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.

Showing 1-4 of 4 results.

A382998 a(n) = Sum_{d|n} phi(n/d) * (-n)^(d-1).

Original entry on oeis.org

1, -1, 11, -66, 629, -7750, 117655, -2097676, 43046889, -999990036, 25937424611, -743008622708, 23298085122493, -793714765724686, 29192926025492783, -1152921504875290680, 48661191875666868497, -2185911559727680349982, 104127350297911241532859
Offset: 1

Views

Author

Seiichi Manyama, Apr 12 2025

Keywords

Crossrefs

Main diagonal of A382993.
Main diagonal of A382995.
Cf. A000010, A382997, A383003.

Programs

  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*(-n)^(d-1));

Formula

a(n) = (1/n) * A382997(n).
a(n) = Sum_{k=1..n} (-n)^(gcd(n,k) - 1).
a(n) = [x^n] Sum_{k>=1} phi(k) * log(1 + n*x^k) / k.
a(n) = [x^n] Sum_{k>=1} phi(k) * x^k / (1 + n*x^k).

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).

A382999 a(n) = Sum_{d|n} phi(n/d) * (-2)^(d-1).

Original entry on oeis.org

1, -1, 6, -8, 20, -30, 70, -136, 270, -500, 1034, -2088, 4108, -8134, 16440, -32912, 65552, -130878, 262162, -524800, 1048740, -2096138, 4194326, -8390976, 16777300, -33550348, 67109418, -134225840, 268435484, -536855640, 1073741854, -2147516704, 4294969404
Offset: 1

Views

Author

Seiichi Manyama, Apr 12 2025

Keywords

Crossrefs

Column k=2 of A382995.
Cf. A000010, A034738, A074763.

Programs

  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*(-2)^(d-1));

Formula

a(n) = Sum_{k=1..n} (-2)^(gcd(n,k) - 1).
G.f.: Sum_{k>=1} phi(k) * x^k / (1 + 2*x^k).

A383000 a(n) = Sum_{d|n} phi(n/d) * (-3)^(d-1).

Original entry on oeis.org

1, -2, 11, -28, 85, -238, 735, -2216, 6585, -19610, 59059, -177428, 531453, -1593606, 4783175, -14351152, 43046737, -129134082, 387420507, -1162281100, 3486785925, -10460294174, 31381059631, -94143360856, 282429536825, -847288078026, 2541865841523, -7625599078020, 22876792454989
Offset: 1

Views

Author

Seiichi Manyama, Apr 12 2025

Keywords

Crossrefs

Column k=3 of A382995.
Cf. A000010, A034754, A343465.

Programs

  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*(-3)^(d-1));

Formula

a(n) = Sum_{k=1..n} (-3)^(gcd(n,k) - 1).
G.f.: Sum_{k>=1} phi(k) * x^k / (1 + 3*x^k).
Showing 1-4 of 4 results.

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