A103438 -id:A103438 - cp's OEIS frontend

A001553 a(n) = 1^n + 2^n + ... + 6^n.

6, 21, 91, 441, 2275, 12201, 67171, 376761, 2142595, 12313161, 71340451, 415998681, 2438235715, 14350108521, 84740914531, 501790686201, 2978035877635, 17706908038281, 105443761093411, 628709267031321, 3752628871164355, 22418196307542441, 134023513204581091

Offset: 0

N. J. A. Sloane

For the o.g.f.s of such sequences see the W. Lang link under A196837. The e.g.f.s are trivial. - Wolfdieter Lang, Oct 14 2011

a(n) is divisible by 7 iff n is not divisible by 6 (see De Koninck & Mercier reference). Example: a(5)= 12201 = 7 * 1743 and a(6) = 67171 = 9595 * 7 + 6. - Bernard Schott, Mar 06 2020

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 289 pp. 45, 194, Ellipses, Paris, (2004).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 366
Index entries for linear recurrences with constant coefficients, signature (21, -175, 735, -1624, 1764, -720).

Column 6 of array A103438, A001552.

Table[Total[Range[6]^n], {n, 0, 40}] (* T. D. Noe, Oct 10 2011 *)

a(n) = Sum_{k=1..6} k^n.
From Wolfdieter Lang, Oct 10 2011: (Start)
E.g.f.: (1-exp(6*x))/(exp(-x)-1) = Sum_{j=1..6} exp(j*x) (trivial).
O.g.f.: (2 - 7*x)*(3 - 42*x + 203*x^2 - 392*x^3 + 252*x^4)/Product_{j=1..6} (1 - j*x).
From the Laplace transformation of the e.g.f. (with argument 1/p, and multiplied with 1/p), which yields the partial fraction decomposition of the given o.g.f., namely Sum_{j=1..6} 1/(1 - j*x).
(End)

More terms from Jon E. Schoenfield, Mar 24 2010

A162298 Faulhaber's triangle: triangle T(k,y) read by rows, giving numerator of the coefficient [m^y] of the polynomial Sum_{x=1..m} x^(k-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, 0, 1, 1, 1, 0, -1, 0, 5, 1, 1, 1, 0, -1, 0, 1, 1, 1, 0, 1, 0, -7, 0, 7, 1, 1, -1, 0, 2, 0, -7, 0, 2, 1, 1, 0, -3, 0, 1, 0, -7, 0, 3, 1, 1, 5, 0, -1, 0, 1, 0, -1, 0, 5, 1, 1, 0, 5, 0, -11, 0, 11, 0, -11, 0, 11, 1, 1, -691, 0, 5, 0, -33, 0, 22, 0, -11, 0, 1, 1, 1, 0, -691, 0, 65, 0, -143, 0, 143, 0, -143, 0, 13, 1, 1

Offset: 0

Juri-Stepan Gerasimov, Jun 30 2009 and Jul 02 2009

There are many versions of Faulhaber's triangle: search the OEIS for his name. For example, A220962/A220963 is essentially the same as this triangle, except for an initial column of 0's. - N. J. A. Sloane, Jan 28 2017
Named after the German mathematician Johann Faulhaber (1580-1653). - Amiram Eldar, Jun 13 2021
From Wolfdieter Lang, Oct 23 2011 (Start):
The sums of the k-th power of each of the first n positive integers, sum(j^k,j=1..n), k>=0, n>=1, abbreviated usually as Sigma n^k, can be written as Sigma n^k = sum(r(k,m)*n^m,m=1..k+1), with the rational number triangle r(n,m)=a(n,m)/A162299(k+1,m). See, e.g., the Graham et al. reference, eq. (6.78), p. 269, where Sigma n^k is S_k(n+1) - delta(k,0), with delta(k,0)=1 if k=0 and 0 else. The formula for r(n,m) given below can be adapted from this reference, and it is found in the given form (for k>0) in the Remmert reference, p. 175.
For sums of powers of integers see the array A103438 with further references and links.
(End)

			The first few polynomials:
    m;
   m/2  + m^2/2;
   m/6  + m^2/2 + m^3/3;
    0   + m^2/4 + m^3/2 + m^4/4;
  -m/30 +   0   + m^3/3 + m^4/2 + m^5/5;
  ...
Initial rows of Faulhaber's triangle of fractions H(n, k) (n >= 0, 1 <= k <= n+1):
    1;
   1/2,  1/2;
   1/6,  1/2,  1/3;
    0,   1/4,  1/2,  1/4;
  -1/30,  0,   1/3,  1/2,  1/5;
    0,  -1/12,  0,   5/12, 1/2,  1/6;
   1/42,  0,  -1/6,   0,   1/2,  1/2,  1/7;
    0,   1/12,  0,  -7/24,  0,   7/12, 1/2,  1/8;
  -1/30,  0,   2/9,   0,  -7/15,  0,   2/3,  1/2,  1/9;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991 (Seventh printing).Second ed. 1994.
R. Remmert, Funktionentheorie I, Zweite Auflage, Springer-Verlag, 1989. English version: Classical topics in complex function theory, Springer, 1998.
Mohammad Torabi-Dashti, Faulhaber's Triangle, College Math. J., Vol. 42, No. 2 (2011), pp. 96-97.
Mohammad Torabi-Dashti, Faulhaber’s Triangle. [Annotated scanned copy of preprint]
Eric Weisstein's MathWorld, Power Sum.

Cf. A000367, A162299 (denominators), A103438, A196837.

A001554 a(n) = 1^n + 2^n + ... + 7^n.

Original entry on oeis.org

7, 28, 140, 784, 4676, 29008, 184820, 1200304, 7907396, 52666768, 353815700, 2393325424, 16279522916, 111239118928, 762963987380, 5249352196144, 36210966447236, 250337422025488, 1733857359003860, 12027604452404464, 83544895168776356, 580964060390826448

Offset: 0

N. J. A. Sloane

Conjectures for o.g.f.s for this type of sequences appear in the PhD thesis by Simon Plouffe. See A001552 for the reference. These conjectures are proved in a link given in A196837. - Wolfdieter Lang, Oct 15 2011

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 367
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (28, -322, 1960, -6769, 13132, -13068, 5040).

Column 7 of array A103438. A196837.

[1+2^n+3^n+4^n+5^n+6^n+7^n : n in [0..30]]; // Wesley Ivan Hurt, Jul 15 2014

A001554:=n->add(i^n, i=1..7): seq(A001554(n), n=0..30); # Wesley Ivan Hurt, Jul 15 2014

Mathematica
```
Table[Total[Range[7]^n], {n, 0, 20}]
```

From Wolfdieter Lang, Oct 15 2011: (Start)
E.g.f.: (1-exp(7*x))/(exp(-x)-1) = Sum_{j=1..7} exp(j*x) (trivial).
O.g.f.: (7 - 168*x + 1610*x^2 - 7840*x^3 + 20307*x^4 - 26264*x^5 + 13068*x^6)/Product_{j=1..7} (1 - j*x). From the e.g.f. via Laplace transformation. See the proof in a link under A196837. (End)

More terms from Jon E. Schoenfield, Mar 24 2010

A057291 Numbers k such that k | 12^k + 11^k + 10^k + 9^k + 8^k + 7^k + 6^k + 5^k + 4^k + 3^k + 2^k + 1^k.

Original entry on oeis.org

1, 2, 3, 9, 10, 13, 26, 27, 39, 50, 81, 110, 117, 130, 169, 243, 250, 279, 310, 338, 351, 470, 507, 550, 650, 663, 729, 1053, 1209, 1250, 1430, 1521, 1550, 1690, 2187, 2197, 2750, 3159, 3250, 3410, 4030, 4043, 4069, 4394, 4509, 4563, 6250, 6561, 6591, 7150

Offset: 1

Robert G. Wilson v, Sep 22 2000

nonn

The only primes in the sequence are 2, 3 and 13. - Robert Israel, Jun 25 2025

Robert Israel, Table of n, a(n) for n = 1..720

Cf. A056739, A057292, A103438.

filter:= n -> 12&^n + 11&^n + 10&^n + 9&^n + 8&^n + 7&^n + 6&^n + 5&^n + 4&^n + 3&^n + 2&^n + 1 mod n = 0:
select(filter, [$1..10^4]); # Robert Israel, Jun 25 2025

Select[ Range[ 10^5 ], Mod[ PowerMod[ 12, #, # ] + PowerMod[ 11, #, # ] + PowerMod[ 10, #, # ] + PowerMod[ 9, #, # ] + PowerMod[ 8, #, # ] + PowerMod[ 7, #, # ] + PowerMod[ 6, #, # ] + PowerMod[ 5, #, # ] + PowerMod[ 4, #, # ] + PowerMod[ 3, #, # ] + PowerMod[ 2, #, # ] + 1, # ] == 0 & ]
Select[Range[7200],Divisible[Total[Range[12]^#],#]&] (* Harvey P. Dale, Aug 05 2017 *)

A057292 Numbers k such that k | 11^k + 10^k + 9^k + 8^k + 7^k + 6^k + 5^k + 4^k + 3^k + 2^k + 1^k.

Original entry on oeis.org

1, 2, 3, 9, 11, 22, 27, 33, 46, 81, 99, 121, 209, 242, 243, 297, 363, 729, 891, 1058, 1089, 1179, 1331, 1702, 2187, 2662, 2673, 3082, 3267, 3993, 6561, 8019, 9801, 11979, 14641, 19683, 20673, 24057, 24334, 25029, 29282, 29403, 30591, 30734

Offset: 1

Robert G. Wilson v, Sep 22 2000

nonn

The only primes in the sequence are 2, 3 and 11. - Robert Israel, Jun 25 2025

Robert Israel, Table of n, a(n) for n = 1..227

Cf. A056739, A057291, A103438.

filter:= n ->  11&^n + 10&^n + 9&^n + 8&^n + 7&^n + 6&^n + 5&^n + 4&^n + 3&^n + 2&^n + 1 mod n = 0:
select(filter, [$1..10^5]); # Robert Israel, Jun 25 2025

Select[ Range[ 10^5 ], Mod[ PowerMod[ 11, #, # ] + PowerMod[ 10, #, # ] + PowerMod[ 9, #, # ] + PowerMod[ 8, #, # ] + PowerMod[ 7, #, # ] + PowerMod[ 6, #, # ] + PowerMod[ 5, #, # ] + PowerMod[ 4, #, # ] + PowerMod[ 3, #, # ] + PowerMod[ 2, #, # ] + 1, # ] == 0 & ]
Select[Range[31000],Mod[Total[PowerMod[Range[0,11],#,#]],#]==0&] (* Harvey P. Dale, Nov 22 2021 *)

A103439 a(n) = Sum_{i=0..n-1} Sum_{j=0..i} (i-j+1)^j.

Original entry on oeis.org

0, 1, 3, 7, 16, 39, 105, 315, 1048, 3829, 15207, 65071, 297840, 1449755, 7468541, 40555747, 231335960, 1381989881, 8623700811, 56078446615, 379233142800, 2662013133295, 19362917622001, 145719550012299, 1133023004941272, 9090156910550109, 75161929739797519

Offset: 0

Ralf Stephan, Feb 11 2005

nonn

Partial sums of A026898.
Antidiagonal sums of array A103438.
Row sums of A123490. - Paul Barry, Oct 01 2006

Alois P. Heinz, Table of n, a(n) for n = 0..600
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.

Cf. A026898, A103438, A123490.

[0] cat [(&+[ (&+[ (k-j+1)^j : j in [0..k]]) : k in [0..n-1]]): n in [1..30]]; // G. C. Greubel, Jun 15 2021

b:= proc(i) option remember; add((i-j+1)^j, j=0..i) end:
a:= proc(n) option remember; add(b(i), i=0..n-1) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 02 2019

Join[{0},Table[Sum[Sum[(i-j+1)^j,{j,0,i}],{i,0,n}],{n,0,30}]] (* Harvey P. Dale, Dec 03 2018 *)

a(n) = sum(i=0, n-1, sum(j=0, i, (i-j+1)^j)); \\ Michel Marcus, Jun 15 2021

[sum(sum((k-j+1)^j for j in (0..k)) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, Jun 15 2021

a(n+1) = Sum_{k=0..n} ((k+2)^(n-k) + k)/(k+1). - Paul Barry, Oct 01 2006

G.f.: (G(0)-1)/(1-x) where G(k) = 1 + x*(2*k*x-1)/(2*k*x+x-1 - x*(2*k*x+x-1)^2/(x*(2*k*x+x-1) + (2*k*x+2*x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 26 2013

Name edited by Alois P. Heinz, Dec 02 2019

A308477 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n, gcd(n,j) = 1} j^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 4, 4, 1, 1, 9, 10, 10, 2, 1, 1, 17, 28, 30, 6, 6, 1, 1, 33, 82, 100, 26, 21, 4, 1, 1, 65, 244, 354, 126, 91, 16, 6, 1, 1, 129, 730, 1300, 626, 441, 84, 27, 4, 1, 1, 257, 2188, 4890, 3126, 2275, 496, 159, 20, 10, 1, 1, 513, 6562, 18700, 15626, 12201, 3108, 1053, 140, 55, 4

Offset: 1

Ilya Gutkovskiy, May 29 2019

			Square array begins:
  1,   1,   1,    1,    1,     1,  ...
  1,   1,   1,    1,    1,     1,  ...
  2,   3,   5,    9,   17,    33,  ...
  2,   4,  10,   28,   82,   244,  ...
  4,  10,  30,  100,  354,  1300,  ...
  2,   6,  26,  126,  626,  3126,  ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..4 give A000010, A023896, A053818, A053819, A053820.

Cf. A103438.

Table[Function[k, Sum[If[GCD[n, j] == 1, j^k, 0], {j, 1, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

A001555 a(n) = 1^n + 2^n + ... + 8^n.

Original entry on oeis.org

8, 36, 204, 1296, 8772, 61776, 446964, 3297456, 24684612, 186884496, 1427557524, 10983260016, 84998999652, 660994932816, 5161010498484, 40433724284976, 317685943157892, 2502137235710736, 19748255868485844, 156142792528260336, 1236466399775623332

Offset: 0

N. J. A. Sloane

Conjectures for o.g.f.s for this type of sequence appear in the PhD thesis by Simon Plouffe. See A001552 for the reference. These conjectures are proved in a link given in A196837. [Wolfdieter Lang, Oct 15 2011]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Robert Israel, Table of n, a(n) for n = 0..1000 (n = 0..200 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 368
Index entries for linear recurrences with constant coefficients, signature (36, -546, 4536, -22449, 67284, -118124, 109584, -40320).

Column 8 of array A103438.

Cf. A001018, A001552, A001554, A196837.

seq(add(j^n,j=1..8), n=0..20); # Robert Israel, Aug 23 2015

Table[Total[Range[8]^n], {n, 0, 20}] (* T. D. Noe, Aug 09 2012 *)

first(m)=vector(m,n,n--;sum(i=1,8,i^n)) \\ Anders Hellström, Aug 23 2015

From Wolfdieter Lang, Oct 15 2011 (Start)
E.g.f.: (1-exp(8*x))/(exp(-x)-1) = Sum_{j=1..8} exp(j*x) (trivial).
O.g.f.: 4*(2-9*x)*(1-27*x+288*x^2-1539*x^3+4299*x^4-5886*x^5+3044*x^6) / Product_{j=1..8} (1-j*x). From the e.g.f. via Laplace transformation. See the proof in a link under A196837. (End)
a(n) = A001554(n) + A001018(n). - Michel Marcus, Jul 26 2013

More terms from Jon E. Schoenfield, Mar 24 2010

A001556 a(n) = 1^n + 2^n + ... + 9^n.

Original entry on oeis.org

9, 45, 285, 2025, 15333, 120825, 978405, 8080425, 67731333, 574304985, 4914341925, 42364319625, 367428536133, 3202860761145, 28037802953445, 246324856379625, 2170706132009733, 19179318935377305, 169842891165484965, 1506994510201252425

Offset: 0

N. J. A. Sloane

nonn

Conjectures for o.g.f.s for this type of sequences appear in the PhD thesis by Simon Plouffe. See A001552 for the reference. These conjectures are proved in the link given in A196837. - Wolfdieter Lang, Oct 15 2011

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 369
Index entries for linear recurrences with constant coefficients, signature (45, -870, 9450, -63273, 269325, -723680, 1172700, -1026576, 362880).

Column 9 of array A103438. A196837.

Table[Total[Range[9]^n], {n, 0, 20}] (* T. D. Noe, Aug 09 2012 *)

a(n) = sum_{j=1..9} j^n, n>=0.
From Wolfdieter Lang, Oct 15 2011: (Start)
E.g.f.: (1-exp(9*x))/(exp(-x)-1) = sum(exp(j*x),j=1..9) (trivial).
O.g.f.: (9 - 360*x + 6090*x^2 - 56700*x^3 + 316365*x^4 - 1077300*x^5 + 2171040*x^6 - 2345400*x^7 + 1026576*x^8)/product_{j=1..9} (1-j*x).
From the e.g.f. via Laplace transformation. See the proof in a link under A196837.
(End)
a(n) = A001555(n) + A001019(n). - Michel Marcus, Jul 26 2013

More terms from Jon E. Schoenfield, Mar 24 2010

A001557 a(n) = 1^n + 2^n + ... + 10^n.

Original entry on oeis.org

10, 55, 385, 3025, 25333, 220825, 1978405, 18080425, 167731333, 1574304985, 14914341925, 142364319625, 1367428536133, 13202860761145, 128037802953445, 1246324856379625, 12170706132009733, 119179318935377305, 1169842891165484965, 11506994510201252425

Offset: 0

N. J. A. Sloane

Conjectures for o.g.f.s for this type of sequences appear in the PhD thesis by Simon Plouffe. See A001552 for the reference. These conjectures are proved in the link given in A196837. - Wolfdieter Lang, Oct 15 2011

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 370.
Index entries for linear recurrences with constant coefficients, signature (55, -1320, 18150, -157773, 902055, -3416930, 8409500, -12753576, 10628640, -3628800).

Column 10 of array A103438. Cf. A196837.

Table[Total[Range[10]^n], {n, 0, 20}] (* T. D. Noe, Aug 09 2012 *)

def A001557(n): return sum(i**n for i in range(1,11)) # Chai Wah Wu, Oct 24 2024

a(n) = Sum_{j=1..10} j^n, n >= 0.
E.g.f.: exp(x) + exp(2*x) + exp(3*x) + exp(4*x) + exp(5*x) + exp(6*x) + exp(7*x) + exp(8*x) + exp(9*x) + exp(10*x). - Vladeta Jovovic, May 08 2002
From Wolfdieter Lang, Oct 15 2011: (Start)
O.g.f.: (2 - 11*x) *(5 - 220*x + 4070*x^2 - 41140*x^3 + 247049*x^4 - 896368*x^5 + 1903836*x^6 - 2143152*x^7 + 966240*x^8)/Product_{j=1..10} (1 - j*x).
From the e.g.f. via Laplace transformation. See the proof in a link under A196837.
(End)
a(n) = A001556(n) + A011557(n). - Michel Marcus, Jul 26 2013

More terms from Jon E. Schoenfield, Mar 24 2010

cp's OEIS Frontend

A001553 a(n) = 1^n + 2^n + ... + 6^n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A162298 Faulhaber's triangle: triangle T(k,y) read by rows, giving numerator of the coefficient [m^y] of the polynomial Sum_{x=1..m} x^(k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A001554 a(n) = 1^n + 2^n + ... + 7^n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A057291 Numbers k such that k | 12^k + 11^k + 10^k + 9^k + 8^k + 7^k + 6^k + 5^k + 4^k + 3^k + 2^k + 1^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A057292 Numbers k such that k | 11^k + 10^k + 9^k + 8^k + 7^k + 6^k + 5^k + 4^k + 3^k + 2^k + 1^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A103439 a(n) = Sum_{i=0..n-1} Sum_{j=0..i} (i-j+1)^j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A308477 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n, gcd(n,j) = 1} j^k.

Views

Author

Keywords

Examples