A009966 -id:A009966 - cp's OEIS frontend

A009965 Powers of 21.

1, 21, 441, 9261, 194481, 4084101, 85766121, 1801088541, 37822859361, 794280046581, 16679880978201, 350277500542221, 7355827511386641, 154472377739119461, 3243919932521508681, 68122318582951682301, 1430568690241985328321, 30041942495081691894741, 630880792396715529789561, 13248496640331026125580781, 278218429446951548637196401

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 21), L(1, 21), P(1, 21), T(1, 21). Essentially same as Pisot sequences E(21, 441), L(21, 441), P(21, 441), T(21, 441). 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 21-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 (21).

Row 10 of A329332.

[21^n: n in [0..100]] // Vincenzo Librandi, Nov 21 2010

21^Range[0,20] (* or *) NestList[21#&,1,20] (* Harvey P. Dale, Aug 31 2023 *)

A009965(n):=21^n$
makelist(A009965(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

a(n)=21^n \\ Charles R Greathouse IV, Nov 18 2011

[lucas_number1(n,21,0) for n in range(1, 17)] # Zerinvary Lajos, Apr 29 2009

For A009966..A009992 we have g.f.: 1/(1-qx), e.g.f.: exp(qx), with q = 21, 22, ..., 48. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
a(n) = 21^n; a(n) = 21*a(n-1), n > 0, a(0)=1. - Vincenzo Librandi, Nov 21 2010
G.f.: 22/G(0) where G(k) = 1 - 2*x*(k+1)/(1 - 1/(1 - 2*x*(k+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 10 2013

A072978 Numbers of the form m*2^Omega(m), where m>1 is odd and Omega(m)=A001222(m), the number of prime factors of m.

Original entry on oeis.org

1, 6, 10, 14, 22, 26, 34, 36, 38, 46, 58, 60, 62, 74, 82, 84, 86, 94, 100, 106, 118, 122, 132, 134, 140, 142, 146, 156, 158, 166, 178, 194, 196, 202, 204, 206, 214, 216, 218, 220, 226, 228, 254, 260, 262, 274, 276, 278, 298, 302, 308, 314, 326, 334, 340, 346

Offset: 1

Reinhard Zumkeller, Aug 20 2002

(number of odd prime factors) = (number of even prime factors).
A000400, A011557, A001023, A001024, A009965, A009966 and A009975 are subsequences. - Reinhard Zumkeller, Jan 06 2008
Subsequence of A028260. - Reinhard Zumkeller, Sep 20 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001222, A007814, A087436.

Cf. A000400, A011557, A001023, A001024, A009965, A009966, A009975, A028260.

Join[{1}, Select[Range[2, 500, 2], First[#] == Total[Rest[#]] & [FactorInteger[#][[All, 2]]] &]] (* Paolo Xausa, Feb 19 2025 *)

isok(k) = {my(v = valuation(k, 2)); bigomega(k >> v) == v;} \\ Amiram Eldar, May 15 2025

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A072978(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def h(x,n): return sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,1,3,1,n))
    def f(x): return int(n+x-primepi(x>>1)-sum(h(x>>m,m) for m in range(2,x.bit_length()+1))) if x>1 else 1
    return bisection(f,n,n) # Chai Wah Wu, Apr 10 2025

A007814(a(n)) = A087436(a(n)). - Reinhard Zumkeller, Jan 06 2008

A100402 Digital root of 4^n.

Original entry on oeis.org

1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7

Offset: 0

Cino Hilliard, Dec 31 2004

Equals A141725 mod 9. - Paul Curtz, Sep 15 2008
Sequence is the digital root of A016777. - Odimar Fabeny, Sep 13 2010
Digital root of the powers of any number congruent to 4 mod 9. - Alonso del Arte, Jan 26 2014
Period 3: repeat [1, 4, 7]. - Wesley Ivan Hurt, Aug 26 2014
From Timothy L. Tiffin, Dec 02 2023: (Start)
The period 3 digits of this sequence are the same as those of A070403 (digital root of 7^n) but the order is different: [1, 4, 7] vs. [1, 7, 4].
The digits in this sequence appear in the decimal expansions of the following rational numbers: 49/333, 490/333, 4900/333, .... (End)

			4^2 = 16, digitalroot(16) = 7, the third entry.

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. Digital roots of powers of c mod 9: c = 2, A153130; c = 5, A070366; c = 7, A070403; c = 8, A010689.

Cf. A000302, A001022, A009966, A009975, A009984, A010872, A010888, A070403, A087752, A121013, A141725, A016777.

Table[PowerMod[4, n, 9], {n, 0, 100}] (* Timothy L. Tiffin, Dec 03 2023 *)
StringRepeat["1, 4, 7, ", 100] (* Timothy L. Tiffin, Dec 03 2023 *)

a(n)=[1,4,7][1+n%3]; \\ Joerg Arndt, Aug 26 2014

[power_mod(4, n, 9) for n in range(0, 105)] # Zerinvary Lajos, Nov 25 2009

a(n) = 4^n mod 9. - Zerinvary Lajos, Nov 25 2009
From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-3) for n>2.
G.f.: (1+4*x+7*x^2)/ ((1-x)*(1+x+x^2)). (End)
a(n) = A010888(A000302(n)). - Michel Marcus, Aug 25 2014
a(n) = 3*A010872(n) + 1. - Robert Israel, Aug 25 2014
a(n) = 4 - 3*cos(2*n*Pi/3) - sqrt(3)*sin(2*n*Pi/3). - Wesley Ivan Hurt, Jun 30 2016
a(n) = A153130(2n). - Timothy L. Tiffin, Dec 01 2023
a(n) = A010888(A001022(n)) = A010888(A009966(n)) = A010888(A009975(n)) = A010888(A009984(n)) = A010888(A087752(n)) = A010888(A121013(n)). - Timothy L. Tiffin, Dec 02 2023
a(n) = A010888(4*a(n-1)). - Stefano Spezia, Mar 20 2025

A009988 Powers of 44.

Original entry on oeis.org

1, 44, 1936, 85184, 3748096, 164916224, 7256313856, 319277809664, 14048223625216, 618121839509504, 27197360938418176, 1196683881290399744, 52654090776777588736, 2316779994178213904384, 101938319743841411792896, 4485286068729022118887424, 197352587024076973231046656

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 44), L(1, 44), P(1, 44), T(1, 44). Essentially same as Pisot sequences E(44, 1936), L(44, 1936), P(44, 1936), T(44, 1936). 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 44-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 (44).

Cf. A000079, A000302, A001020, A008776, A009966.

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

44^Range[0,20] (* Harvey P. Dale, May 22 2017 *)

G.f.: 1/(1-44*x). - Philippe Deléham, Nov 24 2008
a(n) = 44^n; a(n) = 44*a(n-1), a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 08 2025: (Start)
E.g.f.: exp(44*x).
a(n) = A000079(n)*A009966(n) = A000302(n)*A001020(n). (End)

A013727 a(n) = 22^(2*n + 1).

Original entry on oeis.org

22, 10648, 5153632, 2494357888, 1207269217792, 584318301411328, 282810057883082752, 136880068015412051968, 66249952919459433152512, 32064977213018365645815808, 15519448971100888972574851072, 7511413302012830262726227918848, 3635524038174209847159494312722432

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (484).

Bisection of A009966 (22^n).

Cf. A004171, A005408, A013708-A013726, A013728-A013729.

[22^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(22^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 26 2011

NestList[484*# &, 22, 15] (* Paolo Xausa, Jul 12 2025 *)

a(n)=22^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 484*a(n-1); a(0)=22.
G.f.: 22/(1-484*x). (End)
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: 22*exp(484*x).
a(n) = A004171(n)*A013716(n) = A009966(A005408(n)). (End)

A180729 Smallest power of 22 that begins with n.

Original entry on oeis.org

1, 22, 3011361496339065143296, 484, 5153632, 6221821273427820544, 705429498686404044207947776, 851643319086537701956194499721106030592, 9068298061633453450429559033030337013743616, 10648, 113379904

Offset: 1

Daniel Mondot, Sep 18 2010

Cf. A009966, A180727, A180730, A180731.

A013770 a(n) = 22^(3*n + 1).

Original entry on oeis.org

22, 234256, 2494357888, 26559922791424, 282810057883082752, 3011361496339065143296, 32064977213018365645815808, 341427877364219557396646723584, 3635524038174209847159494312722432, 38711059958478986452554295441868455936

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (10648).

Subsequence of A009966.

[22^(3*n+1): n in [0..10]]; // Vincenzo Librandi, Jun 27 2011

a(n)=22^(3*n+1) \\ Charles R Greathouse IV, Jul 10 2016

A013771 a(n) = 22^(3*n + 2).

Original entry on oeis.org

484, 5153632, 54875873536, 584318301411328, 6221821273427820544, 66249952919459433152512, 705429498686404044207947776, 7511413302012830262726227918848, 79981528839832616637508874879893504, 851643319086537701956194499721106030592

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (10648).

Subsequence of A009966.

[22^(3*n+2): n in [0..10]]; // Vincenzo Librandi, Jun 27 2011

A013816 a(n) = 22^(4*n+1).

Original entry on oeis.org

22, 5153632, 1207269217792, 282810057883082752, 66249952919459433152512, 15519448971100888972574851072, 3635524038174209847159494312722432, 851643319086537701956194499721106030592, 199502557355935975909450298726667414302359552

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (234256).

Subsequence of A009966.

[22^(4*n+1): n in [0..15]]; // Vincenzo Librandi, Jul 07 2011

Table[22^(4 n + 1), {n, 0, 10}] (* Wesley Ivan Hurt, Apr 11 2017 *)
NestList[234256#&,22,10] (* Harvey P. Dale, Apr 21 2022 *)

A013902 a(n) = 22^(5*n + 1).

Original entry on oeis.org

22, 113379904, 584318301411328, 3011361496339065143296, 15519448971100888972574851072, 79981528839832616637508874879893504, 412195366437884247746798137865015318806528, 2124303230726006271483826780841554627491524509696, 10947877107572929152919737180202022857988400441953615872

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (5153632).

Cf. A009966.

[22^(5*n+1): n in [0..10]]; // Vincenzo Librandi, May 28 2011

22^(5Range[0,10]+1) (* or *) LinearRecurrence[{5153632},{22},10] (* Harvey P. Dale, Feb 28 2012 *)

a(n) = 5153632*a(n-1), a(0)=22. - Vincenzo Librandi, May 28 2011

G.f.: 22/(1 - 5153632*x). - Wesley Ivan Hurt, May 25 2024

cp's OEIS Frontend

A009965 Powers of 21.

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A072978 Numbers of the form m*2^Omega(m), where m>1 is odd and Omega(m)=A001222(m), the number of prime factors of m.

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A100402 Digital root of 4^n.

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A009988 Powers of 44.

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A013727 a(n) = 22^(2*n + 1).

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A180729 Smallest power of 22 that begins with n.

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A013770 a(n) = 22^(3*n + 1).

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A013771 a(n) = 22^(3*n + 2).

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