A009971 -id:A009971 - cp's OEIS frontend

A002042 a(n) = 7*4^n.

7, 28, 112, 448, 1792, 7168, 28672, 114688, 458752, 1835008, 7340032, 29360128, 117440512, 469762048, 1879048192, 7516192768, 30064771072, 120259084288, 481036337152, 1924145348608, 7696581394432, 30786325577728, 123145302310912, 492581209243648

Offset: 0

N. J. A. Sloane

Subsequence of A000069, the odious numbers. - Reinhard Zumkeller, Aug 26 2007
A rectangular prism with edge lengths 2^n, 2^(n+1) and 2^(n+2) has a surface area 2* (2^n*2^(n+1) + 2^(n+1)*2^(n+2) + 2^n*2^(n+2)) which equals 4*a(n). - J. M. Bergot, Aug 07 2013
x = A306472(n) and y = a(n) satisfy the Lebesgue-Ramanujan-Nagell equation x^2 + 3^(6*n+1) = 4*y^3 (see Theorem 2.1 in Chakraborty, Hoque and Sharma). - Stefano Spezia, Feb 18 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..500
K. Chakraborty, A. Hoque, R. Sharma, Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations, arXiv:1812.11874 [math.NT], 2018.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4).

First differences of A083597. Bisection of A005009.

Cf. A306472 (37*27^n), A009971 (27^n), A000302 (4^n), A000290 (n^2), A000578 (n^3).

[7*4^n: n in [0..30]]; // Vincenzo Librandi, May 31 2011

7*4^Range[0, 100] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
CoefficientList[Series[7/(1-4x), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 25 2015 *)
NestList[4#&,7,30] (* Harvey P. Dale, Mar 19 2021 *)

a(n)=7<<(2*n) \\ Charles R Greathouse IV, Apr 17 2012

[7*4^n for n in (0..30)] # G. C. Greubel, Feb 18 2019

From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 4*a(n-1), n > 0, with a(0) = 7.
G.f.: 7/(1-4*x). (End)
a(n) = 7*A000302(n). - Michel Marcus, Jun 24 2015
E.g.f.: 7*exp(4*x). - G. C. Greubel, Feb 18 2019

A342090 Numbers with at least one prime power p^e in their prime factorization such that p|e.

Original entry on oeis.org

4, 12, 16, 20, 27, 28, 36, 44, 48, 52, 54, 60, 64, 68, 76, 80, 84, 92, 100, 108, 112, 116, 124, 132, 135, 140, 144, 148, 156, 164, 172, 176, 180, 188, 189, 192, 196, 204, 208, 212, 216, 220, 228, 236, 240, 244, 252, 256, 260, 268, 270, 272, 276, 284, 292, 297, 300

Offset: 1

Amiram Eldar, Feb 27 2021

nonn

Numbers with a unitary divisor of the form p^(m*p) where p is a prime and m > 0.
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 19, 188, 1883, 18825, 188244, 1882429, 18824297, 188242957, 1882429628, ...
The asymptotic density of this sequence is 1 - Product_{p prime} 1 - (p - 1)/(p*(p^p - 1)) = 0.18824296270011399086...

			4 = 2^2 is a term since 2 divides 2.
8 = 2^3 is not a term since 2 does not divide 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A013929.
Subsequences: A000302, A000312, A009971, A017113, A051674.
Cf. A072873, A369070 (characteristic function).

q[n_] := AnyTrue[FactorInteger[n], Divisible[Last[#], First[#]] &]; Select[Range[2, 300], q]

Wrong term 1 removed by Amiram Eldar, Jan 16 2024

A227993 Let d(1) < d(2) < ... < d(q) denote the divisors of k. Sequence lists numbers k > 1 such that d(1)/d(2) + d(2)/d(3) + ... + d(q-1)/d(q) is an integer.

Original entry on oeis.org

4, 16, 27, 54, 64, 256, 729, 1024, 1296, 1536, 3125, 4096, 6250, 9375, 12500, 16384, 19683, 30720, 39366, 65536, 262144, 472392, 531441, 823543, 1048576, 1179648, 1647086, 2125764, 3294172, 4194304, 6291456, 6770688, 9765625, 11595672, 14348907, 16777216

Offset: 1

Michel Lagneau, Aug 06 2013

nonn

The sequence is infinite because the powers of 4 (A000302) are in the sequence: the divisors of 2^(2m) are {1, 2, 4, 8, ..., 2^(2m)} and Sum_{i=1..q-1} d(i)/d(i+1) = 1/2 + 2/4 + 4/8 + ... + 2^(2m-1)/2^(2m) = 1/2 + 1/2 + ... + 1/2 = 2m.
The powers of 27 (A009971) are also in the sequence.
In the general case, the numbers of the form p^(p*m) where p is prime are in the sequence.

			54 is in the sequence because the divisors of 54 are {1, 2, 3, 6, 9, 18, 27, 54} and 1/2 + 2/3 + 3/6 + 6/9 + 9/18 + 18/27 + 27/54 = 4 is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..55

Cf. A000302, A009971.

with(numtheory):for n from 1 to 2000000 do: x:=divisors(n):n1:=nops(x): s:=sum('x[i]/x[i+1]', 'i'=1..n1-1): if s=floor(s)then printf(`%d, `, n):else fi:od:

fQ[n_] := Module[{d = Divisors[n]}, IntegerQ[Total[Most[d]/Rest[d]]]]; t = {}; n = 1; While[Length[t] < 40, n++; If[fQ[n], AppendTo[t, n]]]; t (* T. D. Noe, Aug 06 2013 *)

is(n)=my(t,s);fordiv(n,d,s+=t/d;t=d);denominator(s)==1 && n>1 \\ Charles R Greathouse IV, Aug 06 2013

from sympy import divisors
from fractions import Fraction
def ok(n):
    if n < 2: return False
    divs = divisors(n)
    f = sum(Fraction(dn, dd) for dn, dd in zip(divs[:-1], divs[1:]))
    return f.denominator == 1
print([k for k in range(70000) if ok(k)]) # Michael S. Branicky, Feb 06 2022

A067424 Ninth column of triangle A067417.

Original entry on oeis.org

1, 11, 297, 8019, 216513, 5845851, 157837977, 4261625379, 115063885233, 3106724901291, 83881572334857, 2264802453041139, 61149666232110753, 1651040988266990331, 44578106683208738937, 1203608880446635951299

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (27).

Cf. A067417, A067423 (eighth column), A009971 (powers of 27).

[Ceiling(11*(3*9)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

CoefficientList[Series[(1-16x)/(1-27x),{x,0,30}],x] (* or *) LinearRecurrence[{27},{1,11},20] (* Harvey P. Dale, Apr 20 2022 *)

a(n) = A067417(n+8, 8).
a(n) = 11*(3*9)^(n-1), n >= 1, a(0)=1.
G.f.: (1-16*x)/(1-27*x).
E.g.f.: (16 + 11*exp(27*x))/27. - Stefano Spezia, Sep 30 2022

A218730 a(n) = (27^n - 1)/26.

Original entry on oeis.org

0, 1, 28, 757, 20440, 551881, 14900788, 402321277, 10862674480, 293292210961, 7918889695948, 213810021790597, 5772870588346120, 155867505885345241, 4208422658904321508, 113627411790416680717, 3067940118341250379360, 82834383195213760242721, 2236528346270771526553468

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 27 (A009971); q-integers for q=27.

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 (28,-27).

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, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009971.

[n le 2 select n-1 else 28*Self(n-1)-27*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{28, -27}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218730(n):=(27^n-1)/26$
makelist(A218730(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
a(n)=27^n\26
```

G.f.: x/((1-x)*(1-27*x)). - Vincenzo Librandi, Nov 07 2012
a(n) = floor(27^n/26). - Vincenzo Librandi, Nov 07 2012
a(n) = 28*a(n-1) - 27*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(14*x)*sinh(13*x)/13. - Elmo R. Oliveira, Aug 27 2024

A160104 Numerator of Hermite(n, 5/27).

Original entry on oeis.org

1, 10, -1358, -42740, 5512492, 304384600, -37142220680, -3034178687600, 348731717384080, 38877977386007200, -4187277821653825760, -608713688504523233600, 61068424818638825202880, 11260738942261526747094400, -1044883534589865025424443520

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 10/27, -1358/729, -42740/19683, 5512492/531441..

G. C. Greubel, Table of n, a(n) for n = 0..376

Cf. A009971 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(10/27)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 12 2018

Numerator[Table[HermiteH[n, 5/27], {n, 0, 30}]] (* or *) Table[27^n* HermiteH[n, 5/27], {n,0,30}] (* G. C. Greubel, Jul 12 2018 *)

a(n)=numerator(polhermite(n,5/27)) \\ Charles R Greathouse IV, Jan 29 2016

From G. C. Greubel, Jul 12 2018: (Start)
a(n) = 27^n * Hermite(n, 5/27).
E.g.f.: exp(10*x - 729*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(10/27)^(n-2*k)/(k!*(n-2*k)!)). (End)

A089683 a(n) = 3^(4*n).

Original entry on oeis.org

81, 6561, 531441, 43046721, 3486784401, 282429536481, 22876792454961, 1853020188851841, 150094635296999121, 12157665459056928801, 984770902183611232881, 79766443076872509863361, 6461081889226673298932241, 523347633027360537213511521, 42391158275216203514294433201

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jan 05 2004

Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (81).

Cf. A000244, A001019, A008586, A009971.

3^(4*Range[20]) (* or *) NestList[81#&,81,20] (* Harvey P. Dale, Apr 09 2012 *)

my(x='x+O('x^16)); Vec(-81*x/(81*x-1)) \\ Elmo R. Oliveira, Jul 09 2025

a(n) = 81^n.
G.f.: 81*x/(1-81*x). - Philippe Deléham, Nov 25 2008
From Elmo R. Oliveira, Jul 09 2025: (Start)
E.g.f.: exp(81*x) - 1.
a(n) = 81*a(n-1) for n > 1.
a(n) = A001019(n)^2 = A000244(A008586(n)). (End)

More terms from Harvey P. Dale, Apr 09 2012

A160087 Numerator of Hermite(n, 1/27).

Original entry on oeis.org

1, 2, -1454, -8740, 6342316, 63656312, -46108171016, -649081759408, 469281829870480, 8509453301475872, -6140897264957486816, -136349623665433187392, 98215011088057307180224, 2582003037826533660970880, -1856403314087385132972023936

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 2/27, -1454/729, -8740/19683, 6342316/531441..

G. C. Greubel, Table of n, a(n) for n = 0..376

Cf. A009971 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(2/27)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Sep 23 2018

Table[27^n*HermiteH[n, 1/27], {n, 0, 30}] (* G. C. Greubel, Sep 23 2018 *)

a(n)=numerator(polhermite(n, 1/27)) \\ Charles R Greathouse IV, Jan 29 2016

x='x+O('x^30); Vec(serlaplace(exp(2*x - 729*x^2))) \\ G. C. Greubel, Sep 23 2018

From G. C. Greubel, Sep 23 2018: (Start)
a(n) = 27^n * Hermite(n, 1/27).
E.g.f.: exp(2*x - 729*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(2/27)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160088 Numerator of Hermite(n, 2/27).

Original entry on oeis.org

1, 4, -1442, -17432, 6237580, 126613744, -44965503224, -1287479045408, 453768009722512, 16832227624528960, -5887014913080686624, -268961938417954983296, 93340097422316232142528, 5079118464249805316316928, -1748851732685582642764208000

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 4/27, -1442/729, -17432/19683, 6237580/531441...

G. C. Greubel, Table of n, a(n) for n = 0..375

Cf. A009971 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(4/27)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Sep 23 2018

Numerator[HermiteH[Range[0,20],2/27]] (* Harvey P. Dale, Mar 26 2016 *)
Table[27^n*HermiteH[n, 2/27], {n, 0, 30}] (* G. C. Greubel, Sep 23 2018 *)

a(n)=numerator(polhermite(n, 2/27)) \\ Charles R Greathouse IV, Jan 29 2016

x='x+O('x^30); Vec(serlaplace(exp(4*x - 729*x^2))) \\ G. C. Greubel, Sep 23 2018

From G. C. Greubel, Sep 23 2018: (Start)
a(n) = 27^n * Hermite(n, 2/27).
E.g.f.: exp(4*x - 729*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(4/27)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160103 Numerator of Hermite(n, 4/27).

Original entry on oeis.org

1, 8, -1394, -34480, 5821516, 247659488, -40457575736, -2490185806912, 392988531506320, 32189435503872128, -4899280026394954016, -508516209857615258368, 74506523384461350441664, 9493051794744527363939840, -1336252229871124217359780736

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 8/27, -1394/729, -34480/19683, 5821516/531441, ...

G. C. Greubel, Table of n, a(n) for n = 0..375

Cf. A009971 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(8/27)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Sep 24 2018

Table[27^n*HermiteH[n, 4/27], {n, 0, 30}] (* G. C. Greubel, Sep 24 2018 *)

a(n)=numerator(polhermite(n, 4/27)) \\ Charles R Greathouse IV, Jan 29 2016

x='x+O('x^30); Vec(serlaplace(exp(8*x - 729*x^2))) \\ G. C. Greubel, Sep 24 2018

From G. C. Greubel, Sep 24 2018: (Start)
a(n) = 27^n * Hermite(n, 4/27).
E.g.f.: exp(8*x - 729*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(8/27)^(n-2*k)/(k!*(n-2*k)!)). (End)

cp's OEIS Frontend

A002042 a(n) = 7*4^n.

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A342090 Numbers with at least one prime power p^e in their prime factorization such that p|e.

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A227993 Let d(1) < d(2) < ... < d(q) denote the divisors of k. Sequence lists numbers k > 1 such that d(1)/d(2) + d(2)/d(3) + ... + d(q-1)/d(q) is an integer.

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A067424 Ninth column of triangle A067417.

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A218730 a(n) = (27^n - 1)/26.

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Maxima

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A160104 Numerator of Hermite(n, 5/27).

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Magma

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A089683 a(n) = 3^(4*n).

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