A034513 -id:A034513 - cp's OEIS frontend

A001576 a(n) = 1^n + 2^n + 4^n.

3, 7, 21, 73, 273, 1057, 4161, 16513, 65793, 262657, 1049601, 4196353, 16781313, 67117057, 268451841, 1073774593, 4295032833, 17180000257, 68719738881, 274878431233, 1099512676353, 4398048608257, 17592190238721, 70368752566273, 281474993487873, 1125899940397057

Offset: 0

N. J. A. Sloane

Equals A135576, except for the first term. - Omar E. Pol, Nov 18 2008
Conjecture: For n > 1, if a(n) = 1^n + 2^n + 4^n is a prime number then n is of the form 3^h. For example, for h=1, n=3, a(n) = 1^3 + 2^3 + 4^3 = 73 (prime); for h=2, n=9, a(n) = 1^9 + 2^9 + 4^9 = 262657 (prime); for h=3, n=27, a(n) is not prime. - Vincenzo Librandi, Aug 03 2010
The previous conjecture was proved by Golomb in 1978. See A051154. - T. D. Noe, Aug 15 2010
Another more elementary proof can be found in Liu link. - Bernard Schott, Mar 08 2019
Fills in one quarter section of the figurate form of the Sierpinski square curve. See illustration in links and A141725. - John Elias, Mar 29 2023

T. D. Noe, Table of n, a(n) for n = 0..200
John Elias, Illustration of Initial Terms: 1/4 Sierpinski Square Curve
Andy Liu, West German Mathematical Olympiad 1982 - Second round, Problem 4, Crux Mathematicorum, p. 105, Vol. 12, May. 86.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Subsequence of A002061.
Cf. A001550, A034513, A001579, A074501-A074580, A135576, A135577.
See also comments in A051154.
Cf. A006095, A022166.

Mathematica
```
Table[1^n + 2^n + 4^n, {n, 0, 24}]
```

a(n)=1+2^n+4^n \\ Charles R Greathouse IV, Jun 10 2011

[sigma(4,n)for n in range(0,23)] # Zerinvary Lajos, Jun 04 2009

a(n) = 6*a(n-1) - 8*a(n-2) + 3.
O.g.f.: -1/(-1+x) - 1/(-1+2*x) - 1/(-1+4*x) = ( -3+14*x-14*x^2 ) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Feb 29 2008
E.g.f.: e^x + e^(2*x) + e^(4*x). - Mohammad K. Azarian, Dec 26 2008
a(n) = A024088(n)/A000225(n). - Reinhard Zumkeller, Feb 15 2009
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 7*x + 35*x^2 + 155*x^3 + ... is the o.g.f. for the 2nd subdiagonal of triangle A022166, essentially A006095. - Peter Bala, Apr 07 2015

A001550 a(n) = 1^n + 2^n + 3^n.

Original entry on oeis.org

3, 6, 14, 36, 98, 276, 794, 2316, 6818, 20196, 60074, 179196, 535538, 1602516, 4799354, 14381676, 43112258, 129271236, 387682634, 1162785756, 3487832978, 10462450356, 31385253914, 94151567436, 282446313698, 847322163876

Offset: 0

N. J. A. Sloane

a(n)*(-1)^n, n>=0, gives the z-sequence for the Sheffer triangle A049458 ((signed) 3-restricted Stirling1 numbers), which is the inverse triangle of A143495 with offset [0,0] (3-restricted Stirling2 numbers). See the W. Lang link under A006232 for a- and z-sequences for Sheffer matrices. The a-sequence for each (signed) r-restricted Stirling1 Sheffer triangle is A027641/A027642 (Bernoulli numbers). - Wolfdieter Lang, Oct 10 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 363
C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Kai Wang, Girard-Waring Type Formula For A Generalized Fibonacci Sequence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 229-235.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000051, A000079, A000244, A007689, A034472.
Cf. A001576, A001579, A034513, A074501 - A074580.
Column 3 of array A103438.

a001550 n = sum $ map (^ n) [1..3]  -- Reinhard Zumkeller, Mar 01 2012

[1^n + 2^n + 3^n : n in [0..30]]; // Wesley Ivan Hurt, Jun 25 2020

A001550:=-(3-12*z+11*z^2)/(z-1)/(3*z-1)/(2*z-1); # Simon Plouffe in his 1992 dissertation.

Table[1^n + 2^n + 3^n, {n, 0, 30}]
CoefficientList[Series[(3-12x+11x^2)/(1-6x+11x^2-6x^3),{x,0,30}],x] (* or *) LinearRecurrence[{6,-11,6},{3,6,14},31] (* Harvey P. Dale, Apr 30 2011 *)
Total[Range[3]^#]&/@Range[0,30] (* Harvey P. Dale, Sep 23 2019 *)

a(n)=1+2^n+3^n \\ Charles R Greathouse IV, Jun 10 2011

def A001550(n): return 3**n+(1<Chai Wah Wu, Nov 01 2024

From Michael Somos: (Start)
G.f.: (3 -12*x +11*x^2)/(1 -6*x +11*x^2 -6*x^3).
a(n) = 5*a(n-1) - 6*a(n-2) + 2. (End)
E.g.f.: exp(x) + exp(2*x) + exp(3*x). - Mohammad K. Azarian, Dec 26 2008
a(0)=3, a(1)=6, a(2)=14, a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3). - Harvey P. Dale, Apr 30 2011
a(n) = A007689(n) + 1. - Reinhard Zumkeller, Mar 01 2012
From Kai Wang, May 18 2020: (Start)
a(n) = 3*A000392(n+3) - 12*A000392(n+2) + 11*A000392(n+1).
A000392(n) = (3*a(n+1) - 12*a(n) + 10*a(n-1))/2. (End)

Additional terms from Michael Somos

Attribute "conjectured" removed from Simon Plouffe's g.f. by R. J. Mathar, Mar 11 2009

A074501 a(n) = 1^n + 2^n + 5^n.

Original entry on oeis.org

3, 8, 30, 134, 642, 3158, 15690, 78254, 390882, 1953638, 9766650, 48830174, 244144722, 1220711318, 6103532010, 30517610894, 152587956162, 762939584198, 3814697527770, 19073486852414, 95367432689202, 476837160300278

Offset: 0

Robert G. Wilson v, Aug 23 2002

Jason Bard, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Equals A074600(n) + 1.

Cf. A001550, A001576, A034513, A001579, A074502..A074580.

Mathematica
```
Table[1^n + 2^n + 5^n, {n, 0, 24}]
```

a(n)=1+2^n+5^n \\ Charles R Greathouse IV, Jun 10 2011

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-x) + 1/(1-2*x) + 1/(1-5*x).
E.g.f.: e^x + e^(2*x) + e^(5*x). (End)
a(n) = 7*a(n-1) - 10*a(n-2) + 4 with a(0)=3, a(1)=8. - Vincenzo Librandi, Jul 21 2010

A024023 a(n) = 3^n - 1.

Original entry on oeis.org

0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, 59048, 177146, 531440, 1594322, 4782968, 14348906, 43046720, 129140162, 387420488, 1162261466, 3486784400, 10460353202, 31381059608, 94143178826, 282429536480, 847288609442, 2541865828328, 7625597484986, 22876792454960

Offset: 0

N. J. A. Sloane

Number of different directions along lines and hyper-diagonals in an n-dimensional cubic lattice for the attacking queens problem (A036464 in n=2, A068940 in n=3 and A068941 in n=4). The n-dimensional direction vectors have the a(n)+1 Cartesian coordinates (i,j,k,l,...) where i,j,k,l,... = -1, 0, or +1, excluding the zero-vector i=j=k=l=...=0. The corresponding hyper-line count is A003462. - R. J. Mathar, May 01 2006
Total number of sequences of length m=1,...,n with nonzero integer elements satisfying the condition Sum_{k=1..m} |n_k| <= n. See the K. A. Meissner link p. 6 (with a typo: it should be 3^([2a]-1)-1). - Wolfdieter Lang, Jan 21 2008
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x and y are disjoint and either 0) x is a proper subset of y or y is a proper subset of x, or 1) x is not a subset of y and y is not a subset of x. Then a(n) = |R|. - Ross La Haye, Mar 19 2009
Number of neighbors in Moore's neighborhood in n dimensions. - Dmitry Zaitsev, Nov 30 2015
Number of terms in conjunctive normal form of Boolean expression with n variables. E.g., a(2) = 8: [~x, ~y, x, y, ~x|~y, ~x|y, x|~y, x|y]. - Yuchun Ji, May 12 2023
Number of rays of the Coxeter arrangement of type B_n. Equivalently, number of facets of the n-dimensional type B permutahedron. - Jose Bastidas, Sep 12 2023

			From _Zerinvary Lajos_, Jan 14 2007: (Start)
Ternary......decimal:
0...............0
2...............2
22..............8
222............26
2222...........80
22222.........242
222222........728
2222222......2186
22222222.....6560
222222222...19682
2222222222..59048
etc...........etc.
(End)
Sequence combinatorics: n=3: With length m=1: [1],[2],[3] each with 2 signs, with m=2: [1,1], [1,2], [2,1], each 2^2 = 4 times from choosing signs; m=3: [1,1,1] coming in 2^3 signed versions: 3*2 + 3*4 + 1*8 = 26 = a(3). The order is important, hence the M_0 multinomials A048996 enter as factors.
A027902 gives the 384 divisors of a(24). - _Reinhard Zumkeller_, Mar 11 2010

Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Omran Ahmadi and Robert Granger, An efficient deterministic test for Kloosterman sum zeros, Mathematics of Computation, Vol. 83, No. 285 (2014), pp. 347-363; arXiv preprint, arXiv:1104.3882 [math.NT], 2011-2012. See 1st column of Table 2, p. 9.
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
Michael Baake, Franz Gähler, and Uwe Grimm, Examples of Substitution Systems and Their Factors, Journal of Integer Sequences, Vol. 16 (2013), #13.2.14.
R. Samuel Buss, Herbrand's Theorem, University of California, Logic and Computational Complexity pp. 195-209, Lecture Notes in Computer Science, vol 960. Springer.
Jan Draisma, Tyrrell B. McAllister and Benjamin Nill, Lattice width directions and Minkowski's 3^d-theorem, SIAM J. Discrete Math., Vol. 26, No. 3 (2012), pp. 1104-1107; arXiv preprint, arXiv:0901.1375 [math.CO], Jan 10 2009.
Alessandro Farinelli, Herbrand Universe and Herbrand Base.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Krzysztof A. Meissner, Black hole entropy in Loop Quantum Gravity, Classical and Quantum Gravity, Vol. 21, No. 22 (2004), pp. 5245--5251; arXiv preprint, arXiv:gr-qc/0407052, 2004.
Amir Sapir, The Tower of Hanoi with Forbidden Moves, The Computer J. 47 (1) (2004) 20, case three-in-a row, sequence b(n).
Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Wikipedia, Herbrand Structure.
Damiano Zanardini, Computational Logic, Slides, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid, 2009-2010.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. triangle A013609.
Cf. A003462, A007051, A034472, A214369.
Cf. second column of A145901.

a024023 = subtract 1 . a000244  -- Reinhard Zumkeller, Jun 30 2013

[3^n-1: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

3^Range[0,30]-1 (* Paolo Xausa, Jul 15 2023 *)

a(n)=3^n-1 \\ Charles R Greathouse IV, Sep 24 2015

vector(50, n, sum(k=0, n, 2^k*binomial(n-1, k))-1) \\ Altug Alkan, Oct 04 2015

my(x='x+O('x^100)); concat([0], Vec(2*x/(-1+x)/(-1+3*x))) \\ Altug Alkan, Oct 16 2015

a(n) = A000244(n) - 1.
a(n) = 2*A003462(n). - R. J. Mathar, May 01 2006
A128760(a(n)) > 0. - Reinhard Zumkeller, Mar 25 2007
G.f.: 2*x/((-1+x)*(-1+3*x)) = 1/(-1+x) - 1/(-1+3*x). - R. J. Mathar, Nov 19 2007
a(n) = Sum_{k=1..n} Sum_{m=1..k} binomial(k-1,m-1)*2^m, n >= 1. a(0)=0. From the sequence combinatorics mentioned above. Twice partial sums of powers of 3.
E.g.f.: e^(3*x) - e^x. - Mohammad K. Azarian, Jan 14 2009
a(n) = A024101(n)/A034472(n). - Reinhard Zumkeller, Feb 14 2009
a(n) = 3*a(n-1) + 2 (with a(0)=0). - Vincenzo Librandi, Nov 19 2010
E.g.f.: -E(0) where E(k) = 1 - 3^k/(1 - x/(x - 3^k*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
a(n) = A227048(n,A020914(n)). - Reinhard Zumkeller, Jun 30 2013
Sum_{n>=1} 1/a(n) = A214369. - Amiram Eldar, Nov 11 2020
a(n) = Sum_{k=1..n} 2^k*binomial(n,k). - Ridouane Oudra, Jun 15 2025
From Peter Bala, Jul 01 2025: (Start)
For n >= 1, a(2*n)/a(n) = A034472(n) and a(3*n)/a(n) = A034513(n).
Modulo differences in offsets, exp( Sum_{n >= 1} a(k*n)/a(n)*x^n/n ) is the o.g.f. of A003462 (k = 2), A006100 (k = 3), A006101 (k = 4), A006102 (k = 5), A022196 (k = 6), A022197 (k = 7), A022198 (k = 8), A022199 (k = 9), A022200 (k = 10), A022201 (k = 11), A022202 (k = 12) and A022203 (k = 13).
The following are all examples of telescoping series:
Sum_{n >= 1} 3^n/(a(n)*a(n+1)) = 1/2^2; Sum_{n >= 1} 3^n/(a(n)*a(n+1)*a(n+2)) = 1/(2*8^2).
In general, for k >= 1, Sum_{n >= 1} 3^n/(a(n)*a(n+1)*...*a(n+k)) = 1/(a(1)*a(2)*...*a(k)*a(k)).
Sum_{n >= 1} 3^n/(a(n)*a(n+2)) = 5/64; Sum_{n >= 1} (-3)^n/(a(n)*a(n+2)) = -3/64.
Sum_{n >= 1} 3^n/(a(n)*a(n+4)) = 703/83200; Sum_{n >= 1} (-3)^n/(a(n)*a(n+4)) = - 417/83200. (End)

A074580 a(n) = 7^n + 8^n + 9^n.

Original entry on oeis.org

3, 24, 194, 1584, 13058, 108624, 911234, 7703664, 65588738, 561991824, 4843001474, 41948320944, 364990300418, 3188510652624, 27953062038914, 245823065693424, 2167728096132098, 19161612027339024, 169737447404391554

Offset: 0

Robert G. Wilson v, Aug 23 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (24,-191,504).

Cf. A001550, A001576, A034513, A001579, A074501-A074579.

[7^n + 8^n + 9^n: n in [0..20]]; // Vincenzo Librandi, May 20 2011

Table[7^n + 8^n + 9^n, {n, 0, 18}]
LinearRecurrence[{24,-191,504},{3,24,194},20] (* Harvey P. Dale, Jun 09 2019 *)

a(n)=7^n+8^n+9^n \\ Charles R Greathouse IV, Jun 10 2011

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-7*x) + 1/(1-8*x) + 1/(1-9*x).
E.g.f.: e^(7*x) + e^(8*x) + e^(9*x). (End)

A001579 a(n) = 3^n + 5^n + 6^n.

Original entry on oeis.org

3, 14, 70, 368, 2002, 11144, 63010, 360248, 2076802, 12050504, 70290850, 411802328, 2421454402, 14282991464, 84472462690, 500716911608, 2973740844802, 17689728038024, 105375041354530, 628434388600088

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Henri W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, The Fibonacci Quarterly, 37(2):135-140, 1999.
Index entries for linear recurrences with constant coefficients, signature (14,-63,90).

Cf. A001550, A001576, A034513, A074501-A074580.

[3^n + 5^n + 6^n: n in [0..20]]; // Vincenzo Librandi, May 20 2011

Table[3^n + 5^n + 6^n, {n, 0, 20}]
LinearRecurrence[{14,-63,90},{3,14,70},20] (* Harvey P. Dale, Jun 17 2021 *)

a(n)=3^n+5^n+6^n \\ Charles R Greathouse IV, Jun 10 2011

def a(n): return 3**n + 5**n + 6**n
print([a(n) for n in range(20)]) # Michael S. Branicky, Mar 14 2021

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-3*x) + 1/(1-5*x) + 1/(1-6*x).
E.g.f.: e^(3*x) + e^(5*x) + e^(6*x). (End)

A109974 Array read by downwards antidiagonals: sigma_k(n) for n >= 1, k >= 0.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 2, 7, 10, 9, 1, 4, 6, 21, 28, 17, 1, 2, 12, 26, 73, 82, 33, 1, 4, 8, 50, 126, 273, 244, 65, 1, 3, 15, 50, 252, 626, 1057, 730, 129, 1, 4, 13, 85, 344, 1394, 3126, 4161, 2188, 257, 1, 2, 18, 91, 585, 2402, 8052, 15626, 16513, 6562, 513, 1

Offset: 0

Paul Barry, Jul 06 2005

Rows sums are A108639. Antidiagonal sums are A109976. Matrix inverse is A109977.
From Wolfdieter Lang, Jan 29 2016: (Start)
The sum of the (k-1)th power of the divisors of n, sigma_(k-1)(n), appears also as eigenvalue lambda(k, n) of the Hecke operators T_n, n a positive integer, acting on the normalized Eisenstein series E_k(q) = ((2*Pi*i)^k/((k-1)!*Zeta(k))*G_k(q) with even k >= 4 and q = 2*Pi*i*z, where z is from the upper half of the complex plane: T_n E_k = sigma_(k-1)(n)*E_k. These Eisenstein series are entire modular forms of weight k, and each E_k(q) is a simultaneous eigenform of the Hecke operators T_n, for every n >= 1.
This results from the Fourier coefficients of E_k(q) = Sum_{m>=0} E(k, m)*q^m, with E(k, 0) =1 and E(k, m) = ((2*Pi*i)^k / ((k-1)!*Zeta(k))* sigma_(k-1)(m) for m >= 1, together with the Fourier coefficients of T_n E_k. The eigenvalues lambda(n, k) = (Sum_{d | gcd(n,m)} d^{k-1}*E(k, m*n/d^2)) / E(k, m) for each m >= 0. For m=0 this becomes lambda(n, k) = sigma_(k-1)(n).
For Hecke operators, Fourier coefficients and simultaneous eigenforms see, e.g., the Koecher - Krieg reference, p. 207, eqs. (5) and (6) and p. 211, section 4, or the Apostol reference, p. 120, eq. (13), pp. 129 - 134. (End)

			Start of array:
  1,  2,  2,   3,   2,    4, ...
  1,  3,  4,   7,   6,   12, ...
  1,  5, 10,  21,  26,   50, ...
  1,  9, 28,  73, 126,  252, ...
  1, 17, 82, 273, 626, 1394, ...
  ...
The triangle T(m, k) with row offset 1 starts:
  m\k 0  1  2   3    4    5    6    7   8  9 ...
  1:  1
  2:  2  1
  3:  2  3  1
  4:  3  4  5   1
  5:  2  7 10   9    1
  6:  4  6 21  28   17    1
  7:  2 12 26  73   82   33    1
  8:  4  8 50 126  273  244   65    1
  9:  3 15 50 252  626 1057  730  129   1
  10: 4 13 85 344 1394 3126 4161 2188 257  1
  ... - _Wolfdieter Lang_, Jan 14 2016

Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 120, 129 - 134.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 407.
Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 207, 211.

Alois P. Heinz, Antidiagonals k = 0..140, flattened

Rows: A000005, A000203, A001157, A001158, A001159, A001160, A013954 - A013972.
Columns: A000051, A034472, A001576, A034474, A034488, A034491, A034496, A034513, A034517, A034524, A034660.
Row sums A108639.
Diagonals A082245, A023887.
Related sequences: A082771, A109976, A109977, A109978.

A109974:= func< n,k | DivisorSigma(k-1, n-k+1) >;
[A109974(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 18 2023

with(numtheory):
seq(seq(sigma[k](1+d-k), k=0..d), d=0..12);  # Alois P. Heinz, Feb 06 2013

rows=12; Flatten[Table[DivisorSigma[k-n, n], {k,1,rows}, {n,k,1,-1}]] (* Jean-François Alcover, Nov 15 2011 *)

def A109974(n,k): return sigma(n-k+1, k-1)
flatten([[A109974(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 18 2023

Regarded as a triangle, T(n, k) = if(k<=n, sigma(k-1, n-k+1), 0). - Franklin T. Adams-Watters, Jul 17 2006
If the row index (the index of the antidiagonal of the array) is taken as m with offset 1 the triangle is T(m, k) = sigma_k(m-k), 1 <= k+1 <= m, otherwise 0. - Wolfdieter Lang, Jan 14 2016
G.f. for the triangle with offset 1: G(x,y) = Sum_{j>=1} x^j/((1-x^j)*(1-j*x*y)). - Robert Israel, Jan 14 2016

A074527 a(n) = 2^n + 3^n + 5^n.

Original entry on oeis.org

3, 10, 38, 160, 722, 3400, 16418, 80440, 397442, 1973320, 9825698, 49007320, 244676162, 1222305640, 6108314978, 30531959800, 152631002882, 763068724360, 3815084948258, 19074649113880, 95370919473602, 476847620653480

Offset: 0

Robert G. Wilson v, Aug 23 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (10,-31,30).

Cf. A001550, A001576, A034513, A001579, A074501-A074580.

[2^n + 3^n + 5^n: n in [0..25]]; // Vincenzo Librandi, Jun 11 2011

Table[2^n + 3^n + 5^n, {n, 0, 21}]
LinearRecurrence[{10,-31,30},{3,10,38},30] (* Harvey P. Dale, Feb 05 2022 *)

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

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-2*x) + 1/(1-3*x) + 1/(1-5*x).
E.g.f.: exp(2*x) + exp(3*x) + exp(5*x). (End)
a(n) = 10*a(n-1) - 31*a(n-2) + 30*a(n-3). - Wesley Ivan Hurt, May 26 2024

A082771 Triangular array, read by rows: t(n,k) = Sum_{d|n} d^k, 0 <= k < n.

Original entry on oeis.org

1, 2, 3, 2, 4, 10, 3, 7, 21, 73, 2, 6, 26, 126, 626, 4, 12, 50, 252, 1394, 8052, 2, 8, 50, 344, 2402, 16808, 117650, 4, 15, 85, 585, 4369, 33825, 266305, 2113665, 3, 13, 91, 757, 6643, 59293, 532171, 4785157, 43053283, 4, 18, 130, 1134, 10642, 103158, 1015690, 10078254, 100390882, 1001953638

Offset: 1

Reinhard Zumkeller, May 21 2003

			From _R. J. Mathar_, Dec 06 2006 (Start):
The triangle may be extended to a rectangular array (A319278):
  1  1   1    1     1 1 1 1 1 1 1 ...
  2  3   5    9    17 33 65 129 257 513 1025 ...
  2  4  10   28    82 244 730 2188 6562 19684 59050 ...
  3  7  21   73   273 1057 4161 16513 65793 262657 1049601 ...
  2  6  26  126   626 3126 15626 78126 390626 1953126 9765626 ...
  4 12  50  252  1394 8052 47450 282252 1686434 10097892 60526250 ...
  2  8  50  344  2402 16808 117650 823544 5764802 40353608 282475250 ...
  4 15  85  585  4369 33825 266305 2113665 16843009 134480385 1074791425 ...
  3 13  91  757  6643 59293 532171 4785157 43053283 387440173 3486843451 ...
  4 18 130 1134 10642 103158 1015690 10078254 100390882 1001953638... (End)

T. D. Noe, Rows n=1..100, flattened
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000005, A000203, A001157, A001158, A001159, A001160.
Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963.
Cf. A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.

T:= (n,k)-> numtheory[sigma][k](n):
seq(seq(T(n,k), k=0..n-1), n=1..10);  # Alois P. Heinz, Oct 25 2024

T[n_, k_] := DivisorSigma[k, n];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Dec 16 2021 *)

row(n) = {my(f = factor(n)); vector(n, k, sigma(f, k-1));} \\ Amiram Eldar, May 09 2025

t(n, k) = Product(((p^((e(n, p)+1)*k))-1)/(p^k-1): n=Product(p^e(n, p): p prime)), 0<=k

t(n,0) = A000005(n), t(n,n) = A023887(n).

t(n,1) = A000203(n), n>1; t(n,2) = A001157(n), n>2; t(n,3) = A001158(n), n>3.

t(n,4) = A001159(n), n>4; t(n,5) = A001160(n), n>5; t(n,6) = A013954(n), n>6.

From R. J. Mathar, Oct 29 2006: (Start)

t(2,k) = A000051(k); t(3,k) = A034472(k); t(4,k) = A001576(k);

t(5,k) = A034474(k); t(6,k) = A034488(k); t(7,k) = A034491(k);

t(8,k) = A034496(k); t(9,k) = A034513(k); t(10,k) = A034517(k);

t(11,k) = A034524(k); t(12,k) = A034660(k). (End)

Corrected by R. J. Mathar, Dec 05 2006

A074507 a(n) = 1^n + 3^n + 5^n.

Original entry on oeis.org

3, 9, 35, 153, 707, 3369, 16355, 80313, 397187, 1972809, 9824675, 49005273, 244672067, 1222297449, 6108298595, 30531927033, 152630937347, 763068593289, 3815084686115, 19074648589593, 95370918425027, 476847618556329

Offset: 0

Robert G. Wilson v, Aug 23 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000
T. A. Gulliver, Divisibility of sums of powers of odd integers, Int. Math. For. 5 (2010) 3059-3066, eq. (6).
D. Suprijanto, I. W. Suwarno, Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2211 - 2217.
Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Cf. A001550, A001576, A034513, A001579, A074501-A074580.

Table[1^n + 3^n + 5^n, {n, 0, 22}]
LinearRecurrence[{9,-23,15},{3,9,35},30] (* Harvey P. Dale, Mar 02 2022 *)

a(n) = 1 + 3^n + 5^n; \\ Michel Marcus, Aug 07 2017

a(n) = 8*a(n-1) - 15*a(n-2) + 8.

G.f.: 1/(1-x)+1/(1-3*x)+1/(1-5*x). E.g.f.: e^x+e^(3*x)+e^(5*x). [Mohammad K. Azarian, Dec 26 2008]

Showing 1-10 of 88 results. Next

cp's OEIS Frontend

A001576 a(n) = 1^n + 2^n + 4^n.

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A001550 a(n) = 1^n + 2^n + 3^n.

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A074501 a(n) = 1^n + 2^n + 5^n.

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A024023 a(n) = 3^n - 1.

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A074580 a(n) = 7^n + 8^n + 9^n.

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A001579 a(n) = 3^n + 5^n + 6^n.

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