cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A054612 a(n) = Sum_{d|n} phi(d)*5^(n/d).

Original entry on oeis.org

0, 5, 30, 135, 660, 3145, 15810, 78155, 391320, 1953405, 9768870, 48828175, 244157820, 1220703185, 6103593930, 30517584915, 152588282640, 762939453205, 3814699250430, 19073486328215, 95367441415140, 476837158360185, 2384185839844050, 11920928955078235, 59604645020345640
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Links

Crossrefs

Column k=5 of A185651.
Cf. A001869.

Programs

  • Mathematica
    a[n_]:= Sum[5^GCD[n,k],{k,n}]; Array[a,30,0] (* Stefano Spezia, Sep 02 2025 *)
  • PARI
    a(n) = if (n, sumdiv(n, d, eulerphi(d)*5^(n/d)), 0); \\ Michel Marcus, Apr 16 2021

Formula

a(n) = Sum_{k=1..n} 5^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021

A054613 a(n) = Sum_{d|n} phi(d)*6^(n/d).

Original entry on oeis.org

0, 6, 42, 228, 1344, 7800, 46956, 279972, 1681008, 10078164, 60474120, 362797116, 2176832112, 13060694088, 78364444284, 470185001040, 2821111589856, 16926659444832, 101559966840108, 609359740010604, 3656158500550080
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Links

Crossrefs

Column k=6 of A185651.
Cf. A054625.

Programs

  • PARI
    a(n) = if(n==0, 0, sumdiv(n, d, eulerphi(d)*6^(n/d))); \\ Altug Alkan, Mar 16 2018

Formula

a(n) = n * A054625(n).
a(n) = Sum_{k=1..n} 6^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021

A054614 a(n) = Sum_{d|n} phi(d)*7^(n/d).

Original entry on oeis.org

0, 7, 56, 357, 2464, 16835, 118104, 823585, 5767328, 40354335, 282492280, 1977326813, 13841410464, 96889010491, 678223896728, 4747561544985, 33232936339456, 232630513987319, 1628413638500376, 11398895185373269
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Links

Crossrefs

Column k=7 of A185651.
Cf. A054626.

Programs

  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*7^(n/d)); \\ Michel Marcus, Jul 11 2021

Formula

a(n) = Sum_{k=1..n} 7^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021

A054615 a(n) = Sum_{d|n} phi(d)*8^(n/d).

Original entry on oeis.org

0, 8, 72, 528, 4176, 32800, 262800, 2097200, 16781472, 134218800, 1073774880, 8589934672, 68719748256, 549755813984, 4398048608688, 35184372156480, 281474993496384, 2251799813685376, 18014398644225456, 144115188075856016
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Links

Crossrefs

Column k=8 of A185651.
Cf. A054627.

Programs

  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*8^(n/d)); \\ Michel Marcus, Jul 11 2021

Formula

a(n) = Sum_{k=1..n} 8^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021

A054616 a(n) = Sum_{d|n} phi(d)*9^(n/d).

Original entry on oeis.org

0, 9, 90, 747, 6660, 59085, 532350, 4783023, 43053480, 387422001, 3486843810, 31381059699, 282430082700, 2541865828437, 22876797238470, 205891132215735, 1853020231912080, 16677181699666713, 150094635685484490, 1350851717672992251
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Links

Crossrefs

Column k=9 of A185651.
Cf. A054628.

Programs

  • PARI
    a(n) = if(n==0, 0, sumdiv(n, d, eulerphi(d)*9^(n/d))); \\ Altug Alkan, Mar 16 2018

Formula

a(n) = Sum_{k=1..n} 9^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021

A054617 a(n) = Sum_{d|n} phi(d)*10^(n/d).

Original entry on oeis.org

0, 10, 110, 1020, 10120, 100040, 1001220, 10000060, 100010240, 1000002060, 10000100440, 100000000100, 1000001022240, 10000000000120, 100000010000660, 1000000000204080, 10000000100020480, 100000000000000160, 1000000001002002660
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Links

Crossrefs

Column k=10 of A185651.
Cf. A054629.

Formula

Dirichlet convolution of A000010 and A011557. - R. J. Mathar, Jan 11 2013
a(n) = Sum_{k=1..n} 10^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021

A343489 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=1..n} k^(gcd(j, n) - 1).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 3, 2, 0, 1, 4, 6, 4, 4, 0, 1, 5, 11, 12, 5, 2, 0, 1, 6, 18, 32, 20, 6, 6, 0, 1, 7, 27, 70, 85, 42, 7, 4, 0, 1, 8, 38, 132, 260, 260, 70, 8, 6, 0, 1, 9, 51, 224, 629, 1050, 735, 144, 9, 4, 0, 1, 10, 66, 352, 1300, 3162, 4102, 2224, 270, 10, 10
Offset: 0

Views

Author

Seiichi Manyama, Apr 17 2021

Keywords

Examples

			Square array begins:
  0, 0,  0,   0,    0,    0,    0, ...
  1, 1,  1,   1,    1,    1,    1, ...
  1, 2,  3,   4,    5,    6,    7, ...
  2, 3,  6,  11,   18,   27,   38, ...
  2, 4, 12,  32,   70,  132,  224, ...
  4, 5, 20,  85,  260,  629, 1300, ...
  2, 6, 42, 260, 1050, 3162, 7826, ...

Crossrefs

Columns k=0..5 give A000010, A001477, A034738, A034754, A343490, A343492.
Main diagonal gives A056665.
Cf. A185651, A319082.

Programs

  • Mathematica
    T[n_, k_] := Sum[If[k == (g = GCD[j, n] - 1) == 0, 1, k^g], {j, 1, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 17 2021 *)
  • PARI
    T(n, k) = sum(j=1, n, k^(gcd(j, n)-1));
  • PARI
    T(n, k) = if(n==0, 0, sumdiv(n, d, eulerphi(n/d)*k^(d-1)));

Formula

G.f. of column k: Sum_{j>=1} phi(j) * x^j / (1 - k*x^j).
T(n,k) = A185651(n,k)/k for k > 0.
T(n,k) = Sum_{d|n} phi(n/d)*k^(d - 1).

A382994 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -Sum_{d|n} phi(n/d) * (-k)^d.

Original entry on oeis.org

1, 2, 0, 3, -2, 3, 4, -6, 12, 0, 5, -12, 33, -16, 5, 6, -20, 72, -84, 40, 0, 7, -30, 135, -264, 255, -60, 7, 8, -42, 228, -640, 1040, -714, 140, 0, 9, -56, 357, -1320, 3145, -4056, 2205, -272, 9, 10, -72, 528, -2436, 7800, -15540, 16408, -6648, 540, 0
Offset: 1

Views

Author

Seiichi Manyama, Apr 12 2025

Keywords

Examples

			Square array begins:
  1,   2,    3,     4,      5,      6,       7, ...
  0,  -2,   -6,   -12,    -20,    -30,     -42, ...
  3,  12,   33,    72,    135,    228,     357, ...
  0, -16,  -84,  -264,   -640,  -1320,   -2436, ...
  5,  40,  255,  1040,   3145,   7800,   16835, ...
  0, -60, -714, -4056, -15540, -46500, -117390, ...
  7, 140, 2205, 16408,  78155, 279972,  823585, ...

Crossrefs

Main diagonal gives A382997.
Cf. A000010, A185651, A382993, A382995.

Programs

  • PARI
    a(n, k) = -sumdiv(n, d, eulerphi(n/d)*(-k)^d);

Formula

A(n,k) = -Sum_{j=1..n} (-k)^gcd(n,j).
G.f. of column k: k * Sum_{j>=1} phi(j) * x^j / (1 + k*x^j).

A054603 a(n) = Sum_{d|4} phi(d)*n^(4/d).

Original entry on oeis.org

0, 4, 24, 96, 280, 660, 1344, 2464, 4176, 6660, 10120, 14784, 20904, 28756, 38640, 50880, 65824, 83844, 105336, 130720, 160440, 194964, 234784, 280416, 332400, 391300, 457704, 532224, 615496, 708180, 810960, 924544, 1049664
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Links

Crossrefs

Row n=4 of A185651.

Programs

  • Mathematica
    LinearRecurrence[{5,-10,10,-5,1},{0,4,24,96,280},40] (* Harvey P. Dale, Nov 29 2015 *)

Formula

a(n) = n^4 + n^2 + 2n.
G.f.: -4*x*(4*x^2+x+1)/(x-1)^5. [Colin Barker, Dec 21 2012]

A054604 a(n) = Sum_{d|5} phi(d)*n^(5/d).

Original entry on oeis.org

0, 5, 40, 255, 1040, 3145, 7800, 16835, 32800, 59085, 100040, 161095, 248880, 371345, 537880, 759435, 1048640, 1419925, 1889640, 2476175, 3200080, 4084185, 5153720, 6436435, 7962720, 9765725, 11881480, 14349015, 17210480, 20511265
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Links

Crossrefs

Row n=5 of A185651.

Programs

  • Mathematica
    Table[n^5+4n,{n,0,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{0,5,40,255,1040,3145},30] (* Harvey P. Dale, Jul 09 2025 *)

Formula

a(n) = n^5 + 4n.
G.f.: 5*x*(x^4+2*x^3+18*x^2+2*x+1)/(x-1)^6. [Colin Barker, Dec 21 2012]
Previous Showing 11-20 of 26 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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