A001029 -id:A001029 - cp's OEIS frontend

A060222 Number of orbits of length n under the full 19-shift (whose periodic points are counted by A001029).

19, 171, 2280, 32490, 495216, 7839780, 127695960, 2122929090, 35854187880, 613106378136, 10590023536200, 184442905990860, 3234844881712080, 57071906063500860, 1012075135324821024, 18027588346914850290, 322375697516753069760, 5784852794310472599780, 104127350297911241532840

Offset: 1

Thomas Ward, Mar 21 2001

Number of monic irreducible polynomials of degree n over GF(19). - Andrew Howroyd, Dec 10 2017

			a(2)=171 since there are 361 points of period 2 in the full 19-shift and 19 fixed points, so there must be (361-19)/2 = 171 orbits of length 2.

Vincenzo Librandi, Table of n, a(n) for n = 1..799
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
T. Ward, Exactly realizable sequences

Column 19 of A074650.

Cf. A001029.

A060222:= func< n | (1/n)*(&+[MoebiusMu(d)*(19)^Floor(n/d): d in Divisors(n)]) >;
[A060222(n): n in [1..40]]; // G. C. Greubel, Sep 23 2024

a[n_]:=(1/n) Sum[MoebiusMu[d] 19^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 20}] (* Vincenzo Librandi, Sep 19 2017 *)

a001029(n) = 19^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001029(n/d)); \\ Michel Marcus, Sep 11 2017

def A060222(n): return (1/n)*sum(moebius(k)*(19)^(n/k) for k in (1..n) if (k).divides(n))
[A060222(n) for n in range(1, 41)] # G. C. Greubel, Sep 23 2024

a(n) = (1/n)* Sum_{d|n} mu(d)*A001029(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 19*x^k))/k. - Ilya Gutkovskiy, May 20 2019

More terms from Michel Marcus, Sep 11 2017

A008472 Sum of the distinct primes dividing n.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 9, 8, 2, 17, 5, 19, 7, 10, 13, 23, 5, 5, 15, 3, 9, 29, 10, 31, 2, 14, 19, 12, 5, 37, 21, 16, 7, 41, 12, 43, 13, 8, 25, 47, 5, 7, 7, 20, 15, 53, 5, 16, 9, 22, 31, 59, 10, 61, 33, 10, 2, 18, 16, 67, 19, 26, 14, 71, 5, 73

Offset: 1

Olivier Gérard

Sometimes called sopf(n).
Sum of primes dividing n (without repetition) (compare A001414).
Equals A051731 * A061397 = inverse Mobius transform of [0, 2, 3, 0, 5, 0, 7, ...]. - Gary W. Adamson, Feb 14 2008
Equals row sums of triangle A143535. - Gary W. Adamson, Aug 23 2008
a(n) = n if and only if n is prime. - Daniel Forgues, Mar 24 2009
a(n) = n is a new record if and only if n is prime. - Zak Seidov, Jun 27 2009
a(A001043(n)) = A191583(n);
For n > 0: a(A000079(n)) = 2, a(A000244(n)) = 3, a(A000351(n)) = 5, a(A000420(n)) = 7;
a(A006899(n)) <= 3; a(A003586(n)) = 5; a(A033846(n)) = 7; a(A033849(n)) = 8; a(A033847(n)) = 9; a(A033850(n)) = 10; a(A143207(n)) = 10. - Reinhard Zumkeller, Jun 28 2011
For n > 1: a(n) = Sum(A027748(n,k): 1 <= k <= A001221(n)). - Reinhard Zumkeller, Aug 27 2011
If n is the product of twin primes (A037074), a(n) = 2*sqrt(n+1) = sqrt(4n+4). - Wesley Ivan Hurt, Sep 07 2013
From Wilf A. Wilson, Jul 21 2017: (Start)
a(n) + 2, n > 2, is the number of maximal subsemigroups of the monoid of orientation-preserving or -reversing mappings on a set with n elements.
a(n) + 3, n > 2, is the number of maximal subsemigroups of the monoid of orientation-preserving or -reversing partial mappings on a set with n elements.
(End)
The smallest m such that a(m) = n, or 0 if no such number m exists is A064502(n). The only integers that are not in the sequence are 1, 4 and 6. - Bernard Schott, Feb 07 2022

			a(18) = 5 because 18 = 2 * 3^2 and 2 + 3 = 5.
a(19) = 19 because 19 is prime.
a(20) = 7 because 20 = 2^2 * 5 and 2 + 5 = 7.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from Franklin T. Adams-Watters)
Johann Bartel, R. K. Bhaduri, Matthias Brack, and M. V. N. Murthy, On the asymptotic prime partitions of integers, arXiv:1609.06497 [math-ph], 2017.
James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [_Wilf A. Wilson_, Jul 21 2017]

First difference of A024924.
Cf. A001414 (sopfr), A001222, A051731, A061397, A064502, A085020, A143535.
Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), this sequence (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).
Cf. A010051.

a008472 = sum . a027748_row  -- Reinhard Zumkeller, Mar 29 2012

[n eq 1 select 0 else &+[p[1]: p in Factorization(n)]: n in [1..100]]; // Vincenzo Librandi, Jun 24 2017

A008472 := n -> add(d, d = select(isprime, numtheory[divisors](n))):
seq(A008472(i), i = 1..40); # Peter Luschny, Jan 31 2012
A008472 := proc(n)
        add( d, d= numtheory[factorset](n)) ;
end proc: # R. J. Mathar, Jul 08 2012

Prepend[Array[Plus @@ First[Transpose[FactorInteger[#]]] &, 100, 2], 0]
Join[{0}, Rest[Total[Transpose[FactorInteger[#]][[1]]]&/@Range[100]]] (* Harvey P. Dale, Jun 18 2012 *)
(* Requires version 7.0+ *) Table[DivisorSum[n, # &, PrimeQ[#] &], {n, 75}] (* Alonso del Arte, Dec 13 2014 *)
Table[Sum[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)

sopf(n) = local(fac=factor(n)); sum(i=1,matsize(fac)[1],fac[i,1])

vector(100,n,vecsum(factor(n)[,1]~)) \\ Derek Orr, May 13 2015

A008472(n)=vecsum(factor(n)[,1]) \\ M. F. Hasler, Jul 18 2015

from sympy import primefactors
def A008472(n): return sum(primefactors(n)) # Chai Wah Wu, Feb 03 2022

def A008472(n):
    return add(d for d in divisors(n) if is_prime(d))
print([A008472(i) for i in (1..40)]) # Peter Luschny, Jan 31 2012

[sum(prime_factors(n)) for n in range(1,74)] # Giuseppe Coppoletta, Jan 19 2015

Let n = Product_j prime(j)^k(j) where k(j) >= 1, then a(n) = Sum_j prime(j).
Additive with a(p^e) = p.
G.f.: Sum_{k >= 1} prime(k)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
L.g.f.: -log(Product_{k>=1} (1 - x^prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
Dirichlet g.f.: primezeta(s-1)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
a(n) = Sum_{p|n, p prime} p. - Wesley Ivan Hurt, Feb 04 2022
From Bernard Schott, Feb 07 2022: (Start)
For n > 0: a(A001020(n)) = 11, a(A001022(n)) = 13, a(A001026(n)) = 17, a(A001029(n)) = 19, a(A009967(n)) = 23, a(A009973(n)) = 29, a(A009975(n)) = 31, a(A009981(n)) = 37, a(A009985(n)) = 41, a(A009987(n)) = 43, a(A009991(n)) = 47.
For p odd prime, a(2*p) = p+2 <==> a(A100484(n)) = A052147(n) for n > 1. (End)
a(n) = Sum_{d|n} d * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

A000420 Powers of 7: a(n) = 7^n.

Original entry on oeis.org

1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010407, 678223072849, 4747561509943, 33232930569601, 232630513987207, 1628413597910449, 11398895185373143, 79792266297612001, 558545864083284007

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 7), L(1, 7), P(1, 7), T(1, 7). Essentially same as Pisot sequences E(7, 49), L(7, 49), P(7, 49), T(7, 49). See A008776 for definitions of Pisot sequences.
Sum of coefficients of expansion of (1+x+x^2+x^3+x^4+x^5+x^6)^n.
a(n) is number of compositions of natural numbers into n parts < 7.
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 7-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Numbers n such that sigma(7n) = 7n + sigma(n). - Jahangeer Kholdi, Nov 23 2013
Number of ways to assign truth values to n ternary disjunctions connected by conjunctions such that the proposition is true. For example, a(2) = 49, since for the proposition '(a v b v c) & (d v e v f)' there are 49 assignments that make the proposition true. - Ori Milstein, Dec 31 2022
Equivalently, the number of length-n words over an alphabet with seven letters. - Joerg Arndt, Jan 01 2023

			a(2)=49 there are 49 compositions of natural numbers into 2 parts < 7.

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..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 272
Tanya Khovanova, Recursive Sequences
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
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for linear recurrences with constant coefficients, signature (7).

Cf. A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A001018 (powers of 8), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48), A087752 (powers of 49).

a000420 = (7 ^)
a000420_list = iterate (* 7) 1  -- Reinhard Zumkeller, Apr 29 2015

[7^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 27 2016

A000420:=-1/(-1+7*z); # Simon Plouffe in his 1992 dissertation. [This is actually the generating function, so convert(series(...),list) would yield the actual sequence. - M. F. Hasler, Apr 19 2015]
A000420 := n -> 7^n; # M. F. Hasler, Apr 19 2015

Table[7^n, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

makelist(7^n,n,0,20); /* Martin Ettl, Dec 27 2012 */

a(n)=7^n \\ Charles R Greathouse IV, Jul 28 2015

a(n) = 7^n.
a(0) = 1; a(n) = 7*a(n-1).
G.f.: 1/(1-7*x).
E.g.f.: exp(7*x).
4/7 - 5/7^2 + 4/7^3 - 5/7^4 + ... = 23/48. [Jolley, Summation of Series, Dover, 1961]

A001018 Powers of 8: a(n) = 8^n.

Original entry on oeis.org

1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, 35184372088832, 281474976710656, 2251799813685248, 18014398509481984, 144115188075855872, 1152921504606846976, 9223372036854775808, 73786976294838206464, 590295810358705651712, 4722366482869645213696

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 8), L(1, 8), P(1, 8), T(1, 8). Essentially same as Pisot sequences E(8, 64), L(8, 64), P(8, 64), T(8, 64). See A008776 for definitions of Pisot sequences.
If X_1, X_2, ..., X_n is a partition of the set {1..2n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1..2n} -> {1,2,3} such that for fixed y_1,y_2,...,y_n in {1,2,3} we have f(X_i)<>{y_i}, (i=1..n). - Milan Janjic, May 24 2007
This is the auto-convolution (convolution square) of A059304. - R. J. Mathar, May 25 2009
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 8-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
a(n) is equal to the determinant of a 3 X 3 matrix with rows 2^(n+2), 2^(n+1), 2^n; 2^(n+3), 2^(n+4), 2(n+3); 2^n, 2^(n+1), 2^(n+2) when it is divided by 144. - J. M. Bergot, May 07 2014
a(n) gives the number of small squares in the n-th iteration of the Sierpinski carpet fractal. Equivalently, the number of vertices in the n-Sierpinski carpet graph. - Allan Bickle, Nov 27 2022

			For n=1, the 1st order Sierpinski carpet graph is an 8-cycle.

K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2017; p. 15.
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..100
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 273
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Caroline Nunn, A Proof of a Generalization of Niven's Theorem Using Algebraic Number Theory, Rose-Hulman Undergraduate Mathematics Journal: Vol. 22, Iss. 2, Article 3 (2021). See table at p. 9.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Sierpiński Carpet
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A013730, A103333, A013731, A067417, A083233, A055274.
Cf. A008588, A016969, A157176.
Cf. A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001019 (powers of 9), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48), A087752 (powers of 49), A165800 (powers of 50), A159991 (powers of 60).
Cf. A032766 (floor(3*n/2)).
Cf. A271939 (number of edges in the n-Sierpinski carpet graph).

a001018 = (8 ^)
a001018_list = iterate (* 8) 1  -- Reinhard Zumkeller, Apr 29 2015

[8^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 27 2016

seq(8^n, n=0..23); # Nathaniel Johnston, Jun 26 2011
A001018 := n -> 8^n; # M. F. Hasler, Apr 19 2015

Table[8^n, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

makelist(8^n,n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=8^n \\ Charles R Greathouse IV, May 10 2014

print([8**n for n in range(25)]) # Michael S. Branicky, Dec 29 2021

a(n) = 8^n.
a(0) = 1; a(n) = 8*a(n-1) for n > 0.
G.f.: 1/(1-8*x).
E.g.f.: exp(8*x).
Sum_{n>=0} 1/a(n) = 8/7. - Gary W. Adamson, Aug 29 2008
a(n) = A157176(A008588(n)); a(n+1) = A157176(A016969(n)). - Reinhard Zumkeller, Feb 24 2009
From Stefano Spezia, Dec 28 2021: (Start)
a(n) = (-1)^n*(1 + sqrt(-3))^(3*n) (see Nunn, p. 9).
a(n) = (-1)^n*Sum_{k=0..floor(3*n/2)} (-3)^k*binomial(3*n, 2*k) (see Nunn, p. 9). (End)

A218722 a(n) = (19^n-1)/18.

Original entry on oeis.org

0, 1, 20, 381, 7240, 137561, 2613660, 49659541, 943531280, 17927094321, 340614792100, 6471681049901, 122961939948120, 2336276859014281, 44389260321271340, 843395946104155461, 16024522975978953760, 304465936543600121441

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 19 (A001029); q-integers for q=19: diagonal k=1 in triangle A022183.

Partial sums are in A014903. Also, the sequence is related to A014936 by A014936(n) = n*a(n)-sum(a(i), i=0..n-1) for n>0. - Bruno Berselli, Nov 06 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (20,-19).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A001029, A014903, A014936, A022183.

[n le 2 select n-1 else 20*Self(n-1)-19*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{20, -19}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)

A218722(n):=(19^n-1)/18$ makelist(A218722(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218722(n)=19^n\18
```

a(n) = floor(19^n/18).
G.f.: x/((1-x)*(1-19*x)). - Bruno Berselli, Nov 06 2012
a(n) = 20*a(n-1) - 19*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(10*x)*sinh(9*x)/9. - Stefano Spezia, Mar 11 2023

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset: 0

Marc LeBrun

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

			Rows begin:
[1, 0,  0,   0,    0,     0,      0,      0, ...],
[1, 1,  1,   1,    1,     1,      1,      1, ...],
[1, 2,  4,   8,   16,    32,     64,    128, ...],
[1, 3,  9,  27,   81,   243,    729,   2187, ...],
[1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
[1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
[1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

T. D. Noe, Rows n = 0..50 of triangle, flattened

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).

[[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

T(n,k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

A128360 Numbers k such that k divides 20^k - 1.

Original entry on oeis.org

1, 19, 361, 6859, 130321, 2476099, 47045881, 148305659, 893871739, 2817807521, 4234136149, 10350100679, 16983563041, 53538342899, 80448586831, 196651912901, 322687697779, 815211156289, 1017228515081, 1432001198261, 1528523149789

Offset: 1

Alexander Adamchuk, Mar 02 2007

nonn

19 divides a(n) for n > 1. All powers of 19 are terms. a(n) = 19^(n-1) for all to n < 8, while a(8) = A128356(8) = 148305659 = 410819*19^2.

Prime divisors of a(n) in the order of appearance are {19, 410819, 617311, 1508981, ...}. - Alexander Adamchuk, May 16 2010

Cf. A001029, A128358, A014960, A014956, A014951, A014949, A014946, A014945, A067945, A128357, A128356, A128400, A177920.

is(n)=Mod(20,n)^n==1 \\ Charles R Greathouse IV, Nov 04 2016

a(9)-a(11) from Stefan Steinerberger, May 09 2007
a(12)-a(15) from Alexander Adamchuk, May 16 2010
Edited and a(16)-a(21) added by Max Alekseyev, Oct 02 2010

A153653 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2j + 1)prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 8, read by rows.

Original entry on oeis.org

2, 19, 19, 2, 718, 2, 2, 6857, 6857, 2, 2, 7505, 245628, 7505, 2, 2, 8153, 2467944, 2467944, 8153, 2, 2, 8801, 4900212, 84273732, 4900212, 8801, 2, 2, 9449, 7542432, 886319856, 886319856, 7542432, 9449, 2, 2, 10097, 10394604, 2476630764, 28993055148, 2476630764, 10394604, 10097, 2

Offset: 1

Roger L. Bagula, Dec 30 2008

			Triangle begins as:
   2;
  19,    19;
   2,   718,        2;
   2,  6857,     6857,          2;
   2,  7505,   245628,       7505,           2;
   2,  8153,  2467944,    2467944,        8153,          2;
   2,  8801,  4900212,   84273732,     4900212,       8801,        2;
   2,  9449,  7542432,  886319856,   886319856,    7542432,     9449,     2;
   2, 10097, 10394604, 2476630764, 28993055148, 2476630764, 10394604, 10097, 2;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A153652 (j=7), this sequence (j=8), A153654 (j=9), A153655 (j=10).
Cf. A153516, A153518, A153520, A153521, A153648, A153649, A153650, A153651, A153656, A153657.
Cf. A001029 (powers of 19).

f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n,k,j)
  if n eq 2 then return NthPrime(j);
  elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n,j);
  elif (k eq 1 or k eq n) then return 2;
  else return T(n-1,k,j) + T(n-1,k-1,j) + (2*j+1)*NthPrime(j)*T(n-2,k-1,j);
  end if; return T;
end function;
[T(n,k,8): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021

T[n_, k_, j_]:= T[n,k,j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1,k,j] + T[n-1,k-1,j] + (2*j+1)*Prime[j]*T[n-2,k-1,j] ]]];
Table[T[n,k,8], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 2021 *)

@CachedFunction
def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
def T(n,k,j):
    if (n==2): return nth_prime(j)
    elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
    elif (k==1 or k==n): return 2
    else: return T(n-1,k,j) + T(n-1,k-1,j) + (2*j+1)*nth_prime(j)*T(n-2,k-1,j)
flatten([[T(n,k,8) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 03 2021

T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j), T(3, 2, j) = 2*prime(j)^2 - 4, T(4, 2, j) = T(4, 3, j) = prime(j)^2 - 2, T(n, 1, j) = T(n, n, j) = 2 and j = 8.

Sum_{k=0..n} T(n, k, j) = 2*prime(j)^(n-1) for j=8 = 2*A001029(n-1).

Edited by G. C. Greubel, Mar 03 2021

A009992 Powers of 48: a(n) = 48^n.

Original entry on oeis.org

1, 48, 2304, 110592, 5308416, 254803968, 12230590464, 587068342272, 28179280429056, 1352605460594688, 64925062108545024, 3116402981210161152, 149587343098087735296, 7180192468708211294208, 344649238497994142121984, 16543163447903718821855232

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 48), L(1, 48), P(1, 48), T(1, 48). Essentially same as Pisot sequences E(48, 2304), L(48, 2304), P(48, 2304), T(48, 2304). See A008776 for definitions of Pisot sequences.
If X_1, X_2, ..., X_n is a partition of the set {1,2,...,2*n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1,2,..., 2*n}->{1,2,3,4,5,6,7} such that for fixed y_1,y_2,...,y_n in {1,2,3,4,5,6,7} we have f(X_i)<>{y_i}, (i=1,2,...,n). - Milan Janjic, May 24 2007
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 48-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (48).

Cf. A001018 (powers of 8), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009991 (powers of 47), A087752 (powers of 49).

Cf. A000079 (2^n), A000244 (3^n), A000302 (4^n), A000400 (6^n), A001018 (8^n), A001021 (12^n), A001025 (16^n), A009968 (24^n).

List([0..20],n->48^n); # Muniru A Asiru, Nov 21 2018

[48^n: n in [0..20]] // Vincenzo Librandi, Nov 21 2010

A009992 := n -> 48^n: seq(A009992(n), n=0..20); # M. F. Hasler, Apr 19 2015

48^Range[0, 15] (* Michael De Vlieger, Jan 13 2018 *)

A009992(n)=48^n \\ M. F. Hasler, Apr 19 2015

for n in range(0,20): print(48**n, end=', ') # Stefano Spezia, Nov 21 2018

[(48)^n for n in range(20)] # G. C. Greubel, Nov 21 2018

G.f.: 1/(1-48*x). - Philippe Deléham, Nov 24 2008
a(n) = 48^n; a(n) = 48*a(n-1), a(0)=1. - Vincenzo Librandi, Nov 21 2010
E.g.f.: exp(48*x). - Muniru A Asiru, Nov 21 2018

Edited by M. F. Hasler, Apr 19 2015

A087752 Powers of 49.

Original entry on oeis.org

1, 49, 2401, 117649, 5764801, 282475249, 13841287201, 678223072849, 33232930569601, 1628413597910449, 79792266297612001, 3909821048582988049, 191581231380566414401, 9387480337647754305649, 459986536544739960976801, 22539340290692258087863249, 1104427674243920646305299201

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Oct 02 2003

Same as Pisot sequences E(1, 49), L(1, 49), P(1, 49), T(1, 49). Essentially same as Pisot sequences E(49, 2401), L(49, 2401), P(49, 2401), T(49, 2401). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 49-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (49).

Bisection of A000420.
Cf. A001018 (powers of 8), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48).
Cf. A005843, A008776.

[49^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

49^Range[0,20] (* or *) Join[{1},NestList[49#&,49,20]] (* Harvey P. Dale, May 10 2019 *)

makelist(49^n,n,0,20); /* Martin Ettl, Nov 12 2012 */

PARI
```
a(n)=49^n \\ M. F. Hasler, Apr 19 2015
```

G.f.: 1/(1-49*x). - Philippe Deléham, Nov 24 2008
From Vincenzo Librandi, Nov 21 2010: (Start)
a(n) = 49^n.
a(n) = 49*a(n-1), a(0)=1. (End)
From Elmo R. Oliveira, Jul 08 2025: (Start)
E.g.f.: exp(49*x).
a(n) = A000420(A005843(n)). (End)

Edited by M. F. Hasler, Apr 19 2015

cp's OEIS Frontend

A060222 Number of orbits of length n under the full 19-shift (whose periodic points are counted by A001029).

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A008472 Sum of the distinct primes dividing n.

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A000420 Powers of 7: a(n) = 7^n.

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A001018 Powers of 8: a(n) = 8^n.

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A218722 a(n) = (19^n-1)/18.

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A153653 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2j + 1)prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 8, read by rows.