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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A329637 Square array A(n, k) = A329644(prime(n)^k), read by falling antidiagonals: (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

Original entry on oeis.org

1, 1, 1, 4, -1, 1, 0, 4, -5, 1, 24, -16, 4, -13, 1, -8, 40, -48, 4, -29, 1, 104, -88, 72, -112, 4, -61, 1, -48, 184, -248, 136, -240, 4, -125, 1, 352, -400, 344, -568, 264, -496, 4, -253, 1, 80, 544, -1104, 664, -1208, 520, -1008, 4, -509, 1, 1424, -784, 928, -2512, 1304, -2488, 1032, -2032, 4, -1021, 1
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2019

Keywords

Examples

			The top left corner of the array:
   n   p_n |k=1,     2, 3,      4,     5,      6,     7,       8,      9,      10
  ---------+----------------------------------------------------------------------
   1 ->  2 |  1,     1, 4,      0,    24,     -8,   104,     -48,    352,      80,
   2 ->  3 |  1,    -1, 4,    -16,    40,    -88,   184,    -400,    544,    -784,
   3 ->  5 |  1,    -5, 4,    -48,    72,   -248,   344,   -1104,    928,   -2512,
   4 ->  7 |  1,   -13, 4,   -112,   136,   -568,   664,   -2512,   1696,   -5968,
   5 -> 11 |  1,   -29, 4,   -240,   264,  -1208,  1304,   -5328,   3232,  -12880,
   6 -> 13 |  1,   -61, 4,   -496,   520,  -2488,  2584,  -10960,   6304,  -26704,
   7 -> 17 |  1,  -125, 4,  -1008,  1032,  -5048,  5144,  -22224,  12448,  -54352,
   8 -> 19 |  1,  -253, 4,  -2032,  2056, -10168, 10264,  -44752,  24736, -109648,
   9 -> 23 |  1,  -509, 4,  -4080,  4104, -20408, 20504,  -89808,  49312, -220240,
  10 -> 29 |  1, -1021, 4,  -8176,  8200, -40888, 40984, -179920,  98464, -441424,
  11 -> 31 |  1, -2045, 4, -16368, 16392, -81848, 81944, -360144, 196768, -883792,

Links

Crossrefs

Cf. A000079, A000203, A000225, A075708, A182944, A329610, A329644, A329890.
Rows 1-2: A329891, A329892 (from n>=1).
Column 1: A000012, Column 2: -A036563(n) (from n>=1), Column 3: A010709.

Programs

  • PARI
    up_to = 105;
A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));
A329637sq(n,k) = ((2^(n+k-1)) - (((2^n)-1) * A329890(k)));
A329637list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A329637sq(col,(a-(col-1))))); (v); };
v329637 = A329637list(up_to);
A329637(n) = v329637[n];

Formula

A(n, k) = A329644(A182944(n, k)).
A(n, k) = A000079(n+k-1) - (A000225(n) * A329890(k)).

A329890 a(1) = 1; for n > 1, a(n) = sigma((2^n)-1) - sigma((2^(n-1))-1), where sigma is A000203, the sum of divisors.

Original entry on oeis.org

1, 3, 4, 16, 8, 72, 24, 304, 160, 944, 624, 6576, -544, 14336, 16384, 72544, 19616, 342528, 50688, 1475584, 466176, 3443712, 2657376, 29486880, -3340288, 54808448, 65971360, 306781024, 77647680, 1475408064, 132153344, 5157119680, 3054411072, 12548176896, 13343981568, 130039259136, -28235160128, 228451400256, 269821673472
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2019

Keywords

Links

Crossrefs

From second term onwards, the first differences of A075708.
Cf. A000203, A329637, A329891, A329892.

Programs

  • Mathematica
    Join[{1},Table[DivisorSigma[1,2^n-1]-DivisorSigma[1,2^(n-1)-1],{n,2,40}]] (* Harvey P. Dale, Aug 18 2024 *)
  • PARI
    A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));

Formula

a(1) = 1; and for n > 1, a(n) = A075708(n) - A075708(n-1).

A329892 a(0) = 0, a(1) = 1, for n > 1, a(n) = 2^(n+1) - 3*(sigma((2^n)-1) - sigma((2^(n-1))-1)).

Original entry on oeis.org

0, 1, -1, 4, -16, 40, -88, 184, -400, 544, -784, 2224, -11536, 18016, -10240, 16384, -86560, 203296, -503296, 896512, -2329600, 2795776, -1942528, 8805088, -54906208, 77129728, -30207616, 70521376, -383472160, 840798784, -2278740544, 3898507264, -6881424448, 8016635968, -3284792320, 28687532032, -252678823936, 359583387328, -135598386880
Offset: 0

Views

Author

Antti Karttunen, Nov 23 2019

Keywords

Links

Crossrefs

Row 2 of A329637.
Cf. A000203, A000225, A075708, A156552, A323243, A329644, A329890, A329891.

Programs

  • PARI
    A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));
A329892(n) = if(!n,n,2^(n+1) - 3*A329890(n));
  • PARI
    A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323243(n) = if(1==n, 0, sigma(A156552(n)));
A329644(n) = sumdiv(n, d, moebius(n/d)*((2*A156552(d))-A323243(d)));
A329892(n) = A329644(3^n);

Formula

a(n) = A329644(3^n).
a(0) = 0; for n >= 1, a(n) = 2^(n+1) - 3*A329890(n).
Showing 1-3 of 3 results.

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