 -1860*x^2 -1) / ((x -1)^12*(11*x -1)). - Colin Barker, May 09 2013

a(19) from Stefano Spezia, May 03 2025

A366391 Numbers k such that A163511(k) is an eleventh power.

Original entry on oeis.org

0, 1024, 2049, 4099, 8199, 16399, 32799, 65599, 131199, 262399, 524799, 1049599, 2097152, 2099199, 4194305, 4196352, 4198399, 8388611, 8392705, 8394752, 8396799, 16777223, 16785411, 16789505, 16791552, 16793599, 33554447, 33570823, 33579011, 33583105, 33585152, 33587199, 67108895, 67141647, 67158023, 67166211, 67170305

Offset: 1

Antti Karttunen, Oct 09 2023

nonn

Equivalently, numbers k for which A332214(k), and also A332817(k) are eleventh powers.
The sequence is defined inductively as:
 (a) it contains 0 and 1024,
   and
 (b) for any nonzero term a(n), (2*a(n)) + 1 and 2048*a(n) are also included as terms.
When iterating n -> 2n+1 mod 2047, starting from 1024 we get 1024, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535,  and then cycle starts again from 1024 (see A153893), while on the other hand, x^11 mod 2047 obtains values: 0, 1, 230, 322, 344, 368, 390, 482, 622, 712, 713, 942, 967, 1013, 1034, 1080, 1105, 1334, 1335, 1425, 1565, 1657, 1679, 1703, 1725, 1817, 2046. These sets have no terms in common, therefore there are no eleventh powers in this sequence after the initial 0.

Positions of multiples of 11 in A365805.
Sequence A243071(n^11), n >= 1, sorted into ascending order.
Cf. A008455, A153893, A163511, A332214, A332817.
Cf. also A365801, A365802, A365808, A366287.

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
isA366391v(n) = ispower(A163511(n),11);

isA366391(n) = if(n<=1024, !(n%1024), if(n%2, isA366391((n-1)/2), if(n%2048, 0, isA366391(n>>11))));

A016799 a(n) = (3*n + 2)^11.

Original entry on oeis.org

2048, 48828125, 8589934592, 285311670611, 4049565169664, 34271896307633, 204800000000000, 952809757913927, 3670344486987776, 12200509765705829, 36028797018963968, 96549157373046875, 238572050223552512, 550329031716248441, 1196683881290399744, 2472159215084012303

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A016789, A016790, A016791, A016792, A016793, A016794, A016795, A016796, A016797, A016798.

Subsequence of A008455.

(3Range[0,20]+2)^11 (* Harvey P. Dale, Apr 25 2011 *)

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016789(n)^11.
Sum_{n>=0} 1/a(n) = 88573*zeta(11)/177147 - 7388*Pi^11/(2511058725*sqrt(3)). (End)

A024009 a(n) = 1 - n^11.

Original entry on oeis.org

1, 0, -2047, -177146, -4194303, -48828124, -362797055, -1977326742, -8589934591, -31381059608, -99999999999, -285311670610, -743008370687, -1792160394036, -4049565169663, -8649755859374, -17592186044415

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..340

Cf. A008455 (n^11).

[1-n^11: n in [0..35]]; // Vincenzo Librandi, Apr 29 2011

A004917 Numbers that are the sum of at most 11 positive 11th powers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2048, 2049, 2050, 2051, 2052, 2053, 2054, 2055, 2056, 2057, 2058, 4096, 4097, 4098, 4099, 4100, 4101, 4102, 4103, 4104, 4105, 6144, 6145, 6146, 6147, 6148, 6149, 6150, 6151, 6152, 8192, 8193, 8194, 8195, 8196, 8197, 8198

Offset: 1

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A008455 (11th powers).

Column k=11 of A336820.

b:= proc(n, i, t) option remember; n=0 or i>0 and t>0
      and (b(n, i-1, t) or i^11<=n and b(n-i^11, i, t-1))
    end:
a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, -1, a(n-1))
      while not b(k, iroot(k, 11), 11) do od; k
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 16 2016

b[n_, i_, t_] := b[n, i, t] = n == 0 || i > 0 && t > 0 && (b[n, i - 1, t] || i^11 <= n && b[n - i^11, i, t - 1]);
a[n_] := a[n] = Module[{k}, For[k = 1 + If[n == 1, -1, a[n - 1]], !b[k, k^(1/11) // Floor, 11], k++]; k];
Array[a, 60] (* Jean-François Alcover, Dec 03 2020, after Alois P. Heinz *)

More terms from Alois P. Heinz, Sep 16 2016

A016943 a(n) = (6*n + 2)^11.

Original entry on oeis.org

2048, 8589934592, 4049565169664, 204800000000000, 3670344486987776, 36028797018963968, 238572050223552512, 1196683881290399744, 4882812500000000000, 16985107389382393856, 52036560683837093888, 143746751770690322432, 364375289404334925824, 858993459200000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A016787, A016933, A016934, A016935, A016936, A016937, A016938, A016939, A016940, A016941, A016942.

Subsequence of A008455.

[(6*n+2)^11: n in [0..25]]; // Vincenzo Librandi, May 05 2011

(6*Range[0,10]+2)^11 (* Harvey P. Dale, Sep 30 2019 *)

a(n) = A016787(n)*2^11. - Zerinvary Lajos, Jun 22 2009
From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016933(n)^9 = A016935(n)^3.
Sum_{n>=0} 1/a(n) = 1847*Pi^11/(1285662067200*sqrt(3)) + 88573*zeta(11)/362797056. (End)

A016955 a(n) = (6*n + 3)^11.

Original entry on oeis.org

177147, 31381059609, 8649755859375, 350277500542221, 5559060566555523, 50542106513726817, 317475837322472439, 1532278301220703125, 6071163615208263051, 20635899893042801193, 62050608388552823487, 168787390185178426269, 422351360321044921875, 984770902183611232881

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A016763, A016945, A016946, A016947, A016948, A016949, A016950, A016951, A016952, A016953, A016954.

Subsequence of A008455.

[(6*n+3)^11: n in [0..20]]; // Vincenzo Librandi, May 06 2011

a[n_] := (6*n + 3)^11; Array[a, 50, 0] (* Amiram Eldar, Mar 30 2022 *)

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^11.
a(n) = 3^11*A016763(n).
Sum_{n>=0} 1/a(n) = 2047*zeta(11)/362797056.
Sum_{n>=0} (-1)^n/a(n) = 50521*Pi^11/2633035913625600. (End)

A016967 a(n) = (6*n + 4)^11.

Original entry on oeis.org

4194304, 100000000000, 17592186044416, 584318301411328, 8293509467471872, 70188843638032384, 419430400000000000, 1951354384207722496, 7516865509350965248, 24986644000165537792, 73786976294838206464, 197732674300000000000, 488595558857835544576, 1127073856954876807168

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A016799, A016957, A016958, A016959, A016960, A016961, A016962, A016963, A016964, A016965, A016966.

Subsequence of A008455.

[(6*n+4)^11: n in [0..20]]; // Vincenzo Librandi, May 07 2011

(6*Range[0,20]+4)^11 (* Harvey P. Dale, Sep 06 2012 *)

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016957(n)^11.
a(n) = 2^11*A016799(n).
Sum_{n>=0} 1/a(n) = 88573*zeta(11)/362797056 - 1847*Pi^11/(1285662067200*sqrt(3)). (End)

A016979 a(n) = (6*n + 5)^11.

Original entry on oeis.org

48828125, 285311670611, 34271896307633, 952809757913927, 12200509765705829, 96549157373046875, 550329031716248441, 2472159215084012303, 9269035929372191597, 30155888444737842659, 87507831740087890625, 231122292121701565271, 564154396389137449973

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A016969 (6*n + 5), A016970, A016971, A016972, A016973, A016974, A016975, A016976, A016977, A016978.

Subsequence of A008455 (n^11).

[(6*n+5)^11 : n in [0..20]]; // Vincenzo Librandi, May 17 2011

(6Range[0,20]+5)^11 (* or *) LinearRecurrence[{12,-66,220,-495,792,-924,792,-495,220,-66,12,-1},{48828125,285311670611,34271896307633,952809757913927,12200509765705829,96549157373046875,550329031716248441,2472159215084012303,9269035929372191597,30155888444737842659,87507831740087890625,231122292121701565271},20] (* Harvey P. Dale, Dec 17 2024 *)

a(n) = (6*n + 5)^11.
From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^11.
Sum_{n>=0} 1/a(n) = 181308931*zeta(11)/362797056 - 1261501*Pi^11/(428554022400*sqrt(3)). (End)

cp's OEIS Frontend

A123095 Sum of first n 11th powers.

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A220653 a(n) = n^11 + 11*n + 11^n.

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A366391 Numbers k such that A163511(k) is an eleventh power.

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A016799 a(n) = (3*n + 2)^11.

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A024009 a(n) = 1 - n^11.

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A004917 Numbers that are the sum of at most 11 positive 11th powers.

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A016943 a(n) = (6*n + 2)^11.

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A016955 a(n) = (6*n + 3)^11.

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