A036291 -id:A036291 - cp's OEIS frontend

A064751 a(n) = n*5^n - 1.

4, 49, 374, 2499, 15624, 93749, 546874, 3124999, 17578124, 97656249, 537109374, 2929687499, 15869140624, 85449218749, 457763671874, 2441406249999, 12969970703124, 68664550781249, 362396240234374, 1907348632812499, 10013580322265624, 52452087402343749, 274181365966796874

Offset: 1

N. J. A. Sloane, Oct 19 2001

Harry J. Smith, Table of n, a(n) for n = 1..150
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Index entries for linear recurrences with constant coefficients, signature (11,-35,25).

Cf. A003261, A036291.

[n*5^n - 1: n in [1..30]]; // Vincenzo Librandi, Jun 21 2018

Table[n*5^n-1,{n,20}] (* or *) LinearRecurrence[{11,-35,25},{4,49,374},20] (* Harvey P. Dale, Jun 25 2017 *)
CoefficientList[Series[(4 + 5 x - 25 x^2) / ((1 - 5 x)^2 (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 21 2018 *)

a(n) = { n*5^n - 1 } \\ Harry J. Smith, Sep 24 2009

G.f.: x*(4 + 5*x - 25*x^2)/((1 - 5*x)^2*(1 - x)). - Vincenzo Librandi, Jun 21 2018
a(n) = A036291(n) - 1. - Michel Marcus, Jun 21 2018
From Elmo R. Oliveira, May 05 2025: (Start)
E.g.f.: 1 + exp(x)*(5*x*exp(4*x) - 1).
a(n) = 11*a(n-1) - 35*a(n-2) + 25*a(n-3) for n > 3. (End)

A171607 Expressible as A*B^A in a nontrivial way.

Original entry on oeis.org

8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200, 242, 288, 324, 338, 375, 384, 392, 450, 512, 578, 648, 722, 800, 882, 896, 968, 1024, 1029, 1058, 1152, 1215, 1250, 1352, 1458, 1536, 1568, 1682, 1800, 1922, 2048, 2178, 2187, 2312, 2450, 2500, 2592

Offset: 1

Robert Munafo, Dec 12 2009

nonn

			8=2*2^2. 24=3*2^3. 375=3*5^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
R. Munafo, Generalized Cullen and Woodall Numbers

Cf. A171606. Union of the "KN^K" sequences A001105, A117642, A141046, ... or of the "NK^N" sequences A036289, A036290, A018215, A036291, ... but omitting the trivial initial terms.

is(n)=if(n<8, return(0)); for(a=2,logint(n\2,2), if(n%a==0 && ispower(n/a,a), return(1))); 0 \\ Charles R Greathouse IV, Feb 19 2017

list(lim)=my(v=List()); if(lim<8,return([])); for(a=2,logint(lim\2,2), for(b=2,sqrtnint(lim\a,a), listput(v,a*b^a))); Set(v) \\ Charles R Greathouse IV, Feb 19 2017

a(n) = 2n^2 - O(n^(5/3)). - Charles R Greathouse IV, Feb 19 2017

A050916 a(n) = n*5^n + 1.

Original entry on oeis.org

1, 6, 51, 376, 2501, 15626, 93751, 546876, 3125001, 17578126, 97656251, 537109376, 2929687501, 15869140626, 85449218751, 457763671876, 2441406250001, 12969970703126, 68664550781251, 362396240234376, 1907348632812501, 10013580322265626, 52452087402343751

Offset: 0

N. J. A. Sloane, Dec 30 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-35,25).

Cf. A002064, A036291.

[ n*5^n+1: n in [0..20]]; // Vincenzo Librandi, Sep 16 2011

Table[n 5^n+1,{n,0,20}] (* or *) LinearRecurrence[{11,-35,25},{1,6,51},20] (* Harvey P. Dale, Sep 15 2011 *)

a(n) = 11*a(n-1) - 35*a(n-2) + 25*a(n-3); a(0)=1, a(1)=6, a(2)=51. - Harvey P. Dale, Sep 15 2011
G.f.: -(20*x^2-5*x+1)/((x-1)*(5*x-1)^2). - Colin Barker, Oct 14 2012
From Elmo R. Oliveira, May 03 2025: (Start)
E.g.f.: exp(x)*(1 + 5*x*exp(4*x)).
a(n) = A036291(n) + 1. (End)

A158749 a(n) = n*9^n.

Original entry on oeis.org

0, 9, 162, 2187, 26244, 295245, 3188646, 33480783, 344373768, 3486784401, 34867844010, 345191655699, 3389154437772, 33044255768277, 320275094369454, 3088366981419735, 29648323021629456, 283512088894331673, 2701703435345984178, 25666182635786849691, 243153309181138576020

Offset: 0

Zerinvary Lajos, Mar 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Cf. A001019, A018215, A036289, A036290, A036291, A036292, A036293, A036294.

Cf. A038299, A126431.

[n*9^n: n in [0..20]]; // Vincenzo Librandi, Feb 25 2014

Table[n 9^n, {n, 0, 20}] (* Vincenzo Librandi, Feb 25 2014 *)

a(n) = n*9^n; \\ Joerg Arndt, Feb 23 2014

a(n) = n*9^n.
From R. J. Mathar, Mar 26 2009: (Start)
a(n) = 18*a(n-1) - 81*a(n-2) = A038299(n,1).
G.f.: 9*x/(1-9*x)^2. (End)
a(n) = A001019(n)*n. - Omar E. Pol, Mar 26 2009
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(9/8).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(10/9). (End)
E.g.f.: 9*x*exp(9*x). - Elmo R. Oliveira, Sep 09 2024

A219462 a(n) = Sum_{k = 1..2n} binomial(2n,k) * Fibonacci(2*k).

Original entry on oeis.org

0, 5, 75, 1000, 13125, 171875, 2250000, 29453125, 385546875, 5046875000, 66064453125, 864794921875, 11320312500000, 148184814453125, 1939764404296875, 25391845703125000, 332383575439453125, 4350957489013671875, 56954772949218750000, 745547657012939453125

Offset: 0

Reinhard Zumkeller, Nov 20 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
Doron Zeilbeger, An Enquiry Concerning Human (and Computer!) [Mathematical] Understanding, (2007); page 10.
Index entries for linear recurrences with constant coefficients, signature (15,-25).

Cf. A000045, A000351, A001906, A007318, A034870, A036291, A087453.

a219462 = sum . zipWith (*) a001906_list . a034870_row

Table[Sum[Binomial[2n,k]Fibonacci[2k],{k,2n}],{n,0,20}] (* Harvey P. Dale, Aug 26 2017 *)

a(n) = sum(k = 1, 2*n, binomial(2*n,k) * fibonacci(2*k)); \\ Michel Marcus, Jan 26 2022

a(n) = Sum_{k=1..n} A034870(n,k)*A001906(k).
a(n) = 5^n * Fibonacci(2*n) = A000351(n) * A001906(n).
G.f.: 5*x/(25*x^2-15*x+1). - Colin Barker, Dec 03 2012
E.g.f.: 2*exp(15*x/2)*sinh(5*sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Oct 19 2023

A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

Original entry on oeis.org

8, 18, 24, 50, 64, 81, 98, 160, 242, 324, 338, 375, 384, 578, 722, 896, 1029, 1058, 1215, 1682, 1922, 2048, 2500, 2738, 3362, 3698, 3993, 4374, 4418, 4608, 5618, 6591, 6962, 7442, 8978, 9604, 10082, 10240, 10658, 12482, 13778, 14739, 15309, 15625, 15842, 18818

Offset: 1

Amiram Eldar, Apr 16 2022

nonn

Each term in this sequence has a single presentation in the form k*p^k.

			8 is a term since 8 = 2*2^2.
18 is a term since 18 = 2*3^2.
24 is a term since 24 = 3*2^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Lindqvist and Jaak Peetre, On the remainder in a series of Mertens, Expositiones Mathematicae, Vol. 15 (1997), pp. 467-478. See eq. (3).
Eric Weisstein's World of Mathematics, Mertens Constant. See eq. (5).

Subsequences: A036289 \ {0, 2}, A036290 \ {0, 3}, A036291 \ {0, 5}, A036293 \ {0, 7}, A073113 \ {2}, A079704, A100042, A104126.

Cf. A001620, A077761, A143524.

addP[p_, n_] := Module[{k = 2, s = {}, m}, While[(m = k*p^k) <= n, k++; AppendTo[s, m]]; s]; seq[max_] := Module[{m = Floor[Sqrt[max/2]], s = {}, ps}, ps = Select[Range[m], PrimeQ]; Do[s = Join[s, addP[p, max]], {p, ps}]; Sort[s]]; seq[2*10^4]

Sum_{n>=1} 1/a(n) = -A143524 = gamma - B_1, where gamma is Euler's constant (A001620), and B_1 is Mertens's constant (A077761).

A364881 First significant digit of the decimal expansion of n/(2^n).

Original entry on oeis.org

5, 5, 3, 2, 1, 9, 5, 3, 1, 9, 5, 2, 1, 8, 4, 2, 1, 6, 3, 1, 1, 5, 2, 1, 7, 3, 2, 1, 5, 2, 1, 7, 3, 1, 1, 5, 2, 1, 7, 3, 1, 9, 4, 2, 1, 6, 3, 1, 8, 4, 2, 1, 5, 2, 1, 7, 3, 2, 1, 5, 2, 1, 6, 3, 1, 8, 4, 2, 1, 5, 3, 1, 7, 3, 1, 1, 5, 2, 1, 6, 3, 1, 8, 4, 2, 1, 5

Offset: 1

Ejder Aysun, Aug 10 2023

a(n) is also the first digit of n*5^n = A036291(n).

			n     n/(2^n)
1     0.5                            a(1) = 5
2     0.5                            a(2) = 5
3     0.375                          a(3) = 3
4     0.25                           a(4) = 2
5     0.15625                        a(5) = 1
6     0.9375                         a(6) = 9
7     0.0546875                      a(7) = 5
8     0.03125                        a(8) = 3
9     0.017578125                    a(9) = 1
10    0.009765625                    a(10) = 9
...

Cf. A000030, A000079, A000799, A036291, A111395.

a:= n-> parse((""||(n*5^n))[1]):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 18 2023

Table[Floor[n/(2^n)/10^Floor[Log10[n/(2^n)]]], {n, 100000}]

def A364881(n): return (n*5**(m:=len(str((1<>n-m) % 10 # Chai Wah Wu, Aug 24 2023

a(n) = floor(n/(2^n)/10^floor(log_10(n/(2^n)))), for n > 0.
a(n) = floor(n/A000079(n)/10^floor(log_10(n/A000079(n)))).
a(n) = floor(A036291(n)/10^floor(log_10(A036291(n)))).
a(n) = A000030(A036291(n)).

A134574 Array, a(n,k) = total size of all n-length words on a k-letter alphabet, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 24, 18, 4, 5, 64, 81, 32, 5, 6, 160, 324, 192, 50, 6, 7, 384, 1215, 1024, 375, 72, 7, 8, 896, 4374, 5120, 2500, 648, 98, 8, 9, 2048, 15309, 24576, 15625, 5184, 1029, 128, 9, 10, 4608, 52488, 114688, 93750, 38880, 9604, 1536, 162, 10

Offset: 1

Ross La Haye, Jan 22 2008

			a(2,2) = 8 because there are 2^2 = 4 2-length words on a 2 letter alphabet, each of size 2 and 2*4 = 8.
Array begins:
==================================================================
n\k|  1     2       3        4         5         6          7  ...
---|--------------------------------------------------------------
1  |  1     2       3        4         5         6          7  ...
2  |  2     8      18       32        50        72         98  ...
3  |  3    24      81      192       375       648       1029  ...
4  |  4    64     324     1024      2500      5184       9604  ...
5  |  5   160    1215     5120     15625     38880      84035  ...
6  |  6   384    4374    24576     93750    279936     705894  ...
7  |  7   896   15309   114688    546875   1959552    5764801  ...
8  |  8  2048   52488   524288   3125000  13436928   46118408  ...
9  |  9  4608  177147  2359296  17578125  90699264  363182463  ...
... - _Franck Maminirina Ramaharo_, Aug 07 2018

Cf. a(n, 1) = a(1, k) = A000027(n); a(n, 2) = A036289(n); a(n, 3) = A036290(n); a(n, 4) = A018215(n); a(n, 5) = A036291(n); a(n, 6) = A036292(n); a(n, 7) = A036293(n); a(n, 8) = A036294(n); a(2, k) = A001105(k); a(3, k) = A117642(k); a(n, n) = A007778(n); a(n, n+1) = A066274(n+1): sum[a(i-1, n-i+1), {i, 1, n}] = A062807(n).

t[n_, k_] := Sum[k^n, {j, n}]; Table[ t[n - k + 1, k], {n, 10}, {k, n}] // Flatten (* Robert G. Wilson v, Aug 07 2018 *)

a(n,k) = n*k^n.
O.g.f. (by columns): (k*x)/(-1+k*x)^2.
E.g.f. (by columns): k*x*exp(k*x).
a(n,k) = Sum[k^n,{j,1,n}] = n*Sum[C(n,m)*(k-1)^m,{m,0,n}]. - Ross La Haye, Jan 26 2008

cp's OEIS Frontend

A064751 a(n) = n*5^n - 1.

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A171607 Expressible as A*B^A in a nontrivial way.

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A050916 a(n) = n*5^n + 1.

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Magma

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A158749 a(n) = n*9^n.

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Magma

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A219462 a(n) = Sum_{k = 1..2*n} binomial(2*n,k) * Fibonacci(2*k).

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A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

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Mathematica

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A364881 First significant digit of the decimal expansion of n/(2^n).

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A134574 Array, a(n,k) = total size of all n-length words on a k-letter alphabet, read by antidiagonals.

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A219462 a(n) = Sum_{k = 1..2n} binomial(2n,k) * Fibonacci(2*k).