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.

A221855 Number of cyclotomic cosets of 13 mod 10^n.

Original entry on oeis.org

4, 12, 30, 72, 130, 204, 294, 400, 522, 660, 814, 984, 1170, 1372, 1590, 1824, 2074, 2340, 2622, 2920, 3234, 3564, 3910, 4272, 4650, 5044, 5454, 5880, 6322, 6780, 7254, 7744, 8250, 8772, 9310, 9864, 10434, 11020, 11622, 12240, 12874, 13524, 14190, 14872, 15570, 16284, 17014, 17760, 18522, 19300
Offset: 1

Views

Author

V. Raman, Jan 28 2013

Keywords

Examples

			a(2) = 12 because there are 12 cyclotomic cosets of 13 mod 100:
{1, 13, 69, 97, 61, 93, 9, 17, 21, 73, 49, 37, 81, 53, 89, 57, 41, 33, 29, 77}
{3, 39, 7, 91, 83, 79, 27, 51, 63, 19, 47, 11, 43, 59, 67, 71, 23, 99, 87, 31}
{2, 26, 38, 94, 22, 86, 18, 34, 42, 46, 98, 74, 62, 6, 78, 14, 82, 66, 58, 54}
{4, 52, 76, 88, 44, 72, 36, 68, 84, 92, 96, 48, 24, 12, 56, 28, 64, 32, 16, 8}
{5, 65, 45, 85}
{15, 95, 35, 55}
{25}
{75}
{10, 30, 90, 70}
{20, 60, 80, 40}
{50}
{0}

Links

Crossrefs

Cf. A006694, A220468.

Programs

  • Mathematica
    a[n_]:=DivisorSum[10^n,EulerPhi[#]/MultiplicativeOrder[13,#]&]; Array[a,50] (* Ray Chandler, Jul 03 2023, after Jean-François Alcover *)
  • PARI
    for(n=1,50,print1(sumdiv(10^n, d, eulerphi(d)/znorder(Mod(13, d)))", "))

Formula

a(n) = A220018(n) for n = 1.
a(n) = A220018(n) + 1 for all n >= 2.
Conjecture: a(n) = 2*n*(4*n-7) for n>2. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>5. G.f.: 2*x*(4*x^4-7*x^3-3*x^2-2) / (x-1)^3. - Colin Barker, Apr 14 2013

A220019 Number of cyclotomic cosets of 7 mod 10^n.

Original entry on oeis.org

4, 27, 93, 265, 685, 1265, 2005, 2905, 3965, 5185, 6565, 8105, 9805, 11665, 13685, 15865, 18205, 20705, 23365, 26185, 29165, 32305, 35605, 39065, 42685, 46465, 50405, 54505, 58765, 63185, 67765, 72505, 77405, 82465, 87685, 93065, 98605, 104305, 110165, 116185, 122365, 128705, 135205, 141865, 148685
Offset: 1

Views

Author

V. Raman, Jan 27 2013

Keywords

Examples

			a(2) = 27 because there are 27 cyclotomic cosets of 7 mod 100:
{1, 7, 49, 43}
{2, 14, 98, 86}
{3, 21, 47, 29}
{4, 28, 96, 72}
{5, 35, 45, 15}
{6, 42, 94, 58}
{8, 56, 92, 44}
{9, 63, 41, 87}
{10, 70, 90, 30}
{11, 77, 39, 73}
{12, 84, 88, 16}
{13, 91, 37, 59}
{17, 19, 33, 31}
{18, 26, 82, 74}
{20, 40, 80, 60}
{22, 54, 78, 46}
{23, 61, 27, 89}
{24, 68, 76, 32}
{25, 75}
{34, 38, 66, 62}
{36, 52, 64, 48}
{50}
{51, 57, 99, 93}
{53, 71, 97, 79}
{55, 85, 95, 65}
{67, 69, 83, 81}
{0}

Links

Crossrefs

Cf. A006694, A220468.

Programs

  • Mathematica
    a[n_] := DivisorSum[10^n, EulerPhi[#] / MultiplicativeOrder[7, #] & ]; Array[a, 50] (* Jean-François Alcover, Dec 18 2015 *)
  • PARI
    for(n=1,50,print1(sumdiv(10^n, d, eulerphi(d)/znorder(Mod(7, d)))", "))

Formula

Empirical G.f.: x*(88*x^5-142*x^4-63*x^3-24*x^2-15*x-4) / (x-1)^3. [Colin Barker, Feb 03 2013]
Conjecture: a(n) = 5*(16*n^2-60*n+37) for n>3. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>6. [Colin Barker, Apr 14 2013]

A220020 Number of cyclotomic cosets of 9 mod 10^n.

Original entry on oeis.org

6, 20, 56, 140, 256, 404, 584, 796, 1040, 1316, 1624, 1964, 2336, 2740, 3176, 3644, 4144, 4676, 5240, 5836, 6464, 7124, 7816, 8540, 9296, 10084, 10904, 11756, 12640, 13556, 14504, 15484, 16496, 17540, 18616, 19724, 20864, 22036, 23240, 24476, 25744, 27044, 28376, 29740, 31136, 32564, 34024, 35516, 37040, 38596
Offset: 1

Views

Author

V. Raman, Jan 27 2013

Keywords

Examples

			a(2) = 20 because there are 20 cyclotomic cosets of 9 mod 100:
{1, 9, 81, 29, 61, 49, 41, 69, 21, 89}
{3, 27, 43, 87, 83, 47, 23, 7, 63, 67}
{11, 99, 91, 19, 71, 39, 51, 59, 31, 79}
{13, 17, 53, 77, 93, 37, 33, 97, 73, 57}
{2, 18, 62, 58, 22, 98, 82, 38, 42, 78}
{4, 36, 24, 16, 44, 96, 64, 76, 84, 56}
{6, 54, 86, 74, 66, 94, 46, 14, 26, 34}
{8, 72, 48, 32, 88, 92, 28, 52, 68, 12}
{10, 90}
{30, 70}
{20, 80}
{40, 60}
{50}
{5, 45}
{15, 35}
{55, 95}
{65, 85}
{25}
{75}
{0}

Links

Crossrefs

Cf. A006694, A220468.

Programs

  • Mathematica
    a[n_] := DivisorSum[10^n, EulerPhi[#]/MultiplicativeOrder[9, #]&]; Array[a, 50] (* Jean-François Alcover, Dec 10 2015, adapted from PARI *)
LinearRecurrence[{3,-3,1},{6,20,56,140,256},50] (* Harvey P. Dale, Jul 12 2025 *)
  • PARI
    for(n=1,50,print1(sumdiv(10^n, d, eulerphi(d)/znorder(Mod(9, d)))", "))

Formula

Conjecture: a(n) = 4*(4*n^2-7*n-1) for n>2. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>5. G.f.: 2*x*(8*x^4-13*x^3-7*x^2-x-3) / (x-1)^3. - Colin Barker, Apr 13 2013

A220021 Number of cyclotomic cosets of 11 mod 10^n.

Original entry on oeis.org

10, 27, 65, 119, 189, 275, 377, 495, 629, 779, 945, 1127, 1325, 1539, 1769, 2015, 2277, 2555, 2849, 3159, 3485, 3827, 4185, 4559, 4949, 5355, 5777, 6215, 6669, 7139, 7625, 8127, 8645, 9179, 9729, 10295, 10877, 11475, 12089, 12719, 13365, 14027, 14705, 15399, 16109, 16835, 17577, 18335, 19109, 19899
Offset: 1

Views

Author

V. Raman, Jan 27 2013

Keywords

Comments

How is this related to A181890? - R. J. Mathar, Apr 11 2013

Examples

			a(2) = 27 because there are 27 cyclotomic cosets of 11 mod 100:
{1, 11, 21, 31, 41, 51, 61, 71, 81, 91}
{3, 33, 63, 93, 23, 53, 83, 13, 43, 73}
{7, 77, 47, 17, 87, 57, 27, 97, 67, 37}
{9, 99, 89, 79, 69, 59, 49, 39, 29, 19}
{2, 22, 42, 62, 82}
{12, 32, 52, 72, 92}
{4, 44, 84, 24, 64}
{14, 54, 94, 34, 74}
{6, 66, 26, 86, 46}
{16, 76, 36, 96, 56}
{8, 88, 68, 48, 28}
{18, 98, 78, 58, 38}
{5, 55}
{15, 65}
{25, 75}
{35, 85}
{45, 95}
{0}
{10}
{20}
{30}
{40}
{50}
{60}
{70}
{80}
{90}

Links

Crossrefs

Cf. A006694, A220468.

Programs

  • Mathematica
    a[n_] := DivisorSum[10^n, EulerPhi[#] / MultiplicativeOrder[11, #] &]; Array[a, 50] (* Jean-François Alcover, Dec 18 2015 *)
  • PARI
    for(n=1,50,print1(sumdiv(10^n, d, eulerphi(d)/znorder(Mod(11, d)))", "))

Formula

Conjecture: a(n) = 8*n^2-2*n-1 for n>1. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>4. G.f.: x*(5*x^3-14*x^2+3*x-10) / (x-1)^3. - Colin Barker, Apr 13 2013
Showing 1-4 of 4 results.

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