A001015 -id:A001015 - cp's OEIS frontend

A024108 a(n) = 9^n-n^7.

1, 8, -47, -1458, -9823, -19076, 251505, 3959426, 40949569, 382637520, 3476784401, 31361572438, 282393704673, 2541803079812, 22876687041457, 205890961235274, 1853019920416385, 16677181289327896, 150094634684779089, 1350851716779120350

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (17,-100,308,-574,686,-532,260,-73,9).

[9^n-n^7: n in [0..25]]; // Vincenzo Librandi, Jul 06 2011

Table[9^n - n^7, {n, 0, 19}] (* Michael De Vlieger, Mar 20 2015 *)
LinearRecurrence[{17,-100,308,-574,686,-532,260,-73,9},{1,8,-47,-1458,-9823,-19076,251505,3959426,40949569},20] (* Harvey P. Dale, Mar 23 2018 *)

Vec(-(10*x^8+1071*x^7+10627*x^6+20497*x^5+8373*x^4-167*x^3-83*x^2-9*x+1)/((x-1)^8*(9*x-1)) + O(x^100)) \\ Colin Barker, Mar 20 2015

G.f.: -(10*x^8+1071*x^7+10627*x^6+20497*x^5+8373*x^4-167*x^3-83*x^2-9*x+1) / ((x-1)^8*(9*x-1)). - Colin Barker, Mar 20 2015

a(n) = A001019(n) - A001015(n). - Michel Marcus, Mar 20 2015

A024121 a(n) = 10^n - n^7.

Original entry on oeis.org

1, 9, -28, -1187, -6384, 21875, 720064, 9176457, 97902848, 995217031, 9990000000, 99980512829, 999964168192, 9999937251483, 99999894586496, 999999829140625, 9999999731564544, 99999999589661327, 999999999387779968

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (18,-108,336,-630,756,-588,288,-81,10).

Cf. A011557 (10^n), A001015 (n^7).

List([0..20],n->10^n-n^7); # Muniru A Asiru, Oct 16 2018

[10^n-n^7: n in [0..20]]; // Vincenzo Librandi, Jul 01 2011

seq(10^n-n^7,n=0..20); # Muniru A Asiru, Oct 16 2018

a[n_]:=10^n - n^7; Array[a, 50, 0] (* Stefano Spezia, Oct 04 2018 *)
LinearRecurrence[{18,-108,336,-630,756,-588,288,-81,10},{1,9,-28,-1187,-6384,21875,720064,9176457,97902848},20] (* Harvey P. Dale, Sep 16 2021 *)

a(n)=10^n-n^7 \\ Charles R Greathouse IV, Jul 01 2011

From Stefano Spezia, Oct 04 2018: (Start)
a(n) = 18*a(n - 1) - 108*a(n - 2) + 336*a(n - 3) - 630*a(n - 4) + 756*a(n - 5) - 588*a(n - 6) + 288*a(n - 7) - 81*a(n - 8) + 10*a(n - 9) for n > 8.
G.f.: -((1 - 9*x - 82*x^2 - 47*x^3 + 9564*x^4 + 22913*x^5 + 11818*x^6 + 1191*x^7 + 11*x^8)/((-1 + x)^8*(-1 + 10*x))).
E.g.f.: exp(x)*(exp(9*x) - x - 63*x^2 - 301*x^3 - 350*x^4 - 140*x^5 - 21*x^6 - x^7).
(End)

A084641 Binomial transform of n^7.

Original entry on oeis.org

0, 1, 130, 2574, 25904, 183200, 1040112, 5076400, 22171648, 88915968, 333209600, 1181548544, 4001402880, 13033885696, 41061830656, 125666611200, 374947708928, 1093874155520, 3128047828992, 8785866391552, 24280799641600, 66124498599936, 177683966197760

Offset: 0

Paul Barry, Jun 08 2003

The binomial transforms of n, n^2, n^3, n^4, n^5, n^6 are A001787, A001788, A058645, A058649, A059338, A056468 respectively.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Cf. A001015, A001787, A001788, A058645, A058649, A059338, A056468.

[n^2*(n^5+21*n^4+105*n^3+35*n^2-210*n+112)*2^(n-7): n in [0..40]]; // G. C. Greubel, Mar 20 2023

LinearRecurrence[{16,-112,448,-1120,1792,-1792,1024,-256}, {0,1,130, 2574,25904,183200,1040112,5076400}, 41] (* Amiram Eldar, Nov 26 2021 *)

[n^2*(n^5+21*n^4+105*n^3+35*n^2-210*n+112)*2^(n-7) for n in range(41)] # G. C. Greubel, Mar 20 2023

a(n) = n^2*(n^5 + 21*n^4 + 105*n^3 + 35*n^2 - 210*n + 112)*2^(n-7).
a(n) = Sum_{k=0..n} C(n, k)*k^7.
G.f.: x*(1+114*x+606*x^2-1168*x^3-96*x^4+816*x^5-272*x^6)/(1-2*x)^8. - Colin Barker, Sep 20 2012

A085479 Product of three solutions of the Diophantine equation x^3 - y^3 = z^2.

Original entry on oeis.org

728, 93184, 1592136, 11927552, 56875000, 203793408, 599539304, 1526726656, 3482001432, 7280000000, 14186660488, 26085556224, 45680920376, 76741030912, 124385625000, 195421011968, 298726553944, 445696183296, 650738625992

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Aug 15 2003

Parametric representation of the solution is (x,y,z) = (8n^2, 7n^2, 13n^3), thus getting a(n) = 728*n^7.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001015 (n^7), A085377.

728*Range[20]^7 (* Harvey P. Dale, May 27 2012 *)

Vec(728*x*(1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6) / (1 - x)^8 + O(x^25)) \\ Colin Barker, Oct 25 2019

a(n) = 728*n^7.
From Colin Barker, Oct 25 2019: (Start)
G.f.: 728*x*(1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.
(End)

More terms from Matthew Conroy, Jan 16 2006

A134010 a(n) = n^(initial digit of n).

Original entry on oeis.org

1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 2560000, 2825761, 3111696

Offset: 0

Reinhard Zumkeller, Oct 02 2007

a(n) = n^A000030(n).

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A001017, A001016, A001015, A001014, A000584, A000583, A000578, A000290, A000027.

a[n_]:= n^First[IntegerDigits[n]];Join[{1},Array[a,42]] (* James C. McMahon, Mar 30 2025 *)

A168194 a(n) = n^4*(n^3 + 1)/2.

Original entry on oeis.org

0, 1, 72, 1134, 8320, 39375, 140616, 412972, 1050624, 2394765, 5005000, 9750906, 17926272, 31388539, 52725960, 85455000, 134250496, 205211097, 306162504, 447001030, 640080000, 900641511, 1247296072, 1702552644, 2293401600

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 7 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=72, there are 2^7=128 oriented arrangements of two colors. Of these, 2^4=16 are achiral. That leaves (128-16)/2=56 chiral pairs. Adding achiral and chiral, we get 72. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi and Bruno Berselli, Table of n, a(n) for n = 0..1000 (first 480 terms from Vincenzo Librandi).
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Row 7 of A277504.

Cf. A001015 (oriented), A000583 (achiral).

[n^4*(n^3+1)/2: n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

Table[(n^4 (n^3+1))/2,{n,0,40}] (* Harvey P. Dale, Apr 29 2011 *)

vector(50, n, n--; n^4*(n^3+1)/2) \\ G. C. Greubel, Nov 14 2018

[n^4*(n^3+1)/2 for n in (0..50)] # G. C. Greubel, Nov 14 2018

G.f.: x*(1 + 64*x + 586*x^2 + 1208*x^3 + 605*x^4 + 56*x^5)/(1-x)^8. - Colin Barker, Apr 26 2012
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A001015(n) + A000583(n)) / 2 = (n^7 + n^4) / 2.
G.f.: (Sum_{j=1..7} S2(7,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..4} S2(4,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..6} A145882(7,k) * x^k / (1-x)^8.
E.g.f.: (Sum_{k=1..7} S2(7,k)*x^k + Sum_{k=1..4} S2(4,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>7, a(n) = Sum_{j=1..8} -binomial(j-9,j) * a(n-j). (End)
E.g.f.: x*(2 +70*x +307*x^2 +351*x^3 +140*x^4 +21*x^5 +x^6)*exp(x)/2. - G. C. Greubel, Nov 14 2018

A269792 a(n) = 5*n^4.

Original entry on oeis.org

0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805, 50000, 73205, 103680, 142805, 192080, 253125, 327680, 417605, 524880, 651605, 800000, 972405, 1171280, 1399205, 1658880, 1953125, 2284880, 2657205, 3073280, 3536405, 4050000, 4617605, 5242880, 5929605

Offset: 0

Ilya Gutkovskiy, Mar 31 2016

More generally, the ordinary generating function for the sequences of the form k*n^m, is k*Sum_{j>=1}x^j*j^m (when abs(x)<1).

More generally, the ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r, is (r + (p + q + k + m - 4*r)*x + (11*p + 3*q - k - 3*m + 6*r)*x^2 + (11*p - 3*q - k + 3*m - 4*r)*x^3 + (p - q + k - m + r)*x^4)/(1 - x)^5.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quartic polynomial
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. similar sequences of the form k*n^m, for k = 1...5, m = 1...10: A001477(k = 1, m = 1), A005843 (k = 2, m = 1), A008585 (k = 3, m = 1), A008586 (k = 4, m = 1), A008587 (k = 5, m = 1), A000290 (k = 1, m = 2), A001105 (k = 2, m = 2), A033428 (k = 3, m = 2), A016742 (k = 4, m = 2), A033429 (k = 5, m = 2), A000578 (k = 1, m = 3), A033431 (k = 2, m = 3), A117642 (k = 3, m = 3), A033430 (k = 4, m = 3), A244725 (k = 5, m = 3), A000583 (k = 1, m = 4), A244730 (k = 2, m = 4), A219056 (k = 3, m = 4), A141046 (k = 4, m = 4), this sequence(k = 5, m = 4), A000584 (k = 1, m = 5), A001014 (k = 1, m = 6), A106318 (k = 2, m = 6), A001015 (k = 1, m = 7), A001016 (k = 1, m = 8), A001017 (k = 1, m = 9), A008454 (k = 1, m = 10).

A269792:=n->5*n^4: seq(A269792(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[5 n^4, {n, 0, 33}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 80, 405, 1280}, 34]

x='x+O('x^99); concat(0, Vec(5*x*(1+11*x+11*x^2+x^3)/(1-x)^5)) \\ Altug Alkan, Mar 31 2016

G.f.: 5*x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5.
E.g.f.: 5*exp(x)^x*x*(1 + 7*x + 6*x^2 + x^3).
a(n) = 5*a(n-1) - 10*(9n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 5*A000583(n) = A008587(n)*A000578(n).
Sum_{n>=1} 1/a(n) = Pi^4/450 = (1/450)*A092425 = 0.216464646742...

A281678 Numbers k that have no digits in common with k^7.

Original entry on oeis.org

3, 7, 8, 33, 43, 77, 93, 272, 332, 662, 7757, 31333

Offset: 1

Robert Israel, Jan 26 2017

All terms have last digit 2, 3, 7 or 8.
Sequence is likely to be finite. If it exists, a(13) > 10^7.
In this sequence, the only terms with no repeated digits are 3, 7, 8, 43, 93. - Altug Alkan, Jan 26 2017
If it exists, a(13) > 10^17. - David Radcliffe, May 26 2025

			43 is a term because 43^7 = 271818611107 has no digit 4 or 3.

Cf. A001015. Contains A253576.

Cf. A281148.

select(t -> convert(convert(t,base,10),set) intersect convert(convert(t^7,base,10),set) = {},
{seq(seq(10*i+j,j=[2,3,7,8]),i=0..10^4});

Select[Range[40000], Intersection[IntegerDigits[#], IntegerDigits[ #^7]] == {}&] (* Vincenzo Librandi, Jan 27 2017 *)

isok(n) = #setintersect(Set(digits(n)), Set(digits(n^7))) == 0; \\ Michel Marcus, Jan 26 2017

A343526 Number of divisors of n^7.

Original entry on oeis.org

1, 8, 8, 15, 8, 64, 8, 22, 15, 64, 8, 120, 8, 64, 64, 29, 8, 120, 8, 120, 64, 64, 8, 176, 15, 64, 22, 120, 8, 512, 8, 36, 64, 64, 64, 225, 8, 64, 64, 176, 8, 512, 8, 120, 120, 64, 8, 232, 15, 120, 64, 120, 8, 176, 64, 176, 64, 64, 8, 960, 8, 64, 120, 43, 64, 512, 8, 120, 64, 512, 8

Offset: 1

Seiichi Manyama, May 15 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=7 of A343656.

Cf. A000005, A001015.

Table[DivisorSigma[0, n^7], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI
```
a(n) = numdiv(n^7);
```

a(n) = prod(k=1, #f=factor(n)[, 2], 7*f[k]+1);

PARI
```
a(n) = sumdiv(n, d, 7^omega(d));
```

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 7^omega(k)*x^k/(1-x^k)))

for(n=1, 100, print1(direuler(p=2, n, (1 + 6*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 19 2021

a(n) = A000005(A001015(n)).
Multiplicative with a(p^e) = 7*e+1.
a(n) = Sum_{d|n} 7^omega(d).
G.f.: Sum_{k>=1} 7^omega(k) * x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 6/p^s). - Vaclav Kotesovec, Aug 19 2021

A016771 a(n) = (3*n)^7.

Original entry on oeis.org

0, 2187, 279936, 4782969, 35831808, 170859375, 612220032, 1801088541, 4586471424, 10460353203, 21870000000, 42618442977, 78364164096, 137231006679, 230539333248, 373669453125, 587068342272, 897410677851

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 (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A001015 (n^7).

[(3*n)^7: n in [0..25]]; // Vincenzo Librandi, May 09 2011

cp's OEIS Frontend

A024108 a(n) = 9^n-n^7.

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A024121 a(n) = 10^n - n^7.

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A084641 Binomial transform of n^7.

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A085479 Product of three solutions of the Diophantine equation x^3 - y^3 = z^2.

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A134010 a(n) = n^(initial digit of n).

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A168194 a(n) = n^4*(n^3 + 1)/2.

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A269792 a(n) = 5*n^4.

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