A001017 -id:A001017 - cp's OEIS frontend

A103623 a(n) = n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1.

1, 10, 1023, 29524, 349525, 2441406, 12093235, 47079208, 153391689, 435848050, 1111111111, 2593742460, 5628851293, 11488207654, 22250358075, 41189313616, 73300775185, 125999618778, 210027483919, 340614792100, 538947368421, 833994048910, 1264758228163

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Mar 25 2005

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001017.

With[{eq=Total[n^Range[0,9]]},Table[eq,{n,0,20}]] (* Harvey P. Dale, Dec 16 2011 *)

a(n)=n^9+n^8+n^7+n^6+n^5+n^4+n^3+n^2+n+1 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (341*x^8 +11352*x^7 +77440*x^6 +153824*x^5 +99330*x^4 +19624*x^3 +968*x^2 +1)/(x -1)^10. - Colin Barker, Oct 29 2012

a(n) = (n^10-1)/(n-1) with a(1) = 10. - Arkadiusz Wesolowski, Mar 30 2013

A256581 Number of conditions on m under which m^n + (m+1)^n + ... + (m+k)^n is composite for every k>=1 (see comment).

Original entry on oeis.org

2, 3, 2, 7, 5, 7, 7, 11, 5, 7, 7, 31, 23, 11, 9, 15, 17, 31, 31, 47, 23, 15, 29, 47, 23, 15, 7, 15, 11, 31, 47, 95, 47, 15, 11, 127, 95, 47, 39, 63, 71, 63, 63, 95, 47, 31, 71, 95, 71, 47, 31, 31, 47, 63, 39, 47, 23, 15, 23, 255, 191, 127, 111, 95, 71, 127

Offset: 1

Vladimir Shevelev, Apr 02 2015

nonn

We consider a(n), n>=2, conditions of the form: all numbers P_i(m) are composite, i = 1, ..., a(n), where P_i(m) is a polynomial of power n+1. It could be proved that S_k(m)= m^n + (m+1)^n + ... + (m+k)^n, as a polynomial in m of degree n+1, is divisible by k+1. Let S*_k(m) = S_k(m)/(k+1). So we have
S_k(m)=S*_k(m)*(k+1)=(T_k(m)/b(n))*(k+1), (1)
where b(n)=A064538(n) and, by the definition of A064538, T_k(m) = b(n)*S*_k(m) is a polynomial with integer coefficients.
It is clear that (1) could be prime only if k+1>=2 is a divisor of b(n). In this case we should require that (1) be a composite number. We have exactly A000005(b(n))-1 such requirements. In case of n=1, a(n)=2 (see A089306, A077654).
Remark. Sometimes some considered conditions satisfy trivially. For example, both a(3)=2 conditions for every  m>=2 evidently hold, such that every number of the form m^3 + (m+1)^3 + ... +(m+k)^3 is composite.
Note that essentially this method is useful only in case of even n. Indeed, according to our comment in A001017, in case of odd n>=3 the number m^n + (m+1)^n + ... + (m+k)^n is composite for every k>=1. - Vladimir Shevelev, Apr 06 2015

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000005, A064538, A089306 (a(1)=2), A256385 (a(2)=3), A256546 (a(4)=7).

For n>=2, a(n) = A000005(A064538(n))-1.

More terms from Peter J. C. Moses, Apr 02 2015

A036087 Centered cube numbers: a(n) = (n+1)^9 + n^9.

Original entry on oeis.org

1, 513, 20195, 281827, 2215269, 12030821, 50431303, 174571335, 521638217, 1387420489, 3357947691, 7517728043, 15764279725, 31265546157, 59104406159, 107162836111, 187307353233, 316947166865

Offset: 0

N. J. A. Sloane

Never prime nor semiprime, as a(n) = (2n+1) * (n^2 + n + 1) * (n^6 + 3n^5 + 12n^4 + 19n^3 + 15n^2 + 6n + 1). - Jonathan Vos Post, Aug 26 2011

Triprimes (A014612) if n = 2, 5, 6, 14, 21, 75, 90, ... - R. J. Mathar, Aug 27 2011

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Cf. A036085, A100267, A154535, A100266, A152913, A194155, A194185, A194216.

[(n+1)^9+n^9: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

Total/@Partition[Range[0,20]^9,2,1] (* Harvey P. Dale, Jan 31 2015 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,513,20195,281827,2215269,12030821,50431303,174571335,521638217,1387420489},20] (* Harvey P. Dale, Jan 21 2023 *)

a(n)=(n+1)^9+n^9 \\ Charles R Greathouse IV, Jan 31 2017

a(n) = A001017(n+1) + A001017(n).

G.f.: (1+x)*(x^8 + 502*x^7 + 14608*x^6 + 88234*x^5 + 156190*x^4 + 88234*x^3 + 14608*x^2 + 502*x + 1) / (x-1)^10. - R. J. Mathar, Aug 27 2011

A343289 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^(k^9).

Original entry on oeis.org

1, 512, 19683, 393472, 1953125, 20155392, 40353607, 290936320, 581140575, 2000000000, 2357947691, 18064270080, 10604499373, 41322093568, 76886718750, 209122656384, 118587876497, 694262555136, 322687697779, 1792500000000, 1588560093162, 2414538435584, 1801152661463

Offset: 1

Ilya Gutkovskiy, Apr 10 2021

nonn

Cf. A001017, A001055, A023878, A050367, A329125, A343283, A343284, A343285, A343286, A343287, A343288.

A016953 a(n) = (6*n + 3)^9.

Original entry on oeis.org

19683, 387420489, 38443359375, 794280046581, 7625597484987, 46411484401953, 208728361158759, 756680642578125, 2334165173090451, 6351461955384057, 15633814156853823, 35452087835576229, 75084686279296875, 150094635296999121, 285544154243029527, 520411082988487293

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016761, A016945, A016946, A016947, A016948, A016949, A016950, A016951, A016952.

Subsequence of A001017.

[(6*n+3)^9: n in [0..20]]; // Vincenzo Librandi, May 06 2011

(6*Range[0,20]+3)^9  (* Harvey P. Dale, Jan 19 2012 *)

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10). - Harvey P. Dale, Jan 19 2012
From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^9 = A016947(n)^3.
a(n) = 3^9*A016761(n).
Sum_{n>=0} 1/a(n) = 511*zeta(9)/10077696.
Sum_{n>=0} (-1)^n/a(n) = 277*Pi^9/162533081088. (End)

A016965 a(n) = (6*n + 4)^9.

Original entry on oeis.org

262144, 1000000000, 68719476736, 1207269217792, 10578455953408, 60716992766464, 262144000000000, 922190162669056, 2779905883635712, 7427658739644928, 18014398509481984, 40353607000000000, 84590643846578176, 167619550409708032, 316478381828866048, 572994802228616704

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016797, A016957, A016958, A016959, A016960, A016961, A016962, A016963, A016964.

Subsequence of A001017.

[(6*n+4)^9: n in [0..20]]; // Vincenzo Librandi, May 07 2011

(6*Range[0,20]+4)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{262144,1000000000,68719476736,1207269217792,10578455953408,60716992766464,262144000000000,922190162669056,2779905883635712,7427658739644928},20] (* Harvey P. Dale, Mar 04 2016 *)

From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016957(n)^9 = A016958(n)^3.
a(n) = 2^9*A016797(n).
Sum_{n>=0} 1/a(n) = 9841*zeta(9)/10077696 - 809*Pi^9/(14285134080*sqrt(3)). (End)

A016977 a(n) = (6*n + 5)^9.

Original entry on oeis.org

1953125, 2357947691, 118587876497, 1801152661463, 14507145975869, 78815638671875, 327381934393961, 1119130473102767, 3299763591802133, 8662995818654939, 20711912837890625, 45848500718449031, 95151694449171437, 186940255267540403, 350356403707485209, 630249409724609375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016969, A016970, A016971, A016972, A016973, A016974, A016975, A016976.

Subsequence of A001017.

[(6*n+5)^9: n in [0..35]]; // Vincenzo Librandi, May 13 2011

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^9 = A016971(n)^3.
Sum_{n>=0} 1/a(n) = 5028751*zeta(9)/10077696 - 15371*Pi^9/(529079040*sqrt(3)). (End)

A240932 a(n) = n^9 - n^8.

Original entry on oeis.org

0, 0, 256, 13122, 196608, 1562500, 8398080, 34588806, 117440512, 344373768, 900000000, 2143588810, 4729798656, 9788768652, 19185257728, 35880468750, 64424509440, 111612119056, 187339329792, 305704134738, 486400000000, 756457187220, 1152393344256, 1722841676182

Offset: 0

Martin Renner, Aug 03 2014

For n>1 number of 9-digit positive integers in base n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001016, A001017.

Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240931, A240933.

[n^9-n^8: n in [0..40]]; // Vincenzo Librandi, Aug 15 2016

Table[n^9 - n^8, {n, 0, 40}] (* Vincenzo Librandi, Aug 15 2016 *)

vector(100, n, (n-1)^9 - (n-1)^8) \\ Derek Orr, Aug 03 2014

a(n) = n^8*(n-1) = n^9 - n^8.
a(n) = A001017(n) - A001016(n).
G.f.: 2*x^2*(x^7+374*x^6+9327*x^5+49780*x^4+78095*x^3+38454*x^2+5281*x+128) / (x-1)^10. - Colin Barker, Aug 08 2014
Sum_{n>=2} 1/a(n) = 8 - Sum_{k=2..8} zeta(k). - Amiram Eldar, Jul 05 2020

A240933 a(n) = n^10 - n^9.

Original entry on oeis.org

0, 0, 512, 39366, 786432, 7812500, 50388480, 242121642, 939524096, 3099363912, 9000000000, 23579476910, 56757583872, 127253992476, 268593608192, 538207031250, 1030792151040, 1897406023952, 3372107936256, 5808378560022, 9728000000000, 15885600931620, 25352653573632

Offset: 0

Martin Renner, Aug 03 2014

For n>1 number of 10-digit positive integers in base n.

Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A001017, A008454.

Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240931, A240932.

[n^10-n^9 : n in [0..30]]; // Wesley Ivan Hurt, Aug 03 2014

A240933:=n->n^10-n^9: seq(A240933(n), n=0..30); # Wesley Ivan Hurt, Aug 03 2014

Table[n^10 - n^9, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 03 2014 *)
CoefficientList[Series[2 (256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 03 2014 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,0,512,39366,786432,7812500,50388480,242121642,939524096,3099363912,9000000000},40] (* Harvey P. Dale, Oct 19 2022 *)

vector(100, n, (n-1)^10 - (n-1)^9) \\ Derek Orr, Aug 03 2014

a(n) = n^9*(n-1) = n^10 - n^9.
a(n) = A008454(n) - A001017(n). - Michel Marcus, Aug 03 2014
G.f.: 2*(256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11. - Wesley Ivan Hurt, Aug 03 2014
Recurrence: a(n) = 11*a(n-1)-55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+330*a(n-7)-165*a(n-8)+55*a(n-9)-11*a(n-10)+a(n-11). - Wesley Ivan Hurt, Aug 03 2014
Sum_{n>=2} 1/a(n) = 9 - Sum_{k=2..9} zeta(k). - Amiram Eldar, Jul 05 2020

A319358 a(n) = (10^n - 1)^9.

Original entry on oeis.org

0, 387420489, 913517247483640899, 991035916125874083964008999, 999100359916012598740083996400089999, 999910003599916001259987400083999640000899999, 999991000035999916000125999874000083999964000008999999

Offset: 0

Seiichi Manyama, Sep 17 2018

nonn

Number of 9 in a(n) is 5*n-8 for n > 2.

			n|   a(n) can be divided into 9 parts for n > 3.
-+------------------------------------------------------
3|   99   1035   9   16125      874083   9   64008   999
4|  999  10035  99  160125  9  8740083  99  640008  9999
5| 9999 100035 999 1600125 99 87400083 999 6400008 99999

Seiichi Manyama, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (1111111111, -112233445443322110, 1123457901110987543211000, -1123570145779775409653211000000, 112358025801220975197532110000000000, -1123570145779775409653211000000000000000, 1123457901110987543211000000000000000000000, -112233445443322110000000000000000000000000000, 1111111111000000000000000000000000000000000000, -1000000000000000000000000000000000000000000000).

Cf. A001017, A002283, A059988, A272067, A272068.

a[n_]:=(10^n - 1)^9 ; Array[a,  50, 0] (* Stefano Spezia, Sep 17 2018 *)

PARI
```
a(n) = (10^n-1)^9;
```

a(n) = A002283(n)^9.

cp's OEIS Frontend

A103623 a(n) = n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1.

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A256581 Number of conditions on m under which m^n + (m+1)^n + ... + (m+k)^n is composite for every k>=1 (see comment).

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A036087 Centered cube numbers: a(n) = (n+1)^9 + n^9.

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A343289 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^(k^9).

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A016953 a(n) = (6*n + 3)^9.

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A016965 a(n) = (6*n + 4)^9.

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A016977 a(n) = (6*n + 5)^9.

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A240932 a(n) = n^9 - n^8.

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A240933 a(n) = n^10 - n^9.